CS549: Cryptography and Network Security © by Xiang-Yang Li Department of Computer Science, IIT Cryptography and Network Security 1 Notice© This lecture note (Cryptography and Network Security) is prepared by Xiang-Yang Li. This lecture note has benefited from numerous textbooks and online materials. Especially the “Cryptography and Network Security” 2nd edition by William Stallings and the “Cryptography: Theory and Practice” by Douglas Stinson. You may not modify, publish, or sell, reproduce, create derivative works from, distribute, perform, display, or in any way exploit any of the content, in whole or in part, except as otherwise expressly permitted by the author. The author has used his best efforts in preparing this lecture note. The author makes no warranty of any kind, expressed or implied, with regard to the programs, protocols contained in this lecture note. The author shall not be liable in any event for incidental or consequential damages in connection with, or arising out of, the furnishing, performance, or use of these. Cryptography and Network Security 2 About Instructor Associate Professor IIT PhD/MS UIUC 1997-2000 BS, BE Tsinghua University Research Interests: Algorithm design and analysis Wireless networks Game theory Computational geometry Contact Information Phone 312-567-5207 Email: xli@cs.iit.edu Cryptography and Network Security 3 Office and Office hours Office SB 237D Office hours Monday 3:10PM – 4:10PM. Wednesday 3:10PM– 4:10PM. Or by contact: email xli@cs.iit.edu, phone 312 567 5207 Cryptography and Network Security 4 About This Course Textbook Cryptography: Theory and Practice by Douglas R. Stinson CRC press Cryptography and Network Security: Principles and Practice; By William Stallings Prentice Hall Handbook of Applied Cryptography by Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone, CRC Press I have electronic version! Cryptography and Network Security 5 Grading and Others Grading Homework Mid Term Project 30% 25% 20% (select your own topic), 15 pages report Final exam 25% (closed book) Policy Do it yourself Can use library, Internet and so on, but you have to cite the sources when you use this information Cryptography and Network Security 6 Homeworks Do it independently No discussion No copy Can use reference books Write your name also, you could discuss with classmates then write your own report (about 10 pages for the topic you selected) Staple your solution HW1 (Due 2/14/08) HW2 (Due 3/14/08) HW3 (Due 4/11/08) Report (Due 05/05/08) For report, For project (presentation and programming) You SHOULD collaborate with your group member and you SHOULD make enough contributions to get credit Type your solution! And print it then submit Cryptography and Network Security 7 Topics Introduction Number Theory Traditional Methods: secret key system Modern Methods: Public Key System Digital Signature and others Internet Security: DoS, DDoS Other topics: secret sharing, zero-knowledge proof, bit commitment, oblivious transfer,… Cryptography and Network Security 8 Organization Chapters Introduction Number Theory Conventional Encryption Block Ciphers Public Key System Key Management Hash Function and Digital Signature Identification Secret Sharing Pseudo-random number Generation Email Security Internet Security Others Cryptography and Network Security 9 Cryptography and Network Security Introduction Xiang-Yang Li Cryptography and Network Security 10 Introduction The art of war teaches us not on the likelihood of the enemy’s not coming, but on our own readiness to receive him; not on the chance of his not attacking, but rather on the fact that we have made our position unassailable. --The art of War, Sun Tzu Cryptography and Network Security 11 Criteria for Desirable Cryptosystems Confidence in Security established Hard or intractable problems? Practical Efficiency Space, time and so on Explicitness About its environment assumptions, security service offered, special cases in math assumptions, Protection tuned to application needs No less, no more Openness Cryptography and Network Security 12 Most important Security first Efficiency, resource utilization, and security tradeoffs This is especially the case for resource constrained networks such as wireless sensor networks Limited power supply (thus limited communication, and computation), limited storage space Cryptography and Network Security 13 Cryptography Cryptography (from Greek kryptós, "hidden", and gráphein, "to write") is, traditionally, the study of means of converting information from its normal, comprehensible form into an incomprehensible format, rendering it unreadable without secret knowledge — the art of encryption. Past: Cryptography helped ensure secrecy in important communications, such as those of spies, military leaders, and diplomats. In recent decades, cryptography has expanded its remit in two ways mechanisms for more than just keeping secrets: schemes like digital signatures and digital cash, for example. in widespread use by many civilians, and users are not aware of it. Cryptography and Network Security 14 Crypto-graphy, -analysis, -logy The study of how to circumvent the use of cryptography is called cryptanalysis, or codebreaking. Cryptography and cryptanalysis are sometimes grouped together under the umbrella term cryptology, encompassing the entire subject. In practice, "cryptography" is also often used to refer to the field as a whole; crypto is an informal abbreviation. Cryptography is an interdisciplinary subject, linguistics Mathematics: number theory, information theory, computational complexity, statistics and combinatorics engineering Cryptography and Network Security 15 Close, but different fields Steganography the study of hiding the very existence of a message, and not necessarily the contents of the message itself (for example, microdots, or invisible ink) http://en.wikipedia.org/wiki/Steganography Traffic analysis which is the analysis of patterns of communication in order to learn secret information The messages could be encrypted http://en.wikipedia.org/wiki/Traffic_analysis Cryptography and Network Security 16 Stenography Example Last 2 bits Cryptography and Network Security 17 Tools for Stenography http://www.jjtc.com/Steganography/toolm atrix.htm Cryptography and Network Security 18 Network Security Model Trusted Third Party principal principal Security transformation Security transformation attacker Cryptography and Network Security 19 Attacks, Services and Mechanisms Security Attacks Action compromises the information security Could be passive or active attacks Security Services Actions that can prevent, detect such attacks. Such as authentication, identification, encryption, signature, secret sharing and so on. Security mechanism The ways to provide such services Detect, prevent and recover from a security attack Cryptography and Network Security 20 Attacks Passive attacks Interception Release of message contents Traffic analysis Active attacks Interruption, modification, fabrication Masquerade Replay Modification Denial of service Cryptography and Network Security 21 Information Transferring Cryptography and Network Security 22 Attack: Interruption Cut wire lines, Jam wireless signals, Drop packets, Cryptography and Network Security 23 Attack: Interception Wiring, eavesdrop Cryptography and Network Security 24 Attack: Modification intercept Replaced info Cryptography and Network Security 25 Attack: Fabrication Also called impersonation Cryptography and Network Security 26 Attacks, Services and Mechanisms Security Attacks Action compromises the information security Could be passive or active attacks Security Services Actions that can prevent, detect such attacks. Such as authentication, identification, encryption, signature, secret sharing and so on. Security mechanism The ways to provide such services Detect, prevent and recover from a security attack Cryptography and Network Security 27 Important Services of Security Confidentiality, also known as secrecy: Integrity: the recipient should be able to determine if the message has been altered during transmission. Authentication: only an authorized recipient should be able to extract the contents of the message from its encrypted form. Otherwise, it should not be possible to obtain any significant information about the message contents. the recipient should be able to identify the sender, and verify that the purported sender actually did send the message. Non-repudiation: the sender should not be able to deny sending the message. Cryptography and Network Security 28 Secure Communication protecting data locally only solves a minor part of the problem. The major challenge that is introduced by the Web Service security requirements is to secure data transport between the different components. Combining mechanisms at different levels of the Web Services protocol stack can help secure data transport (see figure next page). Cryptography and Network Security 29 Secure Communication Cryptography and Network Security 30 Secure Communication The combined protocol HTTP/TLS or SSL is often referred to as HTTPS (see figure). SSL was originally developed by Netscape for secure communication on the Internet, and was built into their browsers. SSL version 3 was then adopted by IETF and standardized as the Transport Layer Security (TLS) protocol. Use of Public Key Infrastructure (PKI) for session key exchange during the handshake phase of TLS has been quite successful in enabling Web commerce in recent years. TLS also has some known vulnerabilities: it is susceptible to man-in-the-middle attacks and denial-of-service attacks. Cryptography and Network Security 31 SOAP security SOAP (Simple Object Access Protocol) is designed to pass through firewalls as HTTP. This is disquieting from a security point of view. Today, the only way we can recognize a SOAP message is by parsing XML at the firewall. The SOAP protocol makes no distinction between reads and writes on a method level, making it impossible to filter away potentially dangerous writes. This means that a method either needs to be fully trusted or not trusted at all. The SOAP specification does not address security issues directly, but allows for them to be implemented as extensions. As an example, the extension SOAP-DSIG defines the syntax and processing rules for digitally signing SOAP messages and validating signatures. Digital signatures in SOAP messages provide integrity and non-repudiation mechanisms. Cryptography and Network Security 32 PKI PKI key management provides a sophisticated framework for securely exchanging and managing keys. The two main technological features, which a PKI can provide to Web Services, are: Encryption of messages: by using the public key of the recipient Digital signatures: non-repudiation mechanisms provided by PKI and defined in SOAP standards may provide Web Services applications with legal protection mechanisms Note that the features provided by PKI address the same basic needs as those that are recognized by the standardization organizations as being important in a Web Services context. In Web Services, PKI mainly intervenes at two levels: At the SOAP level (non-repudiation, integrity) At the HTTPS level (TLS session negotiation, eventually assuring authentication, integrity and privacy) Cryptography and Network Security 33 Some basic Concepts Cryptography and Network Security 34 Cryptography Cryptography is the study of Secret (crypto-) writing (-graphy) Concerned with developing algorithms: Conceal the context of some message from all except the sender and recipient (privacy or secrecy), and/or Verify the correctness of a message to the recipient (authentication) Form the basis of many technological solutions to computer and communications security problems Cryptography and Network Security 35 Basic Concepts Cryptography encompassing the principles and methods of transforming an intelligible message into one that is unintelligible, and then retransforming that message back to its original form Plaintext The original intelligible message Ciphertext The transformed message Message Is treated as a non-negative integer hereafter Cryptography and Network Security 36 Basic Concepts Cipher An algorithm for transforming an intelligible message into unintelligible by transposition and/or substitution, or some other techniques Keys Some critical information used by the cipher, known only to the sender and/or receiver Encipher (encode) The process of converting plaintext to ciphertext Decipher (decode) The process of converting ciphertext back into plaintext Cryptography and Network Security 37 Basic Concepts cipher an algorithm for encryption and decryption. The exact operation of ciphers is normally controlled by a key — some secret piece of information that customizes how the ciphertext is produced Protocols specify the details of how ciphers (and other cryptographic primitives) are to be used to achieve specific tasks. A suite of protocols, ciphers, key management, userprescribed actions implemented together as a system constitute a cryptosystem; this is what an end-user interacts with, e.g. PGP Cryptography and Network Security 38 Encryption and Decryption Decipher P = D(K2)(C) ciphertext Plaintext Encipher C = E(K1)(P) K1, K2: from keyspace These two keys could be different; could be difficult to get one from the other Cryptography and Network Security 39 What is Security? Two fundamentally different securities Unconditional security No matter how much computer power is available, the cipher cannot be broken Using Shannon’s information theory The entropy of the message I(M) is same as the entropy of the message I(M|C) when known the ciphertext (and possible more) Computational security Given limited computing resources (e.g time needed for calculations is greater than age of universe), the cipher cannot be broken What do we mean “broken”? Proved by some complexity equivalence approach Cryptography and Network Security 40 Cryptography and Network Security Elementary Number Theory Xiang-Yang Li Cryptography and Network Security 41 Number theory Elementary number theory Main topic of this course divisibility, the Euclidean algorithm to compute greatest common divisors, factorization Fermat's little theorem and Euler's theorem, the Chinese remainder theorem and Euler's φ function are investigated; Analytic number theory Algebraic number theory Geometric number theory Computational number theory Cryptography and Network Security 42 Introduction to Number Theory Divisors b|a if a=mb for an integer m b|a and c|b then c|a b|g and b|h then b|(mg+nh) for any integer m,n Prime number P has only positive divisors 1 and p Relatively prime numbers No common divisors for p and q except 1 Cryptography and Network Security 43 Prime numbers Upto 200 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 Largest known so far (till 2008, Jan 22) 232582657-1 with 9808358 digits (found 2006 using proof code G9) When 2n-1is prime it is said to be a Mersenne prime (a French monk 1588-1648, conjecture 1644). Clearly n must be odd. How many prime numbers are there? Infinity ---- Euclid gave simple proof Proof by contradiction They were also irregularly placed (arbitrary gap) How many in the range [0,n] -- Theta( n / log n) Approximately, the nth prime n log n How many primes with d bits approximately? ~ Theta(2d/d) Cryptography and Network Security 44 Determining Primes? How to determine if a given number prime? n is Deterministic Brute force testing Testing whether a number a | n, for a in certain range Random testing A prime number should satisfy some properties If a number x does NOT have any of such properties, then this x is NOT a prime Otherwise, it may be a prime number Properties: for any number a, a does not divide x, • More properties will be studied and used to design efficient methods Cryptography and Network Security 45 Greatest Common Divisor (GCD) Greatest common divisor gcd(a,b) The largest number that divides both a and b Euclid's algorithm Find the GCD of two numbers a and b, a<b Use fact if a and b have divisor d so does a-b, a-2b … d m a nb d a b d a 2b d a 3b d a qb Cryptography and Network Security 46 Cont. GCD (a,b) is given by: let g0=b g1=a gi+1 = gi-1 mod gi when gi =0 then gcd(a,b) = gi-1 The algorithm terminates in O(log b) rounds Why? Every round, the total number of bits of a and b is decreased by at least one What is a more precise complexity bound? Cryptography and Network Security 47 Properties For any two integers a and b Exist integers m and n: gcd(a,b) =ma+bn Example: a=2, b=3; we choose m=-1, n=1 so –2+3=1 a=6, b=11; we choose m=2, n=-1 so 2*6-11=1 Simple proof? Integer n can be factored as n=p1a1 p2a2 p3a3…. pnan where pi is prime number Cryptography and Network Security 48 Extended Euclidean Algorithm input are two integers a and b, computes their greatest common divisor (gcd) as well as integers x and y such that ax + by = gcd(a, b). It later can also be used to compute the inverse of an integer 1 a m od n Cryptography and Network Security 49 Proof Assume we compute gcd(x0,y0), x0>y0 Let Xi=(xi,yi); 0xi-qi+1yi+1<|yi| Then Xi=MiXi-1, where Mi=(0,1; 1,-qi) Assume the gcd algorithm terminates in n steps We have MnMn-1…M1X0=(gcd(x0,y0), 0)T a b … Assume MnMn-1 M1=( ) c d Then ax0+by0=gcd(x0,y0) The above algorithm is to keep track of a,b,c,d, and xi,yi values. Cryptography and Network Security 50 Modular Arithmetic Congruence a b mod n says when divided by n that a and b have the same remainder It defines a relationship between all integers aa a b then b a a b, b c then a c Cryptography and Network Security 51 Cont. addition (a+b) mod n (a mod n) + (b mod n) subtraction a-b mod n a+(-b) mod n multiplication a b mod n derived from repeated addition Possible: a*b 0 where neither a, b 0 mod n Example: 2*3 =0 mod 6 Cryptography and Network Security 52 Addition and Multiplication Integers modulo n with addition and multiplication form a commutative ring with the laws of Associativity (a+b)+c a+(b+c) mod n Commutativity a+b b+a mod n Distributivity (a+b)*c (a*c)+(b*c) mod n Cryptography and Network Security 53 Cont. Division b/a mod n multiplied by inverse of a: b/a = b*a-1 mod n a-1*a 1 mod n 3-1 7 mod 10 because 3*7 1 mod 10 Inverse does not always exist! Only when gcd(a,n)=1 Cryptography and Network Security 54 Euclid's Extended GCD Routine If (a,n)=1 then the inverse always exists Can extend Euclid's algorithm to find inverse by keeping track of gi = ui.n + vi.a Extended Euclid's (or binary GCD) algorithm to find inverse of a number a mod n (where (a,n)=1) is: Cryptography and Network Security 55 Inverse Inverse(a,n) is given by: X=(x1,x2,x3)=(1,0,n); Y=(y1,y2,y3)=(0,1,a) If y3=0 return x3=gcd(a,n); no inverse If y3=1 return y3=gcd(a,n); y2=a-1 mod n Q=[x3/y3] T=X-Q*Y X=Y; Y=T Goto 2nd step Cryptography and Network Security 56 When inverse exists If gcd(a,n)=1 inverse exists We can find x, y such that ax+ny=1 Then x= a-1 mod n If inverse exists gcd(a,n)=1 x be the inverse of a, i.e., ax=1 mod n Then x a=1+q n for some integer q Let gcd(a,n)=d. Then d | (x a-q n ) Obviously d=1 since x a-q n =1 Let Cryptography and Network Security 57 Galois Field If n is constrained to be a prime number then this forms a Galois field modulo p denoted GF(p) and all the normal laws associated with integer arithmetic work Exponentiation p b = ae mod p Discrete Logarithms find x where ax = b mod p Cryptography and Network Security 58 Relative primes Two numbers a and n are relative primes if gcd(a,n)=1 Consider all integers 0<a <n How many are relative prime to n? Equivalently, how many a such that a-1 mod n exists Typically Zn={0,1,2,….,n-1} : all integers 0<= a < n Zn*={a| 0<= a < n, gcd(a,n)=1} All integers in Zn that are co-prime with n Also called reduced residue set mod n Cryptography and Network Security 59 Euler Totient Function If consider arithmetic modulo n, then a reduced set of residues is a subset of the complete set of residues modulo n which are relatively prime to n eg for n=10, the complete set of residues is {0,1,2,3,4,5,6,7,8,9} the reduced set of residues is {1,3,7,9} The number of elements in the reduced set of residues is called the Euler Totient function (n) Cryptography and Network Security 60 cont Compute (n) If factoring of n is known (n)=n (1-1/pi) where pi is its prime factor Otherwise It is expensive! But not proved yet computing (n) when knowing fact n =pq but not the number p and q Conjectured to be a hard question But not proved yet. Equivalent to find p and q Cryptography and Network Security 61 cont Equivalency: finding p,q computing (n) Proof If we found p and q, then (n)=(p-1)(q-1) if we found (n), then solve p, q from equations n pq ( n ) ( p 1)( q 1) Cryptography and Network Security 62 Euler's Theorem Let gcd(a,n)=1 then a(n) mod n = 1 Proof: consider all reduced residues xi in Zn*={x| 0<= x < n, gcd(x,n)=1} Then axi,1<=i <= (n) also form reduced residues set Using axi = xi mod n Using Zn* and aZn* are same sets! have a(n) xi = xi mod n Thus, a(n) =1 mod n We Using the fact that xi has inverse Cryptography and Network Security 63 Fermat's Little Theorem Let p be a prime and gcd(a,p)=1 then ap-1 mod p = 1 Proof: similar to the proof of Euler’s theorem But consider all integers in Zp Generally, for any prime number p ap mod p = a (true for any number a) Generally, for any number n=pq a(n) mod n = a (true for any number a) Need to prove for the case gcd(a,n)>1 Do it yourself Cryptography and Network Security 64 Efficient computing of exponential Compute ab mod n efficiently when b, n large? Example: compute a1024 mod 21024 +1 Simple approach: repetitively time a 1024 times? Efficient computation: Write number b in binary format as xkxk-1xk-2….x2x1x0 Let t1=a mod n. Then compute ti+1= ti * ti mod n for i<k b Then x x x .... x x x a m od n a k k 1 k 2 2 1 0 m od n [a ti i (2 ) ] xi m od n 0i k xi Time complexity? m od n 0i k Cryptography and Network Security 65 Chinese Remainder Theorem By Qin Jiushao Let m1,m2,….mk be pair-wise relative prime numbers Assume integer x= ai mod mi for 1<= I <= k Then x= ai ei mod M Where M= mi; Mi=M/ mi ei= Mi * (Mi-1 mod mi) Proof For each i, the integers mi and M/mi are coprime, and using the extended Euclidean algorithm we can find integers r and s such that r mi + s M/mi = 1. If we set ei = s M/mi, then we have ei =1 mod mi and ei =1 mod mj for j<>i. Cryptography and Network Security 66 General CRT Sometimes, the simultaneous congruences can be solved even if the mi's are not pairwise coprime. a solution x exists if and only if ai ≡ aj (mod gcd(ni, nj)) for all i and j. All solutions x are congruent modulo the least common multiple of the ni. Methods: successive substitution Cryptography and Network Security 67 Example consider the simultaneous congruences x ≡ 3 (mod 4) x ≡ 5 (mod 6) Can be transformed to x ≡ 3 (mod 4) x ≡ 5 (mod 2) x ≡ 1 (mod 2) x ≡ 5 (mod 3) Then transformed to x ≡ 3 (mod 4) x ≡ 2 (mod 3) Using CRT X=11 (mod 12) Cryptography and Network Security 68 Primality Testing To check if exists integer Primary school method a such that a|n Test a=2,3,4,5,6,….,n-1 Test a=2,3,4,5,…, n0.5 Test a=2,3,5,7,11,…., p, where prime number p<=n0.5 Two slow! Check almost n numbers Check n0.5 numbers At least around (n/ln n)0.5 numbers need be checked Example Number n~21024, then (n/ln n)0.5~(21024 /1024) 0.5 ~ 2507 Assume 230 numbers per second, takes about 2507-30*16 = 227 days Any improvement? Cryptography and Network Security 69 Classification of Testing Primes The Quick Tests for Small Numbers and Probable Primes Finding Very Small Primes --- trivial division Fermat, Probable-Primality and Pseudoprimes Strong Probable-Primality and a Practical Test The Classical Tests N-1 Tests (and Pepin's Test for Fermats) N+1 Tests (and the Lucas-Lehmer Test for Mersennes) A Combined Test -- and more The General Purpose Tests Neoclassical Tests, especially APR and APR-CL Using Elliptic Curves, especially the ECPP Test A Polynomial Time Algorithm Cryptography and Network Security 70 Fermat Little Theorem Based Fermat's theorem gives us a powerful test for compositeness: Given n > 1, choose a > 1 and calculate an-1 modulo n (there is a very easy way to do quickly by repeated squaring) If the result is not one modulo n, then n is composite. If it is one modulo n, then n might be prime so n is called a weak probable prime base a (or just an aPRP). Some early articles call all numbers satisfying this test pseudoprimes, but now the term pseudoprime is properly reserved for composite probable-primes. Cryptography and Network Security 71 Carmichael number There may be relatively few pseudoprimes, but there are still infinitely many of them for every base a>1, so we need a tougher test. One way to make this test more accurate is to use multiple bases (check base 2, then 3, then 5,...). But still we run into an interesting obstacle called the Carmichael numbers. The composite integer n is a Carmichael number if an-1=1 (mod n) for every integer a relatively prime to n. Cryptography and Network Security 72 Strong probable-primality and a practical test A better way to make the Fermat test more accurate is to realize that if an odd number n is prime, then the number 1 has just two square roots modulo n: 1 and -1. So the square root of an-1, a(n-1)/2 (since n will be odd), is either 1 or -1. Algorithm Write n-1 = 2sd where d is odd and s is non-negative: n is a strong probable-prime base a (an a-SPRP) if either ad = 1 (mod n) or (ad)2r = -1 (mod n) for some non-negative r less than s. It has been proven ([Monier80] and [Rabin80]) that the strong probable primality test is wrong no more than 1/4th of the time (3 out of 4 numbers which pass it will be prime). Cryptography and Network Security 73 Simple Fact Equation x21 mod p has only solutions 1,-1 If p is prime number Simple proof: (x+1)(x-1) 0 mod p So if we find another solution, then not be prime number! Miller and Rabin 1975,1980 Randomly chosen integer p can a If a21 mod p then p is not prime number Integer a is called the witness Otherwise p maybe, or maybe not a prime number Cryptography and Network Security 74 Witness Algorithm Witness(a,n) Let bkbk-1…b1b0 be the binary code of n-1 Let d=1 For i=k downto 0 x=d; d=d*d mod n If d=1 and x1, and x n-1 return TRUE If bi=1 then d=d*a mod n Endfor If d 1 then return TRUE Return FALSE Cryptography and Network Security 75 Facts Analyze the result of witness If returns TRUE, then n is not prime number Find other solutions for x21 mod n Otherwise, n maybe prime number Given odd n and random a Witness fails with probability less than 0.5 Run witness algorithm s times If one time, it is TRUE Then n is not prime number Otherwise, Pr(n is prime)>1-2-s Cryptography and Network Security 76 Randomized Methods Las Vegas Method Always produces correct results Runs in expected polynomial time Monte Carlo Method Runs in polynomial time May produce incorrect results with bounded probability No-Biased Monte Carlo Method Answer yes is always correct, but the answer no may be wrong Yes-biased Monte Carlo Method Answer no is always correct, but the answer yes may be wrong Cryptography and Network Security 77 Witness Algorithm Witness Algorithm is based on Monte Carlo Method It actually test compositeness, not primality When it reports yes, the number is always composite When it reports no, input may be composite, prime Probability Result Pr(input=composite | ans=composite)= 1 Pr(ans=no | input=composite)<1/2 Pr(input=composite | ans=no) 1/4 Cryptography and Network Security 78 Time Complexity Each round of witness cost O(log n) Unit: integer multiplication and modular arithmetic So the primality testing cost O(s log n) The confidence is 1-2-s if report prime The confidence is 1 if report non-prime If the extended Riemann hypothesis is true, then if n is an a-SPRP for all integers a with 1 < a < 2(log n)2, then n is prime. Miller's Test [Miller76]: Cryptography and Network Security 79 More on proving primes (N-1 test n > 1. If for every prime factor q of n-1 there is an integer a such Theorem 1: Let that an-1 = 1 (mod n), and a(n-1)/q is not 1 (mod n); then n is prime. Cryptography and Network Security 80 N-1 test Theorem 2: Suppose n-1 = FR, where F>R, gcd(F,R) is one and the factorization of F is known. If for every prime factor q of F there is an integer a>1 such that an-1 = 1 (mod n), and gcd(a(n-1)/q-1,n) = 1; then n is prime. Cryptography and Network Security 81 N+1 test n be an odd prime. The Mersenne number M(n) = Lucas-Lehmer Test (1930): Let 2n-1 is prime if and only if S(n-2) = 0 (mod M(n)) where S(0) = 4 and S(k+1) = S(k)2-2. Cryptography and Network Security 82 ECPP method What is the next big leap in primality proving? To switch from Galois groups to some other, perhaps easier to work with groups--in this case the points on Elliptic Curves modulo n. An Elliptic curve is a curve of genus one, that is a curve that can be written in the form E(a,b) : y2 = x3 + ax + b (with 4a3 + 27b2 not zero) http://www.lix.polytechnique.fr/~morain/Prgms/ecpp.english.html for implementation Heuristically, the best version of ECPP is O((log n)4+eps) for some eps>0 Cryptography and Network Security 83 Deterministic Poly-Time Method In 2002 Agrawal, Kayal and Saxena found a relatively simple deterministic algorithm which relies on no unproved assumptions. There has been a long list of research efforts devoted to find deterministic polynomial time methods for testing primes Cryptography and Network Security 84 Basics a and p are relatively prime integers with p > 1. p is prime if and only if Theorem: Suppose that (x-a)p = (xp-a) (mod p) Proof. If p is prime, then p divides the binomial coefficients pCr for r = 1, 2, ... p-1. This shows that (x-a)p = (xp-ap) (mod p), and the equation above follows via Fermat's Little Theorem. On the other hand, if p > 1 is composite, then it has a prime divisor q. Let qk be the greatest power of q that divides p. Then qk does not divide pCq and is relatively prime to ap-q, so the coefficient of the term xq on the left of the equation in the theorem is not zero, but it is on the right. Cryptography and Network Security 85 AKS method Input: Integer n>1 if (n is has the form ab with b > 1) then output COMPOSITE r := 2 while (r < n) { if (gcd(n,r) is not 1) then output COMPOSITE if (r is prime greater than 2) then { let q be the largest factor of r-1 if (q > 4sqrt(r)log n) and (n(r-1)/q is not 1 (mod r)) then break } r := r+1 } for a = 1 to 2sqrt(r)log n { if ( (x-a)n is not (xn-a) (mod xr-1,n) ) then output COMPOSITE } output PRIME; Cryptography and Network Security 86 Time Complexity they proved would run in at most O((log n)12f(log log n)) time where f is a polynomial AKS also showed that if Sophie Germain primes have the expected distribution [HL23] (and they certainly should!), then the exponent 12 in the time estimate can be reduced to 6, bringing it much closer to the (probabilistic) ECPP method. But of course when actually finding primes it is the unlisted constants1 that make all of the difference! We will have to wait for efficient implementations of this algorithm (and hopefully clever restatements of the painful for loop) to see how it compares to the others for integers of a few thousand digits. Until then, at least we have learned that there is a polynomial-time algorithm for all integers that both is deterministic and relies on no unproved conjectures! Cryptography and Network Security 87 Primitive Root Order of integer ordn(a) The order of a modulo n is the smallest positive k such that ak1 mod n Primitive Root Integer a is a primitive root of n if the order of a modulo n is (n) Not all integers have primitive root Example n=pq for primes p and q Prime p has (p-1) primitive roots Cryptography and Network Security 88 cont When primitive root exists Number n in format of p, 2p, pk, 2pk for some integer k and prime number p Otherwise the primitive root does not exist Find a PR for p such that a p 1 q1 1 .... q k ak Let a=2, i=1 If i>k, a is a PR, otherwise go to step 3 1 m od p If a let i=i+1 and go to step 2; otherwise let i=1, and a=a+1 and repeat this step 3. ( p 1 )/ q i Cryptography and Network Security 89 Some “hard” questions Some questions that are assumed to be hard, will be used as bases for cryptography Integer factorization Given n, find all its prime factors Discrete logarithm Given g, y, and p, find x such that gxy mod p Square root Given b, find x such that x2b mod n. Here n is not a prime number Cryptography and Network Security 90 Integer Factorization write an integer as product of prime numbers. For example, given the number 45, the prime factorization would be 32·5. The factorization is always unique, according to the fundamental theorem of arithmetic Given two large prime numbers, it is easy to multiply them. However, given their product, it appears to be difficult to find the factors. This is relevant for many modern systems in cryptography. If a fast method were found for solving the integer factorization problem, then several important cryptographic systems would be broken. Although fast factoring is one way to break these systems, there may be other ways to break them that don't involve factoring. So it is possible that the integer factorization problem is truly hard, yet these systems can still be broken quickly. A rare exception is the BBS generator. It has been proved to be exactly as hard as integer factorization: if you can break the generator in polynomial time then you can factorize integers in polynomial time, and vice versa Cryptography and Network Security 91 Current state of the art If a large, n-bit number is the product of two primes that are roughly the same size, no polynomial time factoring algorithm is known the best known algorithms are sub-exponential, but super-polynomial: asymptotic running time by the general number field sieve (GNFS) algorithm, is Polynomial methods known for quantum computer! Cryptography and Network Security 92 Sub-exponential There are published algorithms that are faster than O((1+ε)b) for all positive ε, i.e., sub-exponential, where b is the number of bits of the input Cryptography and Network Security 93 Factoring algorithms Special purpose its running time depends on the properties of unknown factors: size, special form, etc. Examples Trial division, Pollard's rho algorithm, Pollard's p-1 algorithm, Lenstra elliptic curve factorization, Congruence of squares, Special number field sieve General purpose running time depends solely on the size of the integer to be factored. This is the type of algorithm used to factor RSA numbers. Most general-purpose algorithms are based on the congruence of squares method. Examples: Quadratic sieve, General number field sieve Cryptography and Network Security 94 Factorization for Quantum Computers For an ordinary computer, general number field sieve (GNFS) is the best published algorithm for large n (more than about 100 digits). For a quantum computer, however, Peter Shor discovered an algorithm in 1994 that solves it in polynomial time. This will have significant implications for cryptography if a large quantum computer is ever built. Shor's algorithm takes only O(b3) time and O(b) space on b-bit number inputs. In 2001, the first 7-qubit quantum computer became the first to run Shor's algorithm. It factored the number 15. Cryptography and Network Security 95 List of Algorithms Special-purpose A special-purpose factoring algorithm's running time depends on the properties of its unknown factors: size, special form, etc. Exactly what the running time depends on varies between algorithms. Trial division Pollard's rho algorithm Algebraic-group factorisation algorithms amongst which are Pollard's p − 1 algorithm, Williams' p+1 algorithm and Lenstra elliptic curve factorization Fermat's factorization method Special number field sieve General-purpose A general-purpose factoring algorithm's running time depends solely on the size of the integer to be factored. This is the type of algorithm used to factor RSA numbers. Most general-purpose factoring algorithms are based on the congruence of squares method. Dixon's algorithm Continued fraction factorization (CFRAC) Quadratic sieve General number field sieve Shanks' square forms factorization (SQUFOF) Cryptography and Network Security 96 Discrete Logarithms Y gx mod p Given y, g, and p, compute x as logg(y) 1/3 2/3 Time complexity O(e(ln p) (ln ln p) ) Best known until now In other words, if p is large, then it is very hard to solve the discrete logarithm problem Several protocols are based on this ElGamal discrete log cryptosystem, Diffie-Hellman key exchange and the Digital Signature Algorithm. Current methods: the Pohlig-Hellman algorithm if p-1 is a product of small primes, so this should be avoided in those applications Cryptography and Network Security 97 Methods More sophisticated algorithms exist, usually inspired by similar algorithms for integer factorization. These algorithms run faster than the naive algorithm, but none of them runs in polynomial time. Baby-step giant-step (Also known as 'Little-Step Big-Step') Pollard's rho algorithm for logarithms Pollard's lambda algorithm (aka Pollard's kangaroo algorithm) Pohlig-Hellman algorithm Index calculus algorithm Number field sieve Cryptography and Network Security 98 Quadratic Residue Quadratic Residue Integer b is a quadratic residue of modulo integer n if and only if x2 b mod n has a solution for x Number x is called the square root of b Otherwise b is called quadratic nonresidue Given odd prime p, b is quadratic residue, iff b(p-1)/2 1 mod p b is quadratic nonresidue, iff b(p-1)/2 -1 mod p These facts can be used to test primes with probability Cryptography and Network Security 99 Computing Square root mod p Given number a, find number x, x2 =a mod p If p=3 mod 4, then x=a(p+1)/4 mod p is a solution. If p=5 mod 8, a(p-1)/4 =1 mod p then x= a(p+3)/8 mod p If p=5 mod 8, a(p-1)/4 =-1 mod p then x= 2a(4a)(p-5)/8 mod p h 1 If p=1 mod 8, x a 2 N p 1 2 h k sk Here h is an odd number Cryptography and Network Security 100 Compute square-root mod p Find a solution to x2 =a mod p if exists Let r=0, s=p-1; while s even, {r=r+1; s=s/2;} Choose random n such that n 1 p Let z=ns mod p; x=a(s+1)/2 mod p; b=as mod p; If b=1, return x as a solution Let m=1, y=b2 mod p; while y<>1 {y= y2 mod p; m=m+1;} If r=m then a is Quadratic non-residue; exit; r-m-1 Let x=xz2 mod p and b=bz2r-m mod p and z=z2r-m mod p Go to step 4 The expected running time is O(log4 p) Cryptography and Network Security 101 Complexity Theory The input length of a problem is the number symbols used to characterize it Complexity of a method n of Function f(n) is order O(g(n)) if f(n)<=c*|g(n)|, for all n>=N0, for some c Function f(n) is order (g(n)) if f(n)>=c*|g(n)|, for all n>=N0, for some c Function f(n) is order (g(n)) if c1*|g(n)|<=f(n)<=c2*|g(n)|, for all n>=N0, for some c1 and c2 Polynomial time algorithm (P) solves any instance of a particular problem with input length n in time O(p(n)), where p is a polynomial Cryptography and Network Security 102 Cont. Non-deterministic polynomial time algorithm (NP) is one for which any guess at the solution of an instance of the problem may be checked for validity in polynomial time. NP-complete problems are a subclass of NP problems for which it is known that if any such problem has a polynomial time solution, then all NP problems have polynomial solutions. Co-NP: the complements of NP problems. Cryptography and Network Security 103 Cryptography and Network Security Conventional Methods Xiang-Yang Li Cryptography and Network Security 104 Roadmap of Cryptography classical cryptography (--- 1920s) secret writing required only pen and paper Mostly: transposition, substitution ciphers Easily broken by statistics analysis (e.g., frequency) mechanical devices invented for encryption Rotor machines (e.g. Enigma cipher) 1930s-1950s featured in films, such as in the James Bond adventure From Russia with Love specification of DES and the invention of RSA (1970s) --- modern ciphers Public key system, most notably Quantum Cryptography (future?) Cryptography and Network Security 105 Quantum Cryptography Quantum cryptography currently has two aspects. quantum key exchange (also known as quantum key distribution), a method for secure communications based on quantum mechanics conjectured effect of quantum computing on cryptanalysis, although it is currently, like quantum computing itself, only a theoretical concept. Basic idea of quantum key exchange is to use the "noisy" properties of light to render incoherent an image that acts to complement a secret key. This image can be represented in a number of ways, but the ability to decode that image rests upon an understanding of how it was made. No way to intercept the transmission without changing it is possible, so key information can be exchanged with great confidence it has been transmitted secretly. quantum computing will considerably extend the reach of cryptanalysis, making brute force key space searches much more effective -- if such computers ever become possible in actual practice Cryptography and Network Security 106 History Ancient ciphers Have a history of at least 4000 years Ancient Egyptians enciphered some of their hieroglyphic writing on monuments Ancient Hebrews enciphered certain words in the scriptures 2000 years ago Julius Caesar used a simple substitution cipher, now known as the Caesar cipher Roger bacon described several methods in 1200s Cryptography and Network Security 107 History Ancient ciphers Geoffrey Chaucer included several ciphers in his works Leon Alberti devised a cipher wheel, and described the principles of frequency analysis in the 1460s Blaise de Vigenère published a book on cryptology in 1585, & described the polyalphabetic substitution cipher Increasing use, esp in diplomacy & war over centuries Cryptography and Network Security 108 Classical Cryptographic Techniques Two basic components of classical ciphers: Substitution: letters are replaced by other letters Transposition: letters are arranged in a different order These ciphers may be: Monoalphabetic: only one substitution/ transposition is used, or Polyalphabetic:where several substitutions/ transpositions are used Product cipher: several ciphers concatenated together Cryptography and Network Security 109 Encryption and Decryption Plaintext Encipher C = E(K)(P) ciphertext Decipher P = D(K)(C) Key source Cryptography and Network Security 110 Key Management Using secret channel Encrypt the key Third trusted party The sender and the receiver generate key The key must be same We will talk more about how we can generate keys for two parties who are “unknown” of each other before, and want secure communication Cryptography and Network Security 111 Attacks Recover the message Recover the secret key Thus also the message Thus the number of keys possible must be large! Cryptography and Network Security 112 Possible Attacks Ciphertext only Algorithm, ciphertext Known plaintext Algorithm, ciphertext, plaintext-ciphertext pair Chosen plaintext Algorithm, ciphertext, chosen plaintext and its ciphertext Chosen ciphertext Algorithm, ciphertext, chosen ciphertext and its plaintext Chosen text Algorithm, ciphertext, chosen plaintext and ciphertext Cryptography and Network Security 113 Steganography Conceal the existence of message Character marking Invisible ink Pin punctures Typewriter correction ribbon Cryptography renders message unintelligible! Cryptography and Network Security 114 Contemporary Equiv. Least significant bits of picture frames 2048x3072 pixels with 24-bits RGB info Able to hide 2.3M message Drawbacks Large overhead Virtually useless if system is known Improvement Using some “random” sequence of the last bit for storing the data Challenge: produce such random sequence such that the attacker cannot figure out the sequence! Cryptography and Network Security 115 Caesar Cipher Replace each letter of message by a letter a fixed distance away Reputedly used by Julius Caesar Example: L FDPH L VDZ L FRQTXHUHG I CAME I SAW I CONGUERED The mapping is ABCDEFGHIJKLMNOPQRSTUVWXYZ DEFGHIJKLMNOPQRSTUVWXYZABC Cryptography and Network Security 116 Mathematical Model Description Assume all letters are mapped to integers [0,25] A:-0, B-1, ….., Z25 E(k) : i i + k mod 26 Decryption D(k) : i i - k mod 26 Encryption Cryptography and Network Security 117 Cryptanalysis: Caesar Cipher Key space: 26 Exhaustive key search Example GDUCUGQFRMPCNJYACJCRRCPQ HEVDVHRGSNQDOKZBDKDSSDQR Plaintext: JGXFXJTIUPSFQMBDFMFUUFSTKHYGYKUJV GRNCEGNGVVGTU Ciphertext: LIZHZLVKWRUHSODFHOHWWHUVMJAIAMWX SVITPEGIPIXXIVW Cryptography and Network Security 118 Character Frequencies In most languages letters are not equally common in English e is by far the most common letter Have tables of single, double & triple letter frequencies Use these tables to compare with letter frequencies in ciphertext, a monoalphabetic substitution does not change relative letter frequencies do need a moderate amount of ciphertext (100+ letters) Cryptography and Network Security 119 Letter Frequency Analysis Single Letter A,B,C,D,E,….. Double Letter TH,HE,IN,ER,RE,ON,AN,EN,…. Triple Letter THE,AND,TIO,ATI,FOR,THA,TER,RES,… Cryptography and Network Security 120 Letter Frequencies Cryptography and Network Security 121 Letter Frequencies Cryptography and Network Security 122 N-gram Frequencies Digraph Frequency th he an in er on re ed nd ha at en es of nt ea ti to io le is ou ar as de rt ve Trigraph Frequency the and tha ent ion tio for nde has nce tis oft men For more, see http://www.letterfrequency.org Cryptography and Network Security 123 Modular Arithmetic Cipher Use a more complex equation to calculate the ciphertext letter for each plaintext letter E(a,b) : i ai + b mod 26 Need gcd(a,26) = 1 Otherwise, not reversible So, a2, 13, 26 Caesar cipher: a=1, b=3 Cryptography and Network Security 124 Cryptanalysis Key space:12*26 Brute force search Use letter frequency counts to guess a couple of possible letter mappings frequency pattern not produced just by a shift But it is still a substitution, thus we can use frequency analysis use these mappings to solve 2 simultaneous equations to derive above parameters Cryptography and Network Security 125 Playfair Cipher The Playfair cipher or Playfair square is a manual symmetric encryption technique and was the first literal digraph substitution cipher. The scheme was invented in 1854 by Charles Wheatstone, but bears the name of Lord Playfair who promoted the use of the cipher. Cryptography and Network Security 126 Playfair Cipher Used in WWI and WWII s e i/j a m b p c l d f o v g q w h r x k t y n u z Key: simple Cryptography and Network Security 127 Playfair Cipher Use filler letter to separate repeated letters Encrypt two letters together Same row– followed letters ac--bd Same column– letters under qw--wi Otherwise—square’s corner at same row ar--bq Cryptography and Network Security 128 Analysis Size of diagrams: 25! But the actual different diagrams are not 25! Two diagrams are the same if they derive the same encryption and decryption method Then what is the number of difference diagrams in playfair cipher? 25!/25=24! Difficult using frequency analysis But it still reveals the frequency information Frequency of 2-gram (bi-gram, two-letters) Cryptography and Network Security 129 Playfair Cryptanalysis Like most pre-modern era ciphers, the Playfair cipher can be easily cracked if there is enough text. Obtaining the key is relatively straightforward if both plaintext and ciphertext are known. When only the ciphertext is known, brute force cryptanalysis of the cipher involves searching through the key space for matches between the frequency of occurrence of digrams (pairs of letters) and the known frequency of occurrence of digrams in the assumed language of the original message. Cryptography and Network Security 130 Playfair, cont A different approach to tackling a Playfair cipher is the shotgun hill climbing method. This starts with a random square of letters. Then minor changes are introduced (i.e. switching letters, rows, or reflecting the entire square) to see if the candidate plaintext is more like standard plaintext than before the change (perhaps by comparing the trigrams to a known frequency chart). If the new square is deemed to be an improvement, then it is adopted and then further mutated to find an even better candidate. Eventually, the plaintext or something very close is found to achieve a maximal score by whatever grading method is chosen. Computers can adopt this algorithm to crack Playfair ciphers with a relatively small amount of text. Cryptography and Network Security 131 Hill Cipher Hill cipher is a polygraphic substitution cipher based on linear algebra. Invented by Lester S. Hill in 1929, it was the first polygraphic cipher in which it was practical (though barely) to operate on more than three symbols at once. Each letter is treated as a digit in base 26: A = 0, B =1, and so on. A block of n letters is then considered as a vector of n dimensions, and multiplied by a n × n matrix, modulo 26. The components of the matrix are the key, and should be random provided that the matrix is invertible in (to ensure decryption is possible). The Hill cipher has achieved Shannon's diffusion, and an ndimensional Hill cipher can diffuse fully across n symbols at once. Cryptography and Network Security 132 Hill Cipher Machine Cryptography and Network Security 133 Hill Cipher Machine With fixed Key and patented Triple encryption was recommended for security: a secret nonlinear step, followed by the wide diffusive step from the machine, followed by a third secret nonlinear step. Such a combination was actually very powerful for 1929, and indicates that Hill apparently understood the concepts of a meet-in-the-middle attack as well as confusion and diffusion. Unfortunately, his machine did not sell. Cryptography and Network Security 134 Hill Cipher Encryption Assign each letter an index C=KP mod 26 Matrix K is the key Decryption P=K-1C mod 26 Thus, we can decrypt iff gcd(det(K), 26) =1. Cryptography and Network Security 135 How to Decrypt? Compute K-1 Compute det(K) Check if gcd(det(K), 26) =1 If not, then K-1 do not exist Else K-1 is 1 1 1 K 1 ,1 1 n 1 K 1 , n 1 1 n K n ,1 1 det( K ) 2n 1 K n ,n Cryptography and Network Security 136 cont K i, j k 1 ,1 k 1 , j 1 k 1 , j 1 k 1, n k i 1 ,1 k i 1 , j 1 k i 1 , j 1 k i 1 , n k i 1 ,1 k i 1 ,1 k i 1 ,1 k i 1 ,1 k n ,1 k n , j 1 k n , j 1 k n ,n Cryptography and Network Security 137 Hill Cipher Cryptanalysis Difficult to use frequency analysis But vulnerable to known-plaintext attack Give simple method to attack hill cipher under the known-plaintext assumption? How to attack under the chosen plaintext assumption? The security could be greatly enhanced by combining with some non-linear step to defeat this attack. Cryptography and Network Security 138 Key Sizes How may good keys? One might naïvely think that the key size, in bits, is n2log226 or about 4.7n2. In fact, it is slightly less than this because not all randomly selected matrices are usable. A slightly less naïve view might guess that 1/2 + 1/26 of candidate keys would be unusable, reducing the keyspace by about 54%. In fact, determinants are not uniformly distributed, and the key space reduction is closer to 70%. Additionally it seems to be prudent to avoid too many zeroes in the key matrix, since they reduce diffusion. The net effect is that the effective keyspace of a basic Hill cipher is about 4.64n2. For a 5 × 5 Hill cipher, that is about 114 bits. Of course, key search is not the most efficient known attack Cryptography and Network Security 139 Polyalphabetic Substitution Use more than one substitution alphabet Makes cryptanalysis harder since have more alphabets to guess and flattens frequency distribution same plaintext letter gets replaced by several ciphertext letter, depending on which alphabet is used Cryptography and Network Security 140 Vigenère Cipher Basically multiple Caesar ciphers key is multiple letters long K = k1 k2 ... kd ith letter specifies ith alphabet to use use each alphabet in turn, repeating from start after d letters in message Plaintext THISPROCESSCANALSOBEEXPRESSED Keyword CIPHERCIPHERCIPHERCIPHERCIPHE Ciphertext VPXZTIQKTZWTCVPSWFDMTETIGAHLH Cryptography and Network Security 141 Enigma Machine Enigma was a portable cipher machine used to encrypt and decrypt secret messages. a family of related electro-mechanical rotor machines Japan commercial German military Cryptography and Network Security 142 Enigma Machine Enigma encryption for two consecutive letters — current is passed into set of rotors, around the reflector, and back out through the rotors again. Letter A encrypts differently with consecutive key presses, first to G, and then to C. This is because the right hand rotor has stepped, sending the signal on a completely different route. Cryptography and Network Security 143 Enigma the actual encipherment of a letter is performed electrically. When a key is pressed, the circuit is completed; current flows through the various components and ultimately lights one of many lamps, indicating the output letter. Current flows from a battery through the switch controlled by the depressed key into a fixed entry wheel. This leads into the rotor assembly (or scrambler), where the complex internal wiring of each rotor results in the current passing from one rotor to the next along a convoluted path. After passing through all the rotors, current enters the reflector, which relays the signal back out again through the rotors and the entry wheel — this time via a different path — and, finally, to one of the lamps (the earliest Enigma models do not have the reflector). Cryptography and Network Security 144 Rotors performs a very simple type of encryption a simple substitution cipher Cryptography and Network Security 145 World War II Era Encryption Devices A few here Sigaba (United States) Typex (Britain) Lorenz cipher (Germany) Geheimfernschreiber (Germany) For more, see http://w1tp.com/enigma/ Cryptography and Network Security 146 One-time Pad theoretically unbreakable (Claude Shannon) the plaintext is combined with a random "pad" the same length as the plaintext. Patent by Gilbert Vernam (AT&T) and Joseph Mauborgne Encryption C=PK Decryption P=CK Claude Shannon's work can be interpreted as that any information-theoretically secure cipher will be effectively equivalent to the one-time pad algorithm. Hence one-time pads offer the best possible mathematical security of any encryption scheme, anywhere and anytime. Cryptography and Network Security 147 One-time pad--cont Drawbacks it requires secure exchange of the one-time pad material, which must be as long as the message pad disposed of correctly and never reused In practice Generate a large number of random bits, Exchange the key material securely between the users before sending an one-time enciphered message, Keep both copies of the key material for each message securely until they are used, and Securely dispose of the key material after use, thereby ensuring the key material is never reused. It requires a perfect random numbers as key We will learn how to generate pseudo-random numbers Cryptography and Network Security 148 Random numbers needed If the key material is generated by a deterministic program then it is not actually random should never be used in an one-time pad cipher. If so used, the method becomes a stream cipher; these usually employ a short key that is used to generate a long pseudorandom stream, which is then combined with the message using some such mechanism as those used in one-time pads. Stream ciphers can be secure in practice, but they cannot be absolutely secure in the same provable sense as the one-time pad Cryptography and Network Security 149 Stream ciphers Stream ciphers The most famous: Vernam cipher Invented by Vernam, ( AT&T, in 1917) Process the message bit by bit (as a stream) different from the one-time pad– some call same Simply add bits of message to random key bits Examples A well-known stream cipher is RC4; others include: A5/1, A5/2, Chameleon, FISH, Helix. ISAAC, Panama, Pike, SEAL, SOBER, SOBER-128 and WAKE. Usage Stream ciphers are used in applications where plaintext comes in quantities of unknowable length - for example, a secure wireless connection Cryptography and Network Security 150 Simplest Stream Cipher Key Key Plaintext Ciphertext Ciphertext Plaintext Cryptography and Network Security 151 Pros and Cons Drawbacks Need as many key bits as message, difficult in practice (ie distribute on a mag-tape or CDROM) Strength Is unconditionally secure provided key is truly random Cryptography and Network Security 152 Key Generation Why not to generate keystream from a smaller (base) key? Use some pseudo-random function to do this Although this looks very attractive, it proves to be very very difficult in practice to find a good pseudo-random function that is cryptographically strong This is still an area of much research Cryptography and Network Security 153 Transposition Methods Permutation of plaintext Example Write in a square in row, then read in column order specified by the key Enhance: double or triple transposition Can reapply the encryption on ciphertext Cryptography and Network Security 154 Cryptography and Network Security Block Ciphers Xiang-Yang Li Cryptography and Network Security 155 Block Ciphers The message is broken into blocks, Each of which is then encrypted (Like a substitution on very big characters - 64-bits or more) Cryptography and Network Security 156 Substitution and Permutation In his 1949 paper Shannon also introduced the idea of substitution-permutation (S-P) networks, which now form the basis of modern block ciphers An S-P network is the modern form of a substitutiontransposition product cipher S-P networks are based on the two primitive cryptographic operations we have seen before Cryptography and Network Security 157 Substitution A binary word is replaced by some other binary word The whole substitution function forms the key If use n bit words, The key space is 2n! Can also think of this as a large lookup table, with n address lines (hence 2n addresses), each n bits wide being the output value Will call them s-boxesCryptography and Network Security 158 Cont. Cryptography and Network Security 159 Permutation A binary word has its bits reordered (permuted) The re-ordering forms the key If use n bit words, The key space is n! (Less secure than substitution) This is equivalent to a wire-crossing in practice (Though is much harder to do in software) Will call these p-boxes Cryptography and Network Security 160 Cont. Cryptography and Network Security 161 Substitution-permutation Network Shannon combined these two primitives He called these mixing transformations A special form of product ciphers where S-boxes Provide confusion of input bits P-boxes Provide diffusion across s-box inputs Cryptography and Network Security 162 Confusion and Diffusion Confusion A technique that seeks to make the relationship between the statistics of the ciphertext and the value of the encryption keys as complex as possible. Cipher uses key and plaintext. Diffusion A technique that seeks to obscure the statistical structure of the plaintext by spreading out the influence of each individual plaintext digit over many ciphertext digits. Cryptography and Network Security 163 Desired Effect Avalanche effect A characteristic of an encryption algorithm in which a small change in the plaintext gives rise to a large change in the ciphertext Best: changing one input bit results in changes of approx half the output bits Completeness effect where each output bit is a complex function of all the input bits Cryptography and Network Security 164 Practical Substitutionpermutation Networks In practice we need to be able to decrypt messages, as well as to encrypt them, hence either: Have to define inverses for each of our S & P-boxes, but this doubles the code/hardware needed, or Define a structure that is easy to reverse, so can use basically the same code or hardware for both encryption and decryption Cryptography and Network Security 165 Feistel Cipher Invented by Horst Feistel, working at IBM Thomas J Watson research labs in early 70's, The idea is to partition the input block into two halves, l(i-1) and r(i-1), use only r(i-1) in each round i (part) of the cipher The function f incorporates one stage of the S-P network, controlled by part of the key k(i) known as the ith subkey Cryptography and Network Security 166 Cont. Cryptography and Network Security 167 Cont. This can be described functionally as: L(i) = R(i-1) R(i) = L(i-1) f(k(i), R(i-1)) This can easily be reversed as seen in the above diagram, working backwards through the rounds In practice link a number of these stages together (typically 16 rounds) to form the full cipher Cryptography and Network Security 168 Data Encryption Standard Adopted in 1977 by the National Bureau of Standards, now the National Institute of Standards and Technology Data are encrypted in 64-bit blocks using a 56-bit key The same algorithm is used for decryption. Subject to much controversy Cryptography and Network Security 169 History IBM LUCIFER 60’s Uses 128 bits key Proposal for NBS, 1973 Adopted by NBS, 1977 Uses only 56 bits key Possible brute force attack Design of S-boxes was classified Hidden weak points in in S-Boxes? Wiener (93) claim to be able to build a machine at $100,00 and break DES in 1.5 days Cryptography and Network Security 170 DES DES encrypts 64-bit blocks of data, using a 56-bit key the basic process consists of: an initial permutation (IP) 16 rounds of a complex key dependent calculation f a final permutation, being the inverse of IP Function f can be described as L(i) = R(i-1) R(i) = L(i-1) P(S( E(R(i-1)) K(i) )) Cryptography and Network Security 171 DES Cryptography and Network Security 172 Initial and Final Permutations Inverse Permutations 40 8 48 16 56 24 64 32 39 7 47 15 55 23 63 31 38 6 46 14 54 22 62 30 37 5 45 13 53 21 61 29 36 4 44 12 52 20 60 28 35 3 43 11 51 19 59 27 34 2 42 10 50 18 58 26 33 1 41 9 49 17 57 25 Cryptography and Network Security 173 Function f Cryptography and Network Security 174 Expansion Table Expands the 32 bit data to 48 bits Result(i)=input( array(i)) 32 1 2 3 4 5 4 5 6 7 8 9 8 9 10 11 12 13 12 13 14 15 16 17 16 17 18 19 20 21 20 21 22 23 24 25 24 25 26 27 28 29 28 29 30 31 32 1 Cryptography and Network Security 175 S-Boxes S-Box is a fixed 4 by 16 array Given 6-bits B=b1b2b3b4b5b6, Row r=b1b6 Column c=b2b3b4b5 S(B)=S(r,c) written in binary of length 4 Cryptography and Network Security 176 Example S-Box S1 14 4 13 1 0 15 7 4 1 2 15 11 8 3 10 6 12 5 0 7 9 5 3 8 3 10 5 0 4 14 2 13 1 10 6 14 8 13 6 2 11 15 12 9 7 4 1 7 5 14 10 0 15 12 8 2 9 11 12 11 9 3 6 Cryptography and Network Security 13 177 Permutation Table The permutation after each round 16 7 20 21 29 12 28 17 1 15 23 26 5 18 31 10 2 8 24 14 32 27 3 9 19 13 30 6 22 11 4 25 Cryptography and Network Security 178 Subkey Generation Given a 64 bits key (with parity-check bit) Discard the parity-check bits Permute the remaining bits using fixed table P1 Let C0D0 be the result (total 56 bits) Let Ci =Shifti(Ci-1); Di =Shifti(Di-1) and Ki be another permutation P2 of CiDi (total 56 bits) Where cyclic shift one position left if i=1,2,9,16 Else cyclic shift two positions left Cryptography and Network Security 179 Permutation Tables 57 49 41 33 25 17 9 14 17 11 28 15 24 1 5 6 21 10 4 26 8 1 58 50 42 34 26 18 3 10 2 59 51 23 19 12 19 11 3 16 7 27 20 13 41 52 31 37 47 55 45 33 48 43 35 27 60 52 44 36 63 55 47 39 31 23 15 2 7 62 54 47 38 30 22 30 40 51 14 6 61 53 45 37 29 44 49 39 56 34 53 21 13 5 28 20 12 46 42 50 36 29 32 4 Permutation table P1 Permutation table P2 Cryptography and Network Security 180 DES in Practice DEC (Digital Equipment Corp. 1992) built a chip with 50k transistors Encrypt at the rate of 1G/second Clock rate 250 Mhz Cost about $300 Applications ATM transactions (encrypting PIN and so on) Cryptography and Network Security 181 Model Mode of use The way we use a block cipher Four have been defined for the DES by ANSI in the standard: ANSI X3.106-1983 modes of use) Block modes Splits messages in blocks (ECB, CBC) Stream modes On bit stream messages (CFB, OFB) Cryptography and Network Security 182 Block Modes Electronic Codebook Book (ECB) where the message is broken into independent 64-bit blocks which are encrypted Ci = DESK1 (Pi) Cipher Block Chaining (CBC) again the message is broken into 64-bit blocks, but they are linked together in the encryption operation with an IV Ci = DESK1 (PiCi-1) C-1=IV (initial value) Cryptography and Network Security 183 Stream Model Cipher FeedBack (CFB) where the message is treated as a stream of bits, added to the output of the DES, with the result being feed back for the next stage Ci = Pi DESK1 (Ci-1) C-1=IV (initial value) Cryptography and Network Security 184 Cont. Output FeedBack (OFB) where the message is treated as a stream of bits, added to the message, but with the feedback being independent of the message Ci = Pi Oi Oi = DESK1 (Oi-1) O-1=IV (initial value) Cryptography and Network Security 185 DES Weak Keys With many block ciphers there are some keys that should be avoided, because of reduced cipher complexity These keys are such that the same sub-key is generated in more than one round, and they include: Cryptography and Network Security 186 Cont. Weak keys The same sub-key is generated for every round DES has 4 weak keys Semi-weak keys Only two sub-keys are generated on alternate rounds DES has 12 of these (in 6 pairs) Demi-semi weak keys Have four sub-keys generated Cryptography and Network Security 187 Cont. None of these causes a problem since they are a tiny fraction of all available keys However they MUST be avoided by any key generation program Cryptography and Network Security 188 DES Attacks 1998: The EFF's US$250,000 DES cracking machine contained 1,536 custom chips and could brute force a DES key in a matter of days — the photo shows a DES Cracker circuit board fitted with several Deep Crack chips. Cryptography and Network Security 189 DES Attacks: The COPACOBANA machine, built for US$10,000 by the Universities of Bochum and Kiel, contains 120 low-cost FPGAs and can perform an exhaustive key search on DES in 9 days on average. The photo shows the backplane of the machine with the FPGAs Cryptography and Network Security 190 Attack Faster than Brute Force Differential cryptanalysis was discovered in the late 1980s by Eli Biham and Adi Shamir, although it was known earlier to both IBM and the NSA and kept secret. To break the full 16 rounds, differential cryptanalysis requires 247 chosen plaintexts. DES was designed to be resistant to DC. Linear cryptanalysis was discovered by Mitsuru Matsui, and needs 243 known plaintexts (Matsui, 1993); the method was implemented (Matsui, 1994), and was the first experimental cryptanalysis of DES to be reported. There is no evidence that DES was tailored to be resistant to this type of attack. Cryptography and Network Security 191 Possible Techniques for Improving DES Multiple enciphering with DES Extending DES to 128-bit data paths and 112-bit keys Extending the key expansion calculation Cryptography and Network Security 192 Double DES? Using two encryption stages and two keys C=Ek2(Ek1(P)) P=Dk1(Dk2(C)) It is proved that there is no key k3 such that C=Ek2(Ek1(P))=Ek3(P) But Meet-in-the-middle attack Cryptography and Network Security 193 Meet-in-the-Middle Attack Assume C=Ek2(Ek1(P)) Given the plaintext P and ciphertext C Encrypt P using all possible keys k1 Decrypt C using all possible keys k2 Check the result with the encrypted plaintext lists If found match, they test the found keys again for another plaintext and ciphertext pair If it turns correct, then find the keys Otherwise keep decrypting C Cryptography and Network Security 194 Triple DES DES variant Standardized in ANSI X9.17 & ISO 8732 and in PEM for key management Proposed for general EFT standard by ANSI X9 Backwards compatible with many DES schemes Uses 2 or 3 keys Cryptography and Network Security 195 Cont. No known practical attacks Brute force search impossible (very hard) Meet-in-the-middle attacks need 256 Plaintext-Ciphertext pairs per key Popular current alternative Cryptography and Network Security 196 IDEA: Developed by James Massey & Xuejia Lai at ETH originally in Zurich in 1990, then called IPES: X Lai, J L Massey, "A Proposal for a New Block Encryption Standard" in Advances in Cryptology - Eurocrypt '90, Lecture Notes in Computer Science, vol 473, pp 389-404, X Lai, J L Massey, S Murphy, "Markov Ciphers and Differential Cryptanalysis" in Advances in Cryptology - Eurocrypt '91, Lecture Notes in Computer Science, vol 547, pp 17-38, name changed to IDEA in 1992 Cryptography and Network Security 197 Basic Features Encrypts 64-bit blocks using a 128-bit key Based on mixing operations from different (incompatible) algebraic groups XOR, + mod 2^(16) , X mod 2^(16) +1) On 16-bit sub-blocks, with no permutations used IDEA is patented in Europe & US, however non-commercial use is freely permitted used in the public domain PGP (with agreement) currently no attack against IDEA is known Seem secure against differential cryptanalysis, brute force Cryptography and Network Security 198 Operations Operations XOR, Addition mod 216, multiplication mod 216 +1 Why these special mod for addition, multiplication They do not satisfy the distributive law They do not satisfy the associative law Cryptography and Network Security 199 MA: multiplication/addition Multiplication/addition Basic block to provide diffusion Input of MA Two sub-blocks derived from 4 input sub-blocks, 4 sub-keys Two other sub-keys Output Two sub-blocks Needs four operations Four operations are the minimum to provide full diffusion Cryptography and Network Security 200 Overview Cryptography and Network Security 201 Cont. IDEA encryption works as follows: Use 8-rounds The 64-bit data is divided into: X1 , X2 , X3 , X4 Each round The sub-blocks are added (2,3), multiplied (1,4) with subkeys The results are XORed [1,3] and [2,4] to 2 sub-blocks The XOR results set as input of MA structure, It outputs two subblocks Results are then XORed with 2,4 and 1,3 subblocks respectively The second and third sub-blocks are swapped Finally new sub-keys are combined with the sub-blocks Cryptography and Network Security 202 Sub-Keys Total need 52=6*8+4 sub-keys First are directly from key in order Left shift of 25 bits, and then next 8 sub-keys Each sub-key is a sub-block of the original key Decryption Much more complicated It needs the inverse of the encryption key For addition, multiplication Cryptography and Network Security 203 Decryption The process of decryption is essentially the same as encryption But with different selection of sub-keys Basic Operations K1.1^(-1 ) is the multiplicative inverse mod 2^(16) +1 -K1.2 is the additive inverse mod 2^(16) The original operations are: (+) bit-by-bit XOR + additional mod 2^(16) of 16-bit integers * multiplication mod 2^(16) +1 (where 0 means 2^(16) ) Cryptography and Network Security 204 Decryption Sub-Keys Round Encryption Keys Decryption Keys 1 K1.1 K1.2 K1.3 K1.4 K1.5 K1.6 K9.1-1 -K9.2 -K9.3 K9.4-1 K8.5 K8.6 2 K2.1 K2.2 K2.3 K2.4 K2.5 K2.6 K8.1-1 -K8.3 -K8.2 K8.4-1 K7.5 K7.6 3 K3.1 K3.2 K3.3 K3.4 K3.5 K3.6 K7.1-1 -K7.3 -K7.2 K7.4-1 K6.5 K6.6 4 K4.1 K4.2 K4.3 K4.4 K4.5 K4.6 K6.1-1 -K6.3 -K6.2 K6.4-1 K5.5 K5.6 5 K5.1 K5.2 K5.3 K5.4 K5.5 K5.6 K5.1-1 -K5.3 -K5.2 K5.4-1 K4.5 K4.6 6 K6.1 K6.2 K6.3 K6.4 K6.5 K6.6 K4.1-1 -K4.3 -K4.2 K4.4-1 K3.5 K3.6 7 K7.1 K7.2 K7.3 K7.4 K7.5 K7.6 K3.1-1 -K3.3 -K3.2 K3.4-1 K2.5 K2.6 8 K8.1 K8.2 K8.3 K8.4 K8.5 K8.6 K2.1-1 -K2.3 -K2.2 K2.4-1205 Cryptography and Network Security K1.5 K1.6 Output K9.1 K9.2 K9.3 K9.4 K1.1-1 -K1.2 -K1.3 K1.4-1 Important Feature The size of the sub-block Need 216+1 be prime number To compute the inverse for each possible subkey So sub-block size 8 is also possible 28+1=257 is prime number Cryptography and Network Security 206 CAST-128 By Carlisle Adams, Stafford Tavares Defined in RFC 2144 Use key size varying from 40 to 128 bits Structure of Feistel network 16 rounds on 64-bits data block Four primitive operations Addition, substration (mod 232) Bitwise exclusive-OR Left-circular rotation Cryptography and Network Security 207 Skipjack and Clipper Skipjack used in Clipper escrowed encryption scheme(US govt) Skipjack is a block cipher, 64-bit data hardware only implementation 80-bit key (escrowed in 2 halves) 32 round all design details and descriptions are classified has been very considerable debate over its use attack by Matt Blaze (ATT) on the LEAF component of the Clipper protocol for secure phone communications Cryptography and Network Security 208 Blowfish Scheme Developed by Bruce Schneier Fast, compact, simple and variably secure Two basic operations: addition, XOR Key ranges from 32 bits to 448 bits Similar to Feistel scheme The sub-key and s-boxes are complicated So not suitable when key changes often Function g is very simple, unlike DES Cryptography and Network Security 209 RC5 Developed by R. Rivest Suitable for hardware or software Fast, simple, low memory, data-dependent rotations Adaptable to processors of different word length A family of algorithms determined by word length, number of rounds, size of secret key Decryption and encryption are not the same With little variations Primitive operations Addition, XOR, left circular rotation Cryptography and Network Security 210 Characteristics Key features of advanced sym block cipher Variable key length Mixed operators Data dependent rotation Key dependent rotation Key dependent S-boxes Lengthy key schedule algorithm Variable function F Variable of number of rounds Operation on both halved data each round Cryptography and Network Security 211 AES Advanced Encryption Standard (Rijndael) key size and the block size may be chosen to be any of 128, 192, or 256 bits (later only key, block fixed 128) Rijndael has a variable number of rounds. Not counting an extra round performed at the end of encipherment with one step omitted, the number of rounds in Rijndael is: 9 if both the block and the key are 128 bits long. 11 if either the block or the key is 192 bits long, and neither of them is longer than that. 13 if either the block or the key is 256 bits long. Three big blocks first perform an Add Round Key step (XORing a subkey with the block) by itself, then regular rounds noted above, the final round with the Mix Column step Cryptography and Network Security 212 Advanced Encryption Standard Not “American” Encryption Standard a.k.a Lab #1 Cryptography and Network Security 213 How was AES created? AES competition Started in January 1997 by NIST 4-year cooperation between U.S. Government Private Industry Academia Why? Replace 3DES Provide an unclassified, publicly disclosed encryption algorithm, available royalty-free, worldwide Cryptography and Network Security 214 The Finalists MARS IBM RSA Laboratories Joan Daemen (Proton World International) and Vincent Rijmen (Katholieke Universiteit Leuven) RC6 Rijndael Serpent Ross Anderson (University of Cambridge), Eli Biham (Technion), and Lars Knudsen (University of California San Diego) Twofish Bruce Schneier, John Kelsey, and Niels Ferguson (Counterpane, Inc.), Doug Whiting (Hi/fn, Inc.), David Wagner (University of California Berkeley), and Chris Hall (Princeton University) Wrote the book on crypto Cryptography and Network Security 215 Evaluation Criteria (in order of importance) Security Resistance to cryptanalysis, soundness of math, randomness of output, etc. Cost Computational efficiency (speed) Memory requirements Algorithm / Implementation Characteristics Flexibility, hardware and software suitability, algorithm simplicity Cryptography and Network Security 216 Results Cryptography and Network Security 217 Results Cryptography and Network Security 218 The winner: Rijndael AES adopted a subset of Rijndael Rijndael supports more block and key sizes Cryptography and Network Security 219 Finite Fields AES uses the finite field GF(28) b7x7 + b6x6 + b 5x5 + b4x4 + b3x3 + b 2x2 + b1x + b0 {b7, b6, b5, b4, b3, b2, b1, b0} Byte notation for the element: x6 + x5 + x + 1 {01100011} – binary {63} – hex Has its own arithmetic operations Addition Multiplication Cryptography and Network Security 220 Finite Field Arithmetic Addition (XOR) (x6 + x4 + x2 + x + 1) + (x7 + x + 1) = x7 + x6 + x4 + x2 {01010111} {10000011} = {11010100} {57} {83} = {d4} Multiplication is tricky Cryptography and Network Security 221 Finite Field Multiplication () (x6 + x4 + x2 + x +1) (x7 + x +1) = x13 + x11 + x9 + x8 + x7 + x7 + x5 + x3 + x2 + x + x6 + x4 + x2 + x +1 These cancel = x13 + x11 + x9 + x8 + x6 + x5 + x4 + x3 +1 and x13 + x11 + x9 + x8 + x6 + x5 + x4 + x3 +1 modulo ( x8 + x4 + x3 + x +1) = x7 + x6 +1. Irreducible Polynomial Cryptography and Network Security 222 Efficient Finite field Multiply There’s a better way xtime() – very efficiently multiplies its input by {02} Multiplication by higher powers can be accomplished through repeat application of xtime() Cryptography and Network Security 223 Efficient Finite field Multiply Example: {57} {13} {57} {02} = xtime({57}) = {ae} {57} {04} = xtime({ae}) = {47} {57} {08} = xtime({47}) = {8e} {57} {10} = xtime({8e}) = {07} {57} {13} = {57} ({01} {02} {10}) = ({57} {01}) ({57} {02}) ({57} {10}) = {57} {ae} {07} = {fe} Cryptography and Network Security 224 AES parameters Nb – Number of columns in the State For AES, Nb = 4 Nk – Number of 32-bit words in the Key For AES, Nk = 4, 6, or 8 Nr – Number of rounds (function of Nb and Nk) For AES, Nr = 10, 12, or 14 Cryptography and Network Security 225 AES methods Convert to state array Transformations (and their inverses) AddRoundKey SubBytes ShiftRows MixColumns Key Expansion Cryptography and Network Security 226 Convert to State Array Input block: 0 0 4 8 12 1 5 9 13 1 2 2 6 310 414 5 3 7 11 15 S0,0 S0,1 S0,2 S0,3 = 6 7 8 S1,0 S1,1 S1,2 S1,3 9 10 11 12 13 14 15 S2,0 S2,1 S2,2 S2,3 S3,0 S3,1 S3,2 S3,3 Cryptography and Network Security 227 AddRoundKey XOR each byte of the round key with its corresponding byte in the state array XOR S0,1 S0,0 S0,1 S0,2 S0,3 S1,0 S S1,1 1,1 S1,2 S1,3 S2,0 S2,1 S2,2 S2,3 S2,1 S3,0 S3,1 S3,2 S3,3 S3,1 R0,1 R0,0 R0,1 R0,2 R0,3 1,1 R1,2 R1,3 R1,0 R R1,1 R2,0 R2,1 R2,2 R2,3 R2,1 R3,0 R3,1 R3,2 R3,3 R3,1 S’0,1 S’0,0 S’0,1 S’0,2 S’0,3 S’1,0S’ S’1,1 1,1 S’1,2 S’1,3 S’2,0 S’2,1 S’2,2 S’2,3 S’2,1 S’3,0 S’3,1 S’3,2 S’3,3 S’3,1 Cryptography and Network Security 228 SubBytes Replace each byte in the state array with its corresponding value from the S-Box 00 44 88 CC 11 55 99 DD 55 22 66 AA EE 33 77 BB FF Cryptography and Network Security 229 ShiftRows Last three rows are cyclically shifted S0,0 S0,1 S0,2 S0,3 S1,0 S1,0 S1,1 S1,2 S1,3 S2,0 S2,1 S2,0 S2,1 S2,2 S2,3 S3,0 S3,1 S3,2 S3,0 S3,1 S3,2 S3,3 Cryptography and Network Security 230 MixColumns Apply MixColumn transformation to each column S’0,c = ({02} SMixColumns() 0,c) ({03} S1,c) S2,c S3,c S0,0 S0,1 S’0,1 S S’S1,c =SS0,c ({02} S1,c) ({03} S2,c S’ ) SS ’ 3,cS’ 0,1 0,2 0,3 0,0 0,1 0,2 S’0,3 S1,0 S S1,1 S’1,0S’ S’ 1,1S’S1,2=SS 1,3 S 1,1 )S’1,2 S’1,3 ({02} S ) ({03} S1,1 2,c 0,c 1,c 2,c 3,c S2,0 S2,1 S2,2 S2,3 S’2,0 S’2,1 S’2,2 S’2,3 S3,0 S3,1 S3,2 S3,3 S’3,0 S’3,1 S’3,2 S’3,3 S2,1S’ 3,c S3,1 = ({03} S0,c) S1,c S2,c ({02} S’S2,13,c S’3,1 Cryptography and Network Security 231 Key Expansion Expands the key material so that each round uses a unique round key Generates Nb(Nr+1) words Filled with just the key Filled with a combination of the previous work and the one Nk positions earlier Cryptography and Network Security 232 Encryption byte state[4,Nb] state = in AddRoundKey(state, keySchedule[0, Nb-1]) for round = 1 step 1 to Nr–1 { SubBytes(state) ShiftRows(state) MixColumns(state) AddRoundKey(state, keySchedule[round*Nb, (round+1)*Nb-1]) Prevents an attacker from First and last operations } SubBytes(state) ShiftRows(state) AddRoundKey(state, keySchedule[Nr*Nb, (Nr+1)*Nb-1]) even beginning to key encrypt or involve the decrypt without the key out = state Cryptography and Network Security 233 Decryption byte state[4,Nb] state = in AddRoundKey(state, keySchedule[Nr*Nb, (Nr+1)*Nb-1]) for round = Nr-1 step -1 downto 1 { InvShiftRows(state) InvSubBytes(state) AddRoundKey(state, keySchedule[round*Nb, (round+1)*Nb-1]) InvMixColumns(state) } InvShiftRows(state) InvSubBytes(state) AddRoundKey(state, keySchedule[0, Nb-1]) out = state Cryptography and Network Security 234 Encrypt and Decrypt Encryption Decryption AddRoundKey AddRoundKey SubBytes ShiftRows MixColumns AddRoundKey InvShiftRows InvSubBytes AddRoundKey InvMixColumns SubBytes ShiftRows AddRoundKey InvShiftRows InvSubBytes AddRoundKey Cryptography and Network Security 235 Cryptography and Network Security Public key system Xiang-Yang Li Cryptography and Network Security 236 Public Key Encryption Two difficult problems Key distribution under conventional encryption Digital signature Diffie and Hellman, 1976 Astonishing breakthrough One key for encryption and the other related key for decryption It is computationally infeasible to determine the decryption key using only the encryption key and the algorithm Cryptography and Network Security 237 Public Key Cryptosystem Essential steps of public key cryptosystem Each end generates a pair of keys One for encryption and one for decryption Each system publishes one key, called public key, and the companion key is kept secret It A wants to send message to B Encrypt it using B’s public key When B receives the encrypted message It decrypt it using its own private key Cryptography and Network Security 238 Applications of PKC Encryption/Decryption The sender encrypts the message using the receiver’s public key Q: Why not use the sender’s secret key? Digital signature The sender signs a message by encrypt the message or a transformation of the message using its own private key Key exchange Two sides cooperate to exchange a session key, typically for conventional encryption Cryptography and Network Security 239 Conditions of PKC Computationally easy To generate public and private key pair To encrypt the message using encryption key To decrypt the message using decryption key Computational infeasible To compute the private key using public key To recover the plaintext using ciphertext and public key The encryption and decryption can be applied in either order Cryptography and Network Security 240 One Way Function PKC boils down to one way function Maps a domain into a range with unique inverse The calculation of the function is easy The calculation of the inverse is infeasible Easy The problem can be solved in polynomial time Infeasible The effort to solve it grows faster than polynomial time For example: 2n It requires infeasible for all inputs, not just worst case Cryptography and Network Security 241 Trapdoor One-way Function Trapdoor one way function Maps a domain into a range with unique inverse Y=fk(X) The calculation of the function is easy The calculation of the inverse is infeasible if the key is not known The calculation of the inverse is easy if the key is known Cryptography and Network Security 242 Possible Attacks Brute force Use large keys Trade-off: speed (not linearly depend on key size) Confined to small data encryption: signature, key management Compute the private key from public key Not proven that is not feasible for most protocols! Probable message attack Encrypt all possible messages using encryption key Compare with the ciphertext to find the matched one! If data is small, feasible, regardless of key size of PKC Cryptography and Network Security 243 History http://www.research.att.com/~smb/nsam- 160/ British National Security Action Memorandum 160 Kennedy Nuclear Weapon http://www.research.att.com/~smb/nsam-160/pg1.html Cryptography and Network Security 244 RSA Algorithm R. Rivest, A. Shamir, L. Adleman (1977) James Ellis came up with the idea in 1970, and proved that it was theoretically possible. In 1973, Clifford Cocks a British mathematician invented a variant on RSA; a few months later, Malcom Williamson invented a Diffie-Hellman analog Only revealed till 1997 Patent expired on September 20, 2000. Block cipher using integers 0~n-1 Thus block size k is less than log2n Algorithm: Encryption: C=Me mod n Decryption: M=Cd mod n Both sender and the receiver know n Cryptography and Network Security 245 RSA (public key encryption) Alice wants Bob to send her a message. She: selects two (large) primes p, q, TOP SECRET, computes n = pq and (n) = (p-1)(q-1), (n) also TOP SECRET, selects an integer e, 1 < e < (n), such that gcd(e, (n)) = 1, computes d, such that de 1 (mod (n)), d also TOP SECRET, gives public key (e, n), keeps private key (d, n). Cryptography and Network Security 246 Requirements Possible to find e and d such that M=Mde mod n for all message M Easy to conduct encryption and decryption Infeasible to compute d Given n and e Cryptography and Network Security 247 RSA Example 1. 2. 3. 4. 5. 6. 7. Select primes: p=17 & q=11 Compute n = pq =17×11=187 Compute ø(n)=(p–1)(q1)=16×10=160 Select e : gcd(e,160)=1; choose e=7 Determine d: de=1 mod 160 and d < 160 Value is d=23 since 23×7=161= 10×160+1 Publish public key KU={7,187} Keep secret private key KR={23,17,11} Cryptography and Network Security 248 RSA Example cont sample RSA encryption/decryption is: given message M = 88 (nb. 88<187) encryption: C = 887 mod 187 = 11 decryption: M = 1123 mod 187 = 88 Cryptography and Network Security 249 Key Generation Recall Euler Theorem a(n)+1 =a mod n for all 0<a<n and gcd(a,n)=1 Then ed=1 mod (n) is sufficient to make algorithm correct (need more proofs) RSA chooses the following Integer n=pq for two primes p and q Select e, such that gcd(e, (n))=1 Compute the inverse of e mod (n) The result is set as d Cryptography and Network Security 250 Key Generation The prime numbers sufficiently large p and q must be They are chosen by applying primality testing of randomly chosen large numbers About n/ln n prime numbers less than n Implies needs to check about 2ln n random numbers to find 2 primes numbers around n Compute n=pq, keep p and q secret! Select random number e Test gcd(e, (n))=1, and get d if equation holds Cryptography and Network Security 251 Exponentiation can use the Square and Multiply Algorithm a fast, efficient algorithm for exponentiation concept is based on repeatedly squaring base and multiplying in the ones that are needed to compute the result look at binary representation of exponent only takes O(log2 n) multiples for number n eg. 75 = 74.71 = 3.7 = 10 mod 11 eg. 3129 = 3128.31 = 5.3 = 4 mod 11 Cryptography and Network Security 252 Exponentiation Cryptography and Network Security 253 More on Exponention (PGP) To compute Cd mod n, we compute Cd mod p and Cd mod q Remember that the receiver could keep p,q Then Chinese Remainder Theorem to find Cd mod n Cryptography and Network Security 254 Security of RSA Brute force: try all possible private keys Factoring integer n, then know Not proven to be NPC Determine (n) directly without factoring Equivalent to factoring! (1996) Determine (n) d directly without knowing (n) Currently appears as hard as factoring But not proven, so it may be easier! Cryptography and Network Security 255 Practical Considerations Testing p, q using probability first, then deterministic methods A good random number generator is needed for p,q 'random' and 'unpredictable' p and q should be in similar scale Both p-1 and q-1 should have large prime factor Primes The gcd(p-1,q-1) should be small e = 2 should not be used The decryption key d should larger then n1/4 The encryption key RSA is much slower than symmetric cryptosystems. In practice, typically encrypts a secret message with a symmetric algorithm, encrypts the (comparatively short) symmetric key with RSA, and transmits both the RSA-encrypted symmetric key and the symmetrically-encrypted message to Alice. Cryptography and Network Security 256 Fixed point of RSA How many m such that me=m mod n assume that gcd(m, n)=1 It is same as me-1=1 mod n Thus, me-1=1 mod p and me-1=1 mod q Solutions gcd(e-1,p-1)*gcd(e-1,q-1) Need more proofs. Cryptography and Network Security 257 Cyclic Attack Compute me mod n, me2 mod n, me3 mod n…till it reaches some message readable. Need period large Let r be the largest prime of p-1, L be the largest prime of r-1 Then period is at least L with high probability Implies that we often need find a large prime x Based on this, find a large prime of y=kx+1 format (by trying k=2,3,…) Based on y, then find a large prime p=t y+1 format Try difference values for t=2,3,4… Cryptography and Network Security 258 How to deal with p, q Delete them securely Or used for speed-up calculation from CRT Compute Me mod p and Me mod q Then find using Me mod n based on CRT Cryptography and Network Security 259 Timing Attacks Keep track of how long a computer takes to decrypt a message! Paul Kocher, 1995, Dec-7 Stunning attack strategy and cipher only attack! Guessing the key bit by bit Countermeasures (Rivest 11 Dec 1995) Constant exponentiation time Random delay Blinding (add a random number for encryption and decryption) Cryptography and Network Security 260 Chosen Ciphertext Attack Collect ciphertext c (send to Alice), want to find m=cd mod n Attacker chooses random r Compute x= re mod n; y=xc mod n; and t= r-1 mod n Attacker gets Alice to sign y with private key using RSA: yd mod n That is why not use the same key for encryption and digital signature Alice sends u= yd mod n to Attacker Attacker then computes tu mod nm Cryptography and Network Security 261 Other attacks on RSA Comprised decryption key If the private key d (for decryption of received ciphertext) of a user is comprised, then the user has to reselect n and e and d It cannot use the old number n to produce the key-pairs! Otherwise attacker already can factor n almost surely! The number n can only be used by one person If two user uses the same n, even they do not know the factoring of n, they still could figure out the factoring of n with probability almost one. Similar as above Cryptography and Network Security 262 Bit security of RSA Given ciphertext C, We may want to find the last bit of M, denoted by parity(C) We may want to find if M>n/2, denoted by half(C) We may want to find all bits of M The above three attacks are the same! If we can solve one, we can solve the other two! Cryptography and Network Security 263 Other Public Key Systems Rabin Cryptosystem Decryption is not unique Elgamal Cryptosystem Expansion of the plaintext (double) Knapsack System Already broken Elliptic Curve System If directly implement Elgamal on elliptic curve Expansion of plaintext by 4; Restricted plaintext Menezes-Vanston system is more efficient Cryptography and Network Security 264 Rabin Cryptosystem Procedure Let n=pq and p=3 mod 4, q=3 mod 4 Publish n, and a number b<n For message m C=m(m+b) mod n The receiver decrypts ciphertext C (b2/4+C)1/2-b/2 Cryptography and Network Security 265 Analysis For receiver, need solve equation x2+xb=C mod n Let x1=x+b/2, c=b2/4+C, then need Solve x12 =c mod n Chinese Remainder Theorem implies that x12 =c mod p x12 =c mod q When p=3 and q=3 mod 4 Solution x1=c(p+1)/4 mod p and x1=c(q+1)/4 mod q Then Chinese Remainder Theorem again to combine solution Cryptography and Network Security 266 Security Breaking it < factoring n Secure against Chosen plaintext attack Not secure against Chosen ciphertext attack Decoding produces three false results in addition to the correct one, so that the correct result must be guessed. This is the major disadvantage of the Rabin cryptosystem and one of the factors which have prevented it from finding widespread practical use. It has been proven that decoding the Rabin cryptosystem is equivalent to the integer factorization problem, which is rather different than for RSA. Cryptography and Network Security 267 Dealing with 4 solutions By adding redundancies, for example, the repetition of the last 64 bits, the system can be made to produce a single root. If this technique is applied, the proof of the equivalence with the factorization problem fails. Cryptography and Network Security 268 ElGamal Cryptosystem Based on Discrete Logarithm Find unique integer a such that gx=y mod p Here x is a primitive element in Zp, p is prime Procedure Make p, g, y public, keep x secret Encryption: Ek(m)=(gk mod p, m y k mod p) Decryption Dk(y1,y2)=y2(y1x)-1 mod p Cryptography and Network Security 269 Security of ElGamal ElGamal is a simple example of a semantically secure asymmetric key encryption algorithm (under reasonable assumptions). ElGamal's security rests, in part, on the difficulty of solving the discrete logarithm problem in G. Specifically, if the discrete logarithm problem could be solved efficiently, then ElGamal would be broken. However, the security of ElGamal actually relies on the so-called Decisional DiffieHellman (DDH) assumption. This assumption is often stronger than the discrete log assumption, but is still believed to be true for many classes of groups. Cryptography and Network Security 270 Semantic Security Semantic security is a widely-used definition for security in an PKS. For a cryptosystem to be semantically secure, it must be infeasible for a computationally-bounded adversary to derive significant information about a message (plaintext) when given only its ciphertext and the corresponding public encryption key. Semantic security considers only the case of a "passive" attacker, i.e., one who observes ciphertexts and generates chosen ciphertexts using the public key Indistinguishability definition is used more commonly than the original definition of semantic security. Cryptography and Network Security 271 Indistinguishability: semantic security. Indistinguishability under Chosen Plaintext Attack (IND-CPA) is commonly defined by the following game: A probabilistic polynomial time-bounded adversary is given a public key, which it may use to generate any number of ciphertexts (within polynomial bounds). The adversary generates two equal-length messages m0 and m1, and transmits them to a challenge oracle along with the public key. The challenge oracle selects one of the messages by flipping a uniformly-weighted coin, encrypts the message under the public key, and returns the resulting ciphertext c to the adversary. The underlying cryptosystem is IND-CPA (and thus semantically secure under chosen plaintext attack) if the adversary cannot determine which of the two messages was chosen by the oracle, with probability significantly greater than 1 / 2 (the success rate of random guessing). a semantically secure encryption scheme must by definition be probabilistic, possessing a component of randomness; if this were not the case, the adversary could simply compute the deterministic encryption of m0 and m1 and compare these encryptions with the returned ciphertext c to successfully guess the oracle's choice. Cryptography and Network Security 272 Deal with deterministic PKS RSA, can be made semantically secure (under stronger assumptions) through the use of random encryption padding schemes such as Optimal Asymmetric Encryption Padding (OAEP). ElGamal scheme is semantically secure Cryptography and Network Security 273 Bit security of Discrete Log Given gx=y mod p We may want to find the value of x Find some bits of x Assume that p-1 = 2st We can find the last s bits of x for sure But to find the other bits of x is same as to find all bits of x! Example, the last bit of x is 0 y is QR iff y(p-1)/2=1 mod p 1 y is NQR iff y(p-1)/2=-1 mod p Cryptography and Network Security 274 DH Assumption Consider a cyclic group G of order q. The DDH assumption states that, given (g,ga,gb) for a randomly-chosen generator g and random , the value gab "looks like" a perfectly random element of G. This intuitive notion is formally stated by saying that the following two ensembles are computationally indistinguishable: (g,ga,gb,gab), where g,a,b are chosen at random as described above (this input is called a "DDH tuple"); (g,ga,gb,gc), where g,a,b are chosen at random and c is chosen at random. Diffie-Hellman problem computing gab from (g,ga,gb) Cryptography and Network Security 275 Knapsack Cryptosystem Based on subset sum problem Given a set, find a subset with half summation value It is NPC problem generally Superincreasing set if si>j<isj The subset problem over superincreasing set can be solved in polynomial time! Been broken by Shamir, 1984 Using integer programming tech by Lenstra Cryptography and Network Security 276 Solve Subset Problem Let T be the half summation, t=T; For i=n downto 1 do If tsi then t=t-si Set xi=1 Else xi=0 If xisi=T then (x1, x2,… xn) is the solution Else, there is no solution Cryptography and Network Security 277 Knapsack System Procedure Select a superincreasing set s Let p be prime larger than set summation of s, Select integer a, keep s, a, p secret Make t=(as1, as2,…asn) mod p public Encryption Ciphertext C = E(x1,x2,…xn)=xiti mod p Decryption Solve the subset summation problem (s, a-1C mod p) Cryptography and Network Security 278 Elliptic Curve Cryptography majority of public-key crypto (RSA, D-H) use either integer or polynomial arithmetic with very large numbers/polynomials imposes a significant load in storing and processing keys and messages an alternative is to use elliptic curves offers same security with smaller bit sizes Cryptography and Network Security 279 Real Elliptic Curves an elliptic curve is defined by an equation in two variables x & y, with coefficients consider a cubic elliptic curve of form y2 = x3 + ax + b where x,y,a,b are all real numbers also define zero point O have addition operation for elliptic curve geometrically sum of Q+R is reflection of intersection R Cryptography and Network Security 280 Real Elliptic Curve Example Cryptography and Network Security 281 Finite Elliptic Curves Elliptic curve cryptography uses curves whose variables & coefficients are finite have two families commonly used: prime curves Ep(a,b) defined over Zp use integers modulo a prime p best in software binary curves E2m(a,b) defined over GF(2n) use polynomials with binary coefficients best in hardware Cryptography and Network Security 282 Elliptic Curve Cryptography ECC addition is analog of modulo multiply ECC repeated addition is analog of modulo exponentiation need “hard” problem equiv to discrete log Q=kP, where Q,P belong to a prime curve is “easy” to compute Q given k,P but “hard” to find k given Q,P known as the elliptic curve logarithm problem Certicom example: E23(9,17) Cryptography and Network Security 283 ECC Diffie-Hellman can do key exchange analogous to D-H users select a suitable curve Ep(a,b) select base point G=(x1,y1) with large order n s.t. n*G=O A & B select private keys nA<n, nB<n compute public keys: PA=nA×G, PB=nB×G compute shared key: K=nA×PB, K=nB×PA same since K=nA×nB×G Cryptography and Network Security 284 ECC Encryption/Decryption several alternatives, will consider simplest must first encode any message M as a point on the elliptic curve Pm select suitable curve & point G as in D-H each user chooses private key nA<n and computes public key PA=nA×G to encrypt Pm : Cm={kG, Pm+k PA}, k random decrypt Cm compute: Pm+kPA–nA(kG) = Pm+k(nAG)–nA(kG) = Pm Cryptography and Network Security 285 ECC Security relies on elliptic curve logarithm problem fastest method is “Pollard rho method” compared to factoring, can use much smaller key sizes than with RSA etc for equivalent key lengths computations are roughly equivalent hence for similar security ECC offers significant computational advantages Cryptography and Network Security 286 Cryptography and Network Key Management and generation Xiang-Yang Li Cryptography and Network Security 287 Key Exchange Public key systems are much slower than private key system Public key system is then often for short data Signature, key distribution Key distribution One party chooses the key and transmits it to other user Key agreement Protocol such two parties jointly establish secret key over public communication channel Key is the function of inputs of two users Cryptography and Network Security 288 Distribution of Public Keys can be considered as using one of: Public announcement Publicly available directory Public-key authority Public-key certificates Cryptography and Network Security 289 Public Key Management Simple one: publish the public key Such as newsgroups, yellow-book, etc. But it is not secure, although it is convenient Anyone can forge such a announcement Ex: user B pretends to be A, and publish a key for A Then all messages sent to A, readable by B! Let trusted authority maintain the keys Need to verify the identity, when register keys User can replace old keys, or void old keys Cryptography and Network Security 290 Possible Attacks Observe all messages over the channel So assume that all plaintext messages are available to all Save messages for reuse later So have to avoid replay attack Masquerade various users in the network So have to be able to verify the source of the message Cryptography and Network Security 291 Public Announcement users distribute public keys to recipients or broadcast to community at large eg. append PGP keys to email messages or post to news groups or email list major weakness is forgery anyone can create a key claiming to be someone else and broadcast it until forgery is discovered can masquerade as claimed user Cryptography and Network Security 292 Publicly Available Directory can obtain greater security by registering keys with a public directory directory must be trusted with properties: contains {name,public-key} entries participants register securely with directory participants can replace key at any time directory is periodically published directory can be accessed electronically still vulnerable to tampering or forgery Cryptography and Network Security 293 Public-Key Authority improve security by tightening control over distribution of keys from directory has properties of directory and requires users to know public key for the directory then users interact with directory to obtain any desired public key securely does require real-time access to directory when keys are needed Cryptography and Network Security 294 Public-Key Authority Cryptography and Network Security 295 Cont. More advanced distribution A sends request-for-key(B) to authority with timestamp, that is, Ida|Idb|Time Authority replies with key(B) (encrypted by its private key), that is EKTta(KUb| Ida|Idb|Time) A initiates a message to B, including a random number Na, its IDA B then ask authority to get key(A) B sends A (encrypted by A’s public key) Na and Nb A then replies B Nb encrypted by B’s public key Cryptography and Network Security 296 Cont. In above scheme, the authority is bottleneck New approach: certificate Any user can read certificate, determine name and public key of the certificate’s owner Any user can verify the authority of certificate Only the authority can create and update certificate Any user can verify the time-stamp of certificate The certificate is CA=EKRauth[T,IDA, KUA] Time-stamp is to avoid reuse of voided key Cryptography and Network Security 297 Public-Key Certificates certificates allow key exchange without real-time access to public-key authority a certificate binds identity to public key usually with other info such as period of validity, rights of use etc with all contents signed by a trusted Public-Key or Certificate Authority (CA) can be verified by anyone who knows the public-key authorities public-key To validate the certificate, we need another certificate, one that matches the Issuer (of CA) in the first certificate. Then we take the RSA public key from the second (CA) certificate, use it to decode the signature on the first certificate to obtain an MD5 hash, which must match an actual MD5 hash computed over the rest of the certificate. Cryptography and Network Security 298 X.509 The structure of a X.509 v3 digital certificate is as follows: Certificate Version Serial Number Algorithm ID Issuer Validity Not Before Not After Subject Subject Public Key Info Public Key Algorithm Subject Public Key Issuer Unique Identifier (Optional) Subject Unique Identifier (Optional) Extensions (Optional) ... Certificate Signature Algorithm Certificate Signature Cryptography and Network Security 299 Sample Certificate Certificate: Data: Version: 1 (0x0) Serial Number: 7829 (0x1e95) Signature Algorithm: md5WithRSAEncryption Issuer: C=ZA, ST=Western Cape, L=Cape Town, O=Thawte Consulting cc, OU=Certification Services Division, CN=Thawte Server CA/emailAddress=server-certs@thawte.com Validity Not Before: Jul 9 16:04:02 1998 GMT Not After : Jul 9 16:04:02 1999 GMT Subject: C=US, ST=Maryland, L=Pasadena, O=Brent Baccala, OU=FreeSoft, CN=www.freesoft.org/emailAddress=baccala@freesoft.org Subject Public Key Info: Public Key Algorithm: rsaEncryption RSA Public Key: (1024 bit) Modulus (1024 bit): 00:b4:31:98:0a:c4:bc:62:c1:88:aa:dc:b0:c8:bb: 33:35:19:d5:0c:64:b9:3d:41:b2:96:fc:f3:31:e1: 66:36:d0:8e:56:12:44:ba:75:eb:e8:1c:9c:5b:66: 70:33:52:14:c9:ec:4f:91:51:70:39:de:53:85:17: 16:94:6e:ee:f4:d5:6f:d5:ca:b3:47:5e:1b:0c:7b: c5:cc:2b:6b:c1:90:c3:16:31:0d:bf:7a:c7:47:77: 8f:a0:21:c7:4c:d0:16:65:00:c1:0f:d7:b8:80:e3: d2:75:6b:c1:ea:9e:5c:5c:ea:7d:c1:a1:10:bc:b8: e8:35:1c:9e:27:52:7e:41:8f Exponent: 65537 (0x10001) Signature Algorithm: md5WithRSAEncryption 93:5f:8f:5f:c5:af:bf:0a:ab:a5:6d:fb:24:5f:b6:59:5d:9d: 92:2e:4a:1b:8b:ac:7d:99:17:5d:cd:19:f6:ad:ef:63:2f:92: ab:2f:4b:cf:0a:13:90:ee:2c:0e:43:03:be:f6:ea:8e:9c:67: d0:a2:40:03:f7:ef:6a:15:09:79:a9:46:ed:b7:16:1b:41:72: 0d:19:aa:ad:dd:9a:df:ab:97:50:65:f5:5e:85:a6:ef:19:d1: 5a:de:9d:ea:63:cd:cb:cc:6d:5d:01:85:b5:6d:c8:f3:d9:f7: 8f:0e:fc:ba:1f:34:e9:96:6e:6c:cf:f2:ef:9b:bf:de:b5:22: 68:9f Cryptography and Network Security 300 Security In 2005, Arjen Lenstra and Benne de Weger demonstrated "how to use hash collisions to construct two X.509 certificates that contain identical signatures and that differ only in the public keys," achieved using a collision attack on the MD5 hash function See http://www.win.tue.nl/~bdeweger/Colliding Certificates/ddl-full.pdf Cryptography and Network Security 301 Public-Key Certificates Cryptography and Network Security 302 Public-Key Distribution of Secret Keys use previous methods to obtain public-key can use for secrecy or authentication but public-key algorithms are slow so usually want to use private-key encryption to protect message contents hence need a session key have several alternatives for negotiating a suitable session Cryptography and Network Security 303 Simple Secret Key Distribution proposed by Merkle in 1979 A generates a new temporary public key pair A sends B the public key and their identity B generates a session key K sends it to A encrypted using the supplied public key A decrypts the session key and both use problem is that an opponent can intercept and impersonate both halves of protocol Cryptography and Network Security 304 Secret key Distribution Simple secret key distribution A generates KUA and KRA, sends KUA to B B generates a secret key ks B sends ks to A using A’s public key KUA A decrypts the message to get the secret key ks To get more security, the public/private keys can be regenerated when needed But vulnerable to the active attack! Attacker E can compromise the communication between A and B as follows Cryptography and Network Security 305 Cont. Attacking A generates KUA and KRA, sends IDA, KUA to B E intercepts the message, transmits IDA, KUE to B B generates a secret key ks B sends ks to A using A’s “public key” KUE E intercepts the message, decrypt it and get ks E sends A the message Ks, encrypted by KUA A decrypts the message to get the secret key ks Now E knows Ks, but A, B are unaware of it Cryptography and Network Security 306 Secret Key Distribution So need confidentiality and authentication A and B need to use a secure method to exchange their public keys Schemes A initiates a message to B, EKUB(Na,IDa) B replies it with EKUA(Na,Nb) A then replies it with EKUB(Nb) A sends B the message EKUB (EKRA(Ks)) Security The first 3 steps are used to assure that A is A, B is B Cryptography and Network Security 307 Public-Key Distribution of Secret Keys if have securely exchanged public-keys: Cryptography and Network Security 308 Key Predistribution Trusted Authority (TA) generates keys for all pair of users and transmits to them Large overhead (for TA and user) Blom Scheme Keys are chosen from a finite field Zp P is public prime number TA transmits k+1 elements of Zp to each user over secure channel Secure condition: any set of at most k users (not U,V) can not determine any information about Ku,v Cryptography and Network Security 309 Blom Scheme Scheme (when k=1) Each user u has distinct element ru from Zp TA choose a,b,c and defines f(x,y)=a+b(x+y)+cxy mod p For each u, TA computes gu(x)=f(x, ru) mod p TA transmits gu(x) to user u Two users u and v compute the common key f(ru, rv)= a+b(ru + rv)+c ru rv mod p Here f(ru, rv)= gv(ru)= gu(rv) Cryptography and Network Security 310 Security of Blom Scheme Less than k users can not determine keys However, more than k users can compute any keys Solving equations to get a,b,c for k=1 Generally Function f(x,y)=Sum ai,jxiyj mod p Here ai,j=aj,i Cryptography and Network Security 311 Diffie-Hellman Key Predist. Computationally secure if discrete logarithm is intractable Scheme Assume prime number p public and an integer c public Each user u has secret component au User u computes bu=c au mod p TA certifies it by computing (ID(u), bu, sigTA(ID(u), bu)) The common key of two users u and v is K=c au av mod p Cryptography and Network Security 312 Diffie Hellman Around September 1974, Diffie (Graduate student) had been traveling USA with his wife, Mary, discussing cryptography with anyone who was available. At the time, there was very little published material about modern methods and much was classified. Very few people were interested in the topic and Marty Hellman even says that many of his colleagues felt that it was "born classified," like secrets about the atomic bomb, because it was so important to national security. John Gill gave the idea of exponential Cryptography and Network Security 313 Diffie-Hellman Problem Diffie-Hellman problem definition Given bu=gau mod p, bv=gav mod p, how to compute gavau mod p? Here g is a primitive element of mod p The problem is not harder than the discrete logarithmetic problem, because the later one can always be used to solve it It can be proved that it has the same difficulty as the ElGamal encryption system Cryptography and Network Security 314 Diffie-Hellman Key Exchange Computationally secure if discrete logarithm is intractable Scheme Assume prime number p public and an integer c public Each user u chooses a secret component au (new!) User u computes bu=c au mod p User v computes bv=c av mod p The common key of two users u and v is K=c au av mod p Cryptography and Network Security 315 Middle Attack Intruder w intercept the communications Intruder w communications with u Intruder w communications with v The key computed by u is K=c au av’ mod p c au’ c au v w u c av’ c av Cryptography and Network Security 316 Authenticated Key Agreement Introducing the identification scheme before key exchange does not help The attacker remains inactive until identification done Simplified station to station protocol Key agreement protocol itself authenticates the user’s identity at the same time the key being defined Cryptography and Network Security 317 Station-to-station Protocol Scheme Each user has a certificate C(v)=(Idv,verv,sigTA(Idv,verv)) User u selects au and computes bu=c au mod p User v selects av and computes Value bv=c av mod p Key K=c au av mod p Signature yv=sigv(bu,bv) User v sends (C(V), bv, yv) to U User u computes K=c au av mod p, verifies yv, and C(V) User u computes yu=sigu(bu,bv), sends (C(u),yu) to V User v verifies yu, and C(u) Cryptography and Network Security 318 MTI Agreement Protocol Scheme Assume prime number p public and an integer c public Each user has certificate c(u)=(Idu,bu, sigTA(Idu,bu)) Here bu= c au mod p Each user u chooses a secret component ru (new!) User u computes su=c ru mod p, sends (c(u),su) User v computes sv=c rv mod p, sends (c(v),sv) The common key of two users u and v is K=c rvau+ ru av mod p= sv aubv ru mod p= su avbu rv mod p Cryptography and Network Security 319 Cryptography and Network Security Authentication Xiang-Yang Li Cryptography and Network Security 320 Message Authentication Digital Signature Authentication Authentication requirements Authentication functions Mechanisms MAC: message authentication code Hash functions, security in hash functions Hash and MAC algorithms MD5, SHA, RIPEMD-160, HMAC Digital signatures Cryptography and Network Security 321 Message Attacks Possible attacks Disclosure Traffic analysis Masquerade Content modification Sequence modification Time modification Repudiation Denial of the receipt of message by the destination or Denial of the transmitting by the source Cryptography and Network Security 322 Authentication Enables receiver to verify message authenticity Using some lower level functions as primitive Three types of functions Message encryption Message authentication code (MAC) Hash function Cryptography and Network Security 323 Authentication Goal: Bob wants Alice to “prove” her identity to him Protocol ap1.0: Alice says “I am Alice” “I am Alice” Failure scenario?? Cryptography and Network Security 324 Authentication Goal: Bob wants Alice to “prove” her identity to him Protocol ap1.0: Alice says “I am Alice” “I am Alice” in a network, Bob can not “see” Alice, so Trudy simply declares herself to be Alice Cryptography and Network Security 325 Authentication: another try Protocol ap2.0: Alice says “I am Alice” in an IP packet containing her source IP address Alice’s IP address “I am Alice” Failure scenario?? Cryptography and Network Security 326 Authentication: another try Protocol ap2.0: Alice says “I am Alice” in an IP packet containing her source IP address Trudy can create a packet “spoofing” Alice’s Alice’s address IP address “I am Alice” Cryptography and Network Security 327 Authentication: another try Protocol ap3.0: Alice says “I am Alice” and sends her secret password to “prove” it. Alice’s Alice’s “I’m Alice” IP addr password Alice’s IP addr OK Failure scenario?? Cryptography and Network Security 328 Authentication: another try Protocol ap3.0: Alice says “I am Alice” and sends her secret password to “prove” it. Alice’s Alice’s “I’m Alice” IP addr password Alice’s IP addr OK playback attack: Trudy records Alice’s packet and later plays it back to Bob Alice’s Alice’s “I’m Alice” IP addr password Cryptography and Network Security 329 Authentication: yet another try Protocol ap3.1: Alice says “I am Alice” and sends her encrypted secret password to “prove” it. Alice’s encrypted “I’m Alice” IP addr password Alice’s IP addr OK Failure scenario?? Cryptography and Network Security 330 Authentication: another try Protocol ap3.1: Alice says “I am Alice” and sends her encrypted secret password to “prove” it. Alice’s encrypted “I’m Alice” IP addr password Alice’s IP addr record and playback still works! OK Alice’s encrypted “I’m Alice” IP addr password Cryptography and Network Security 331 Authentication: yet another try Goal: avoid playback attack Nonce: number (R) used only once –in-a-lifetime ap4.0: to prove Alice “live”, Bob sends Alice nonce, R. Alice must return R, encrypted with shared secret key “I am Alice” R KA-B(R) drawbacks? Alice is live, and only Alice knows key to encrypt nonce, so it must be Alice! Cryptography and Network Security 332 Authentication: ap5.0 ap4.0 requires shared symmetric key can we authenticate using public key techniques? ap5.0: use nonce, public key cryptography “I am Alice” R Bob computes + - - K A (R) “send me your public key” + KA KA(KA (R)) = R and knows only Alice could have the private key, that encrypted R such that + K (K (R)) = R A A Cryptography and Network Security 333 ap5.0: security hole Man (woman) in the middle attack: Trudy poses as Alice (to Bob) and as Bob (to Alice) I am Alice R I am Alice R K (R) T K (R) A Send me your public key + K T Send me your public key + K A - + m = K (K (m)) A A + K (m) A Trudy gets - + m = K (K (m)) T Alice sends T m to + K (m) T encrypted with Alice’s public key Cryptography and Network Security 334 ap5.0: security hole Man (woman) in the middle attack: Trudy poses as Alice (to Bob) and as Bob (to Alice) Difficult to detect: Bob receives everything that Alice sends, and vice versa. (e.g., so Bob, Alice can meet one week later and recall conversation) problem is that Trudy receives all messages as well! Cryptography and Network Security 335 Message Encryption Conventional Encryption Authentication provided due to the secret key But the message need to be meaningful What happened it message is not readable? How to determine intelligible automatically? Approach Checksum or frame check sequence(FCS) to message Encrypt the message and the appending FCS Receiver decrypt the ciphertext Computes FCS of message, compare with received one Cryptography and Network Security 336 Public Key Encryption Direct encryption by receiver’s public key Only confidentiality, no authentication For authentication Encrypt using sender’s private key Assume the message is intelligible No confidentiality: everyone can decrypt Confidentiality and authentication Encrypt by sender’s, then receiver’s public key But too time-consuming: 4 rounds RSA on large data Cryptography and Network Security 337 Message Authentication Code Assume both uses share secret key k Procedure Sender computes MAC=Ck(M) for M Sent M and MAC of it to receiver Receiver computes the MAC on received M Compare it with received MAC If match, then accepts the message MAC is similar to encryption, but not need be reversible! Cryptography and Network Security 338 MAC with Confidentiality Two options Using another key to encrypt M and MAC Using another key to encrypt M only Requirements of MAC Size of MAC: n Size of key: k Need 2n computations of MAC and n/k pairs of Mi and MACi Cryptography and Network Security 339 Why not Conventional Encrypt Possible situations Broadcast a message (one destination can verify) Authentication is done selectively Authentication of computer program Authentication may be important than secrecy Architecture flexibility Authentication lasts longer than secret protection Cryptography and Network Security 340 MAC Requirements Computationally infeasible to construct M’ such that Ck(M’)=Ck(M) Ck(M) uniformly distributed Cryptography and Network Security 341 Data Authentication Algorithm ANSI standard X9.17 Based on DES Using Cipher Block Chaining mode Data is grouped into 64 bits blocks Padding 0’s if necessary Outputi=Ek(DiOutputi-1) 0<i, and Output0=0’s The data authentication code DAC consists of the leftmost m bits of the last output, m16 Cryptography and Network Security 342 Authentication Protocols Central issues Confidentiality: prevent masqueraded and compromised Timeliness: prevent replay attacks Simple replay, repetition within timestamp, replay arrives but not the true messages,backward replay attack to the sender Mutual authentication One-way authentication Cryptography and Network Security 343 Coping with Replay Time stamps Party A accepts a message only if has valid timestamp within a valid time Need synchronized clock How to set the synchronized clock? Network delay consideration? Challenge/response Party A, (receiver), sends B a nonce (challenge) and requires the subsequent message contains it Cryptography and Network Security 344 Challenge-Response To ensure a password is never sent in the clear. Given a client and a server share a key server sends a random challenge vector client encrypts it with private key and returns this server verifies response with copy of private key can repeat protocol in other direction to authenticate server to client (2-way authentication) Secret key management physically distributed before secure communications keys are stored in a central trusted key server Cryptography and Network Security 345 Conventional Encryption App. Each user shares a secret master key with KDC (Key Distribution Center) Kerberos is an example Needham-Schroeder protocol Party A KDC Ida|Idb|Na KDCA Eka(Ks|Idb|Na|Ekb(Ks|Ida)) AB Ekb(Ks|Ida) BA Eks(Nb) AB Eks(f(Nb)) Cryptography and Network Security 346 Analysis Step 4 and 5 prevent the replay of step 3 Assume that Ks is not compromised If Ks is compromised Vulnerable to replay attack Attacker can replay step 3 Unless B remembers all previous session keys with A, it can not tell that it is a replay! Cryptography and Network Security 347 Denning Protocol Denning Protocol Party A KDC KDCA AB BA AB Ida|Idb Eka(Ks|Idb|T|Ekb(Ks|Ida|T)) Ekb(Ks|Ida|T) Eks(Nb) Eks(f(Nb)) Here T is timestamp assures the freshness of the key Ks Rely on synchronized clock Cryptography and Network Security 348 Public-key Encryption App. The simple one proposed by Denning AS: authentication server AAS Ida|Idb ASA Ekras(KUa|Ida|T)|Ekras(Kub|Idb|T) AB Ekras(KUa|Ida|T)|Ekras(Kub|Idb|T)| Ekub(Ekra(Ks|T)) It needs clock synchronization Cryptography and Network Security 349 Cont. Protocol by Woo and Lam, using nonce AKDC KDCA AB BKDC KDCB BA AB Ida|Idb EKRau(Idb|KUb) EKUb(Na|Ida) Idb|Ida|EKUau(Na) EKRau(Ida|KUa)|EKUb(EkRau(Na|Ks|Ida|Idb)) EKUa(EkRau(Na|Ks|Ida|Idb) | Nb) Eks(Nb) Cryptography and Network Security 350 One-way Authentication Using Public Key approach If confidentiality is main concern AB: EKUb(Ks) | Eks(M) If authentication is main concern AB: M|EKRa(H(M)) This can not avoid the interception and replay attack Sign the message then EKUb(M|EKRa(H(M)) ) Or EKUb(Ks) | Eks(M|EKRa(H(M)) ) Also A can sends the digital certificate EKRau(T|Ida|KUa) Cryptography and Network Security 351 Authentication Applications will consider authentication functions developed to support application-level authentication & digital signatures will consider Kerberos – a private-key authentication service then X.509 directory authentication service Cryptography and Network Security 352 Kerberos Trusted key server system developed by MIT Provides centralized third-party authentication in a distributed network access control may be provided for each computing resource in either a local or remote network (realm) A Key Distribution Centre (KDC), containing database: principles (customers and services) encryption keys KDC provides non-corruptible authentication credentials (tickets or tokens) Cryptography and Network Security 353 Kerberos Two Kerberos versions 4 : restricted to a single realm 5 : allows inter-realm authentication, in beta test Kerberos v5 is an Internet standard specified in RFC1510 To use Kerberos need to have a KDC on your network need to have Kerberised applications running on all participating systems US export restrictions Cannot be directly distributed outside US in source format Crypto libraries must be re-implemented locally Cryptography and Network Security 354 Kerberos Requirements first published report identified its requirements as: security reliability transparency scalability implemented using an authentication protocol based on Needham-Schroeder Cryptography and Network Security 355 Kerberos 4 Overview a basic third-party authentication scheme have an Authentication Server (AS) users initially negotiate with AS to identify self AS provides a non-corruptible authentication credential (ticket granting ticket TGT) have a Ticket Granting server (TGS) users subsequently request access to other services from TGS on basis of users TGT Cryptography and Network Security 356 Kerberos 4 Overview Cryptography and Network Security 357 Kerberos Realms a Kerberos environment consists of: a Kerberos server a number of clients, all registered with server application servers, sharing keys with server this is termed a realm typically a single administrative domain if have multiple realms, their Kerberos servers must share keys and trust Cryptography and Network Security 358 Kerberos Version 5 developed in mid 1990’s provides improvements over v4 addresses environmental shortcomings encryption alg, network protocol, byte order, ticket lifetime, authentication forwarding, interrealm auth and technical deficiencies double encryption, non-std mode of use, session keys, password attacks specified as Internet standard RFC 1510 Cryptography and Network Security 359 Authentication Protocols used to convince parties of each others identity and to exchange session keys may be one-way or mutual key issues are confidentiality – to protect session keys timeliness – to prevent replay attacks Cryptography and Network Security 360 Replay Attacks where a valid signed message is copied and later resent simple replay repetition that can be logged repetition that cannot be detected backward replay without modification countermeasures include use of sequence numbers (generally impractical) timestamps (needs synchronized clocks) challenge/response (using unique nonce) Cryptography and Network Security 361 Using Symmetric Encryption as discussed previously can use a two-level hierarchy of keys usually with a trusted Key Distribution Center (KDC) each party shares own master key with KDC KDC generates session keys used for connections between parties master keys used to distribute these to them Cryptography and Network Security 362 Needham-Schroeder Protocol original third-party key distribution protocol for session between A B mediated by KDC protocol overview is: 1. A→KDC: IDA || IDB || N1 2. KDC→A: EKa[Ks || IDB || N1 || EKb[Ks||IDA] ] 3. A→B: EKb[Ks||IDA] 4. B→A: EKs[N2] 5. A→B: EKs[f(N2)] Cryptography and Network Security 363 Needham-Schroeder Protocol used to securely distribute a new session key for communications between A & B but is vulnerable to a replay attack if an old session key has been compromised then message 3 can be resent convincing B that is communicating with A modifications to address this require: timestamps (Denning 81) using an extra nonce (Neuman 93) Cryptography and Network Security 364 Using Public-Key Encryption have a range of approaches based on the use of public-key encryption need to ensure have correct public keys for other parties using a central Authentication Server (AS) various protocols exist using timestamps or nonces Cryptography and Network Security 365 Denning AS Protocol Denning 81 presented the following: 1. A→AS: IDA || IDB 2. AS→A: EKRas[IDA||KUa||T] || EKRas[IDB||KUb||T] 3. A→B: EKRas[IDA||KUa||T] || EKRas[IDB||KUb||T] || EKUb[EKRas[Ks||T]] note session key is chosen by A, hence AS need not be trusted to protect it timestamps prevent replay but require synchronized clocks Cryptography and Network Security 366 One-Way Authentication required when sender & receiver are not in communications at same time (eg. email) have header in clear so can be delivered by email system may want contents of body protected & sender authenticated Cryptography and Network Security 367 Using Symmetric Encryption can refine use of KDC but can’t have final exchange of nonces, vis: 1. A→KDC: IDA || IDB || N1 2. KDC→A: EKa[Ks || IDB || N1 || EKb[Ks||IDA] ] 3. A→B: EKb[Ks||IDA] || EKs[M] does not protect against replays could rely on timestamp in message, though email delays make this problematic Cryptography and Network Security 368 Public-Key Approaches have seen some public-key approaches if confidentiality is major concern, can use: A→B: EKUb[Ks] || EKs[M] has encrypted session key, encrypted message if authentication needed use a digital signature with a digital certificate: A→B: M || EKRa[H(M)] || EKRas[T||IDA||KUa] with message, signature, certificate Cryptography and Network Security 369 Differences between Authentication and Digital Signature Two authentications: Data authentication is comparable to stamping a document in a way disallowing all future modifications to it. Data authentication is usually accompanied with data origin authentication that bounds a concrete person to this document Digital signature is a cryptographic technique that enables to protect digital information (represented as a bit-stream) from undesirable modification. Since signature cannot just be appended to a digital bitstream, more sophisticated methods (also known as signatures schemes) for signing have been elaborated. Signature scheme is a function Sig of a key pair (SA,VA) and a bitstring M, such that for anyone who knows the secret key SA, it is easy to compute for any plaintext M the signature C=Sig(PA,M). for anyone who knows VA (the public key), C and M, it is easy to verify if C=Sig(SA,M). for a randomly chosen C, it is intractable for anyone who does not know SA to find a value M for which C=Sig(SA,M). Cryptography and Network Security 370 Cryptography and Network Security Hash Algorithms Xiang-Yang Li Cryptography and Network Security 371 Hash Function Map a message to a smaller value Requirements Be applied to a block of data of any size Produced a fixed length output H(x) is easy to compute (by hardware, software) One-way: given code h, it is computationally infeasible to find x: H(x)=h Weak collision resistance: given x, computationally infeasible to find y so H(x)=H(y) Strong collision resistance: Computationally infeasible to find x, y so H(x)=H(y) Cryptography and Network Security 372 Hash Algorithms see similarities in the evolution of hash functions & block ciphers increasing power of brute-force attacks leading to evolution in algorithms from DES to AES in block ciphers from MD4 & MD5 to SHA-1 & RIPEMD-160 in hash algorithms likewise tend to use common iterative structure as do block ciphers Cryptography and Network Security 373 Basic Uses of Hash Function Six basics usages Ek(M||H(M)) Confidentiality and authentication M|| Ek(H(M)) Authentication M|| EKRa(H(M)) Authentication and digital signature Ek(M|| EKRa(H(M))) Authentication, digital signature and confidentiality M||H(M||S) Authentication (S shared by both sides) Ek(M||H(M||S)) Confidentiality and authentication Cryptography and Network Security 374 Birthday Attacks If 64-bits hash code is used On average, how many messages need to try to find one match the intercepted hash code? Birthday paradox A will sign a message appended with m-bits hash code Attacker generates some variations of fraud message, also variations of good message Find pair of message each from the two sets messages Such that they have the same hash code Give good message to A to get signature Replace good message with fraud message Cryptography and Network Security 375 Analysis Using birthday attack, given 64-bits hash code How many message variations needed so the success probability is large, say 90%? Cryptography and Network Security 376 Examples Simple hash functions XOR of the input message H(M)=X1 X2 … Xm-1 Xm But not secure Ym=H(M) Y1 Y2 … Ym-1 has same hash value as (X1X2 … Xm-1 Xm), where Yi is any value Cryptography and Network Security 377 Cont. Based on DES, block chaining technique Rabin, 1978 Divide message M into fix-sized blocks Mi Assume total n data blocks H0=initial value Hi=Emi[Hi-1] Hn is the hash value Birthday attack still applies If still 64-bits code used Cryptography and Network Security 378 More Attacks Birthday attack applied if chosen plaintext Meet in the middle attack if known plaintext Known signed hash code G Construct n-2 desired message block Qi Compute Hi=EQi[Hi-1] Generate 2m/2 random blocks X For each X, Compute Hn-1=EX[Hn-2] Generate 2m/2 random blocks Y For each Y, Compute H’n-1=DY[G] Find X, Y such that Hn-1= H’n-1 Then Q1, Q2,…Qn-2, X,Y is a fraud message Cryptography and Network Security 379 Security The size of hash code determines security 128bits is not secure Currently, most use 160 bits hash code Now recommend 256 bits Attack MAC Objective is to find valid (x, Ck(x)) pair Attack the key space: roughly 2k, k =key size Attack the MAC value Cryptography and Network Security 380 More Hash Algorithms Algorithms Message Digest:MD5 (was mostly widely used) Secure Hash Algorithm: SHA-1 (from MD4) RIPEMD-160 HMAC Cryptography and Network Security 381 MD5 designed by Ronald Rivest (the R in RSA) latest in a series of MD2, MD4 produces a 128-bit hash value until recently was the most widely used hash algorithm in recent times have both brute-force & cryptanalytic concerns specified as Internet standard RFC1321 Cryptography and Network Security 382 MD5 Overview pad message so its length is 448 mod 512 2. append a 64-bit length value to message 3. initialise 4-word (128-bit) MD buffer (A,B,C,D) 4. process message in 16-word (512-bit) blocks: 1. using 4 rounds of 16 bit operations on message block & buffer add output to buffer input to form new buffer value 5. output hash value is the final buffer value Cryptography and Network Security 383 MD5 Overview Cryptography and Network Security 384 MD5 Compression Function each round has 16 steps of the form: a = b+((a+g(b,c,d)+X[k]+T[i])<<<s) a,b,c,d refer to the 4 words of the buffer, but used in varying permutations note this updates 1 word only of the buffer after 16 steps each word is updated 4 times where g(b,c,d) is a different nonlinear function in each round (F,G,H,I) T[i] is a constant value derived from sin Cryptography and Network Security 385 MD5 Compression Function Cryptography and Network Security 386 MD4 precursor to MD5 also produces a 128-bit hash of message has 3 rounds of 16 steps vs 4 in MD5 design goals: collision resistant (hard to find collisions) direct security (no dependence on "hard" problems) fast, simple, compact favours little-endian systems (eg PCs) Cryptography and Network Security 387 Strength of MD5 MD5 hash is dependent on all message bits Rivest claims security is good as can be known attacks are: Berson 92 attacked any 1 round using differential cryptanalysis (but can’t extend) Boer & Bosselaers 93 found a pseudo collision (again unable to extend) Dobbertin 96 created collisions on MD compression function (but initial constants prevent exploit) conclusion is that MD5 looks vulnerable soon Cryptography and Network Security 388 Bad news Chinese authors (Wang, Feng, Lai, and Yu) reported a family of collisions in MD5 (fixing the previous bug in their analysis), and also reported that their method can efficiently (2^40 hash steps) find a collision in SHA-0. August Crypto 2004, MD5 is fatally wounded; its use will be phased out. SHA-1 is still alive but the vultures are circling. A gradual transition away from SHA-1 will now start. The first stage will be a debate about alternatives, leading to a consensus among practicing cryptographers about what the substitute will be. Cryptography and Network Security 389 Why collisions are bad An example of what you might do with this. You could request an SSL certificate (for your real identity) from a certificate authority. After the response comes back, you can then use that response (which is based on the MD5 of your identity+key) to "authenticate" a carefully chosen different certificate, one which claims that you are LargeBankOrSoftwareCorp., but which has the same MD5 as your real identity. You can then present this to other people in order to convince them that you are someone whom you are not. Another example, core internet routers use md5 to exchange passwords. I simply sniff the md5sum, and if I can find a string that generates the same sum, easily, I can send my own routing update that takes down the internet. More examples, since a LOT of applications use md5, but you get the idea. Cryptography and Network Security 390 Further detail Obviously the above attack isn't quite so simple, but this research makes it *possible*. Before, it was believed to be sufficiently difficult to find a collision, that nobody worried about it. Now they are saying its feasible to do it in hours. The question hanging around right now is that these researchers managed to find collisions easily, but not for an artbitrary string. The questions is how long before someone modifies this method to find any colllision. That is how much time the world has to move away. More at http://www.freedom-to-tinker.com/archives/000664.html Cryptography and Network Security 391 What to do next The U.S. National Institute of Standards and Technology is having a competition for a new cryptographic hash function. The phrase "one-way hash function" might sound arcane and geeky, but hash functions are the workhorses of modern cryptography. Submissions will be due in fall 2008, and a single standard is scheduled to be chosen by the end of 2011. we have an interim solution in SHA-256. Cryptography and Network Security 392 Secure Hash Algorithm (SHA-1) SHA was designed by NIST & NSA in 1993, revised 1995 as SHA-1 US standard for use with DSA signature scheme standard is FIPS 180-1 1995, also Internet RFC3174 nb. the algorithm is SHA, the standard is SHS produces 160-bit hash values now the generally preferred hash algorithm based on design of MD4 with key differences Cryptography and Network Security 393 SHA Overview pad message so its length is 448 mod 512 2. append a 64-bit length value to message 3. initialise 5-word (160-bit) buffer (A,B,C,D,E) to 1. (67452301,efcdab89,98badcfe,10325476,c3d2e1f0) 4. process message in 16-word (512-bit) chunks: expand 16 words into 80 words by mixing & shifting use 4 rounds of 20 bit operations on message block & buffer add output to input to form new buffer value 5. output hash value is the final buffer value Cryptography and Network Security 394 SHA-1 Compression Function each round has 20 steps which replaces the 5 buffer words thus: (A,B,C,D,E) <(E+f(t,B,C,D)+(A<<5)+Wt+Kt),A,(B<<30),C,D) a,b,c,d refer to the 4 words of the buffer t is the step number is nonlinear function for round Wt is derived from the message block Kt is a constant value derived from sin f(t,B,C,D) Cryptography and Network Security 395 SHA-1 Compression Function Cryptography and Network Security 396 SHA-1 verses MD5 brute force attack is harder (160 vs 128 bits for MD5) not vulnerable to any known attacks (compared to MD4/5) a little slower than MD5 (80 vs 64 steps) both designed as simple and compact optimised for big endian CPU's (vs MD5 which is optimised for little endian CPU’s) Cryptography and Network Security 397 Revised Secure Hash Standard NIST have issued a revision FIPS 180-2 adds 3 additional hash algorithms SHA-256, SHA-384, SHA-512 designed for compatibility with increased security provided by the AES cipher structure & detail is similar to SHA-1 hence analysis should be similar Cryptography and Network Security 398 RIPEMD-160 RIPEMD-160 was developed in Europe as part of RIPE project in 96 by researchers involved in attacks on MD4/5 initial proposal strengthen following analysis to become RIPEMD-160 somewhat similar to MD5/SHA uses 2 parallel lines of 5 rounds of 16 steps creates a 160-bit hash value slower, but probably more secure, than SHA Cryptography and Network Security 399 RIPEMD-160 Overview 1. 2. 3. pad message so its length is 448 mod 512 append a 64-bit length value to message initialise 5-word (160-bit) buffer (A,B,C,D,E) to (67452301,efcdab89,98badcfe,10325476,c3d2e1f0) 4. process message in 16-word (512-bit) chunks: 5. use 10 rounds of 16 bit operations on message block & buffer – in 2 parallel lines of 5 add output to input to form new buffer value output hash value is the final buffer value Cryptography and Network Security 400 RIPEMD-160 Round Cryptography and Network Security 401 RIPEMD-160 Compression Function Cryptography and Network Security 402 RIPEMD-160 Design Criteria use 2 parallel lines of 5 rounds for increased complexity for simplicity the 2 lines are very similar step operation very close to MD5 permutation varies parts of message used circular shifts designed for best results Cryptography and Network Security 403 RIPEMD-160 verses MD5 & SHA-1 brute force attack harder (160 like SHA-1 vs 128 bits for MD5) not vulnerable to known attacks, like SHA-1 though stronger (compared to MD4/5) slower than MD5 (more steps) all designed as simple and compact SHA-1 optimised for big endian CPU's vs RIPEMD-160 & MD5 optimised for little endian CPU’s Cryptography and Network Security 404 Keyed Hash Functions as MACs have desire to create a MAC using a hash function rather than a block cipher because hash functions are generally faster not limited by export controls unlike block ciphers hash includes a key along with the message original proposal: KeyedHash = Hash(Key|Message) some weaknesses were found with this eventually led to development of HMAC Cryptography and Network Security 405 HMAC specified as Internet standard RFC2104 uses hash function on the message: HMACK = Hash[(K+ XOR opad) || Hash[(K+ XOR ipad)||M)]] where K+ is the key padded out to size and opad, ipad are specified padding constants overhead is just 3 more hash calculations than the message needs alone any of MD5, SHA-1, RIPEMD-160 can be used Cryptography and Network Security 406 HMAC Overview Cryptography and Network Security 407 HMAC Security know that the security of HMAC relates to that of the underlying hash algorithm attacking HMAC requires either: brute force attack on key used birthday attack (but since keyed would need to observe a very large number of messages) choose hash function used based on speed verses security constraints Cryptography and Network Security 408 Summary have considered: some current hash algorithms: MD5, SHA-1, RIPEMD160 HMAC authentication using hash function Cryptography and Network Security 409 Cryptography and Network Security Digital Signature Xiang-Yang Li Cryptography and Network Security 410 Digital Signature Cryptography and Network Security 411 Digital Signatures have looked at message authentication but does not address issues of lack of trust digital signatures provide the ability to: verify author, date & time of signature authenticate message contents be verified by third parties to resolve disputes hence include authentication function with additional capabilities Cryptography and Network Security 412 Digital Signature Properties must depend on the message signed must use information unique to sender to prevent both forgery and denial must be relatively easy to produce must be relatively easy to recognize & verify be computationally infeasible to forge with new message for existing digital signature with fraudulent digital signature for given message be practical save digital signature in storage Cryptography and Network Security 413 Securities A total break results in the recovery of the signing key. A universal forgery attack results in the ability to forge signatures for any message. A selective forgery attack results in a signature on a message of the adversary's choice. An existential forgery merely results in some valid message/signature pair not already known to the adversary. Cryptography and Network Security 414 Classification of Digital Signature Undeniable Fail-Stop Blind One-time Multi-party (group signature) (n,k)-multi-party Oblivious Multi-undeniable Cryptography and Network Security 415 Algorithm and legal concerns several prior requirements quality algorithms. Some public key algorithms are known to be insecure, practicable attacks against them having been identified. quality implementations. An implementation of a good algorithm with mistake(s) will not work. (about 1 defect per 1,000 lines). the private key must remain actually secret; if it becomes known to some other party, that party can produce perfect digital signatures of anything whatsoever. distribution of public keys must be done in such a way that the public key claimed to belong to Bob actually belongs to Bob, and vice versa. This is commonly done using a public key infrastructure and the public key user association is attested by the operator of the PKI (called a certificate authority). For 'open' PKIs in which anyone can request such an attestation, the possibility of mistake is non trivial. users (and their software) must carry out the signature protocol properly. Legal concerns Cryptography and Network Security 416 Direct Digital Signatures involve only sender & receiver assumed receiver has sender’s public-key digital signature made by sender signing entire message or hash with private-key can encrypt using receivers public-key important that sign first then encrypt message & signature security depends on sender’s private-key Cryptography and Network Security 417 Arbitrated Digital Signatures involves use of arbiter A validates any signed message then dated and sent to recipient requires suitable level of trust in arbiter can be implemented with either private or public-key algorithms arbiter may or may not see message Cryptography and Network Security 418 RSA signature N=p q,where p and q are large primes Alice’s private key (e,n), Alice’s public key (d,n) Signature of message m by Alice S=H(m)e mod n Verification of signature by Bob Check if h(m) = Sd mod n Cryptography and Network Security 419 From wikipedia Cryptography and Network Security 420 Cont. Typically d is chosen small (3 or 216+1) Problem: Easy to create the signature of h(m1)h(m2) RSA-PSS Use some more randomization to enhance security It was added in version 2.1 of PKCS #1 (see RFC 3447). Cryptography and Network Security 421 ElGamal Signature Global public components Prime number p with 512-1024 bits Primitive element g in Zp Users private key Random integer x less than p Users public key Integer y=gx mod p Cryptography and Network Security 422 Elgamal Signature For each message M, generates random k Computes r=gk mod p Computes s=k-1(H(M)-xr) mod (p-1) Signature is (r,s) Verifying Computes v1=gH(M) mod p Computes v2=yrrs mod p Test if v1= v2 Cryptography and Network Security 423 Proof of Correctness Computes v2=yrrs mod q So v2=yrrs mod q =gxr gks mod p -1 = gxr+k k (H(M)-xr) mod (p-1) mod p =gH(M) mod p=v1 Notice that here it uses Fermat theorem to show That g (H(M)-xr) mod (p-1) mod p = g (H(M)-xr) mod p Cryptography and Network Security 424 Cont. The main disadvantage of ElGamal is the need for randomness (sometimes it is good), and its slower speed (especially for signing). Another potential disadvantage of the ElGamal system is that message expansion by a factor of two takes place during encryption. However, such message expansion is negligible if the cryptosystem is used only for exchange of secret keys. Cryptography and Network Security 425 Digital Signature Standard FIPS PUB 186 by NIST, 1991 Final announcement 1994 It uses Secure Hashing Algorithm (SHA) for hashing Digital Signature Algorithm (DSA) for signature The hash code is set as input of DSA The signature consists of two numbers DSA Based on the difficulty of discrete logarithm Based on Elgamal and Schnorr system Cryptography and Network Security 426 DSA Global public components Prime number p with 512-1024 bits Prime divisor q of (p-1) with 160 bits Integer g=h(p-1)/q mod p Users private key Random integer x less than q Users public key Integer y=gx mod p Cryptography and Network Security 427 DSA Signature For each message M, generates random k Computes r=(gk mod p) mod q Computes s=k-1(H(M)+xr) mod q Signature is (r,s) Verifying Computes w=s-1 mod q, u1=H(M)w mod q Computes u2=rw mod q,v=(gu1yu2 mod p) mod q Test if v=r Cryptography and Network Security 428 Proof of Correctness Notice that v=(gu1yu2 mod p) mod q =(gH(M)w mod q yrw mod q mod p) mod q =(gH(M)w mod q gxrw mod q mod p) mod q =(gH(M)w +xrw mod q mod p) mod q =(g(H(M)+xr)w mod q mod p) mod q -1 =(g(H(M)+xr)k(H(M)+xr) mod q mod p) mod q =(gk mod p) mod q =r Cryptography and Network Security 429 In practice (Sun Java Library) g = F7E1A085D69B3DDE CBBCAB5C36B857B9 7994AFBBFA3AEA82 F9574C0B3D078267 5159578EBAD4594F E67107108180B449 167123E84C281613 B7CF09328CC8A6E1 3C167A8B547C8D28 E0A3AE1E2BB3A675 916EA37F0BFA2135 62F1FB627A01243B CCA4F1BEA8519089 A883DFE15AE59F06 928B665E807B5525 64014C3BFECF492A p = FD7F53811D751229 52DF4A9C2EECE4E7 F611B7523CEF4400 C31E3F80B6512669 455D402251FB593D 8D58FABFC5F5BA30 F6CB9B556CD7813B 801D346FF26660B7 6B9950A5A49F9FE8 047B1022C24FBBA9 D7FEB7C61BF83B57 E7C6A8A6150F04FB 83F6D3C51EC30235 54135A169132F675 F3AE2B61D72AEFF2 2203199DD14801C7 q = 9760508F15230BCC B292B982A2EB840B F0581CF5 Here g and p have 1024 bits, while q has 160 bits. They fulfill the requirement that gq = 1 mod p, Cryptography and Network Security 430 Note Can we use the random number k twice? What will happen if k used twice? We have r=(gk mod p) mod q s1=k-1(H(M1)+xr) mod q and s2=k-1(H(M2)+xr) mod q We have s1 - s2 =k-1(H(M1)-H(M2)) mod q Another attack (for OpenPGP) Replace p and g http://www.tigertools.net/board/?topic=topic4&msg=14 http://www.orlingrabbe.com/DSAflaw_OpenPGP.htm Cryptography and Network Security 431 Cont. We cannot use small k Cryptography and Network Security 432 Non-deterministic Non-determined signatures For each message, many valid signatures exist DSA, Elgamal Deterministic signatures For each message, one valid signature exists RSA Cryptography and Network Security 433 Comparisons Speed DSS has faster signing than verifying RSA could have faster verifying than signing Message be signed once, but verified many times This prefers the faster verification But the signer may have limited computing power Example: smart card This prefers the faster siging Cryptography and Network Security 434 Blind Signature (digital cash) first introduced by Chaum, allow a person to get a message signed by another party without revealing any information about the message to the other party. Suppose Alice has a message m that she wishes to have signed by Bob, and she does not want Bob to learn anything about m. Let (n,e) be Bob's public key and (n,d) be his private key. e Alice generates a random value r such that gcd(r, n) = 1 and sends x = (r m) mod n to Bob. The value x is ``blinded'' by the random value r; hence Bob can derive no useful information from it. d Bob returns the signed value t = x mod n to Alice. d e d d Since x (r m) r m mod n, Alice can obtain the true signature s of m by computing -1 s = r t mod n. Cryptography and Network Security 435 Security Concerns GnuPG permits creating ElGamal keys are usable for both encryption and signing. It is even possible to have one key (the primary one) used for both operations. This is not considered good cryptographic practice, but is permitted by the OpenPGP standard. signature is much larger than a RSA or DSA signature verification and creation takes far longer and the use of ElGamal for signing has always been problematic due to a couple of cryptographic weaknesses when not done properly. Cryptography and Network Security 436 Applications of Blind Signature In an online context the blind signature works as follows. Voters encrypt their ballot with a secret key and then blinds it. Then the voter signs the encrypted vote and sends it to the validator. The validator checks to see if the signature is valid (the signature acts as a I.D. tag and will have to be registered with the voter before the voting process has started) and if it is the validator signs it and returns it to the voter. The voter removes the blinding encryption layer, which then leaves behind an encrypted ballot with the validator's signature. Cryptography and Network Security 437 Cont. This is then sent to the tallier who checks to make sure the validator's signature is present on the votes. He then waits until all votes haven been collected and then publishes all the encrypted votes so that the voters can verify their votes have been received. The voters then send their keys to the tallier to decrypt their ballots. Once the vote has been counted the tallier publishes the encrypted votes and the decryption keys so that voters can then verify the results. Next we illustrate the transfer of ballots between the various parties. Cryptography and Network Security 438 Cont. Cryptography and Network Security 439 Cont, This protocol has been implemented used in reality and has been found that the entire voting process can be completed in a matter of minutes despite the complex nature of the voting procedure. Most of the tasks can be automated with the only user interaction needed being the actual vote casting. Encryption, blinding and all the verification needed can be performed by software in the background. Of course we'd have to trust this software to handle the voting procedures correctly and accurately and to assume it has not been compromised in some way. Cryptography and Network Security 440 Cryptography and Network Security Certificate Xiang-Yang Li Cryptography and Network Security 441 Certificate A public-key certificate is a digitally signed statement from one entity, saying that the public key (and some other information) of another entity has some specific value. Cryptography and Network Security 442 More terms Digitally Signed Identity If some data is digitally signed it has been stored with the "identity" of an entity, and a signature that proves that entity knows about the data. The data is rendered unforgeable by signing with the entitys' private key. A known way of addressing an entity. In some systems the identity is the public key, in others it can be anything from a Unix UID to an Email address to an X.509 Distinguished Name. Entity An entity is a person, organization, program, computer, business, bank, or something else you are trusting to some degree. Cryptography and Network Security 443 More about CA Why need it In a large-scale networked environment it is impossible to guarantee that prior relationships between communicating entities have been established or that a trusted repository exists with all used public keys. Certificates were invented as a solution to this public key distribution problem. Now a Certification Authority (CA) can act as a Trusted Third Party. CAs are entities (e.g., businesses) that are trusted to sign (issue) certificates for other entities. It is assumed that CAs will only create valid and reliable certificates as they are bound by legal agreements. There are many public Certification Authorities, such as VeriSign, Thawte, Entrust, and so on. You can also run your own Certification Authority using products such as the Netscape/Microsoft Certificate Servers or the Entrust CA product for your organization. Cryptography and Network Security 444 Who uses Certificate? Probably the most widely visible application of X.509 certificates today is in web browsers (such as Netscape Navigator and Microsoft Internet Explorer) that support the SSL protocol. SSL (Secure Socket Layer) is a security protocol that provides privacy and authentication for your network traffic. These browsers can only use this protocol with web servers that support SSL. Other technologies that rely on X.509 certificates include: Various code-signing schemes, such as signed Java Archives, and Microsoft Authenticode. Various secure E-Mail standards, such as PEM and S/MIME. E-Commerce protocols, such as SET. Cryptography and Network Security 445 How to create certificate? There are two basic techniques used to get certificates: you can create one yourself (using the right tools, such as keytool) Not everyone will accept self-signed certificates, you can ask a Certification Authority to issue you one (either directly or using a tool such as keytool to generate the request). The main inputs to the certificate creation are: Matched public and private keys, generated using some special tools (such as keytool), or a browser. information about the entity being certified (e.g., you). This normally includes information such as your name and organizational address. If you ask a CA to issue a certificate for you, you will normally need to provide proof to show correctness of the information. Cryptography and Network Security 446 business Many companies sale the service of creating the certificate (such as SSL certificate) Comodo Verisign Thawte Entrust Geotrust Cryptography and Network Security 447 X.509 Authentication Service Public key certificate associated with user The certificates are created by Trusted Authority Then placed in the directory by TA or user Itself is not responsible for creating certificate It includes Version, serial number, signature algorithm identifier, Issuer name, issuer identifier, validity period, the user, user identifier, user’s public key, extensions, signature by TA The signature by TA guarantees the authority Certificates can be used to certify other TAs Y<<X>>: certificate of user X issued by TA Y Cryptography and Network Security 448 What is inside X.509 certificate? Version Thus far, three versions are defined. Serial Number distinguish it from other certificates it issues. This information is used in numerous ways, for example when a certificate is revoked its serial number is placed in a Certificate Revocation List (CRL). Signature Algorithm Identifier This identifies the algorithm used by the CA to sign the certificate. Issuer Name The X.500 name of the entity that signed the certificate. This is normally a CA. Using this certificate implies trusting the entity that signed this certificate. root or top-level CA certificates, the issuer signs its own certificate. Cryptography and Network Security 449 cont Validity Period This period is described by a start date and time and an end date and time, and can be as short as a few seconds or almost as long as a century. It depends on a number of factors, such as the strength of the private key used to sign the certificate or the amount one is willing to pay for a certificate. This is the expected period that entities can rely on the public value, if the associated private key has not been compromised. Subject Name The name of the entity whose public key the certificate identifies. This name uses the X.500 standard, so it is intended to be unique across the Internet. Subject Public Key Information together with an algorithm identifier Cryptography and Network Security 450 Certificate Revocation Need the private key together with the certificate to revoke it The revocation is recorded at the directory Each time a certificate is arrived, check the directory to see if it is revoked Cryptography and Network Security 451 X.509 Authentication Service part of CCITT X.500 directory service standards distributed servers maintaining some info database defines framework for authentication services directory may store public-key certificates with public key of user signed by certification authority also defines authentication protocols uses public-key crypto & digital signatures algorithms not standardised, but RSA recommended Cryptography and Network Security 452 X.509 Certificates issued by a Certification Authority (CA), containing: version (1, 2, or 3) serial number (unique within CA) identifying certificate signature algorithm identifier issuer X.500 name (CA) period of validity (from - to dates) subject X.500 name (name of owner) subject public-key info (algorithm, parameters, key) issuer unique identifier (v2+) subject unique identifier (v2+) extension fields (v3) signature (of hash of all fields in certificate) notation CA<<A>> denotes certificate for A signed by CA Cryptography and Network Security 453 X.509 Certificates Cryptography and Network Security 454 Obtaining a Certificate any user with access to CA can get any certificate from it only the CA can modify a certificate because cannot be forged, certificates can be placed in a public directory Cryptography and Network Security 455 CA Hierarchy if both users share a common CA then they are assumed to know its public key otherwise CA's must form a hierarchy use certificates linking members of hierarchy to validate other CA's each CA has certificates for clients (forward) and parent (backward) each client trusts parents certificates enable verification of any certificate from one CA by users of all other CAs in hierarchy Cryptography and Network Security 456 CA Hierarchy Use Cryptography and Network Security 457 Certificate Revocation certificates have a period of validity may need to revoke before expiry, eg: 1. 2. 3. CA’s maintain list of revoked certificates user's private key is compromised user is no longer certified by this CA CA's certificate is compromised the Certificate Revocation List (CRL) users should check certs with CA’s CRL Cryptography and Network Security 458 Authentication Procedures X.509 includes three alternative authentication procedures: One-Way Authentication Two-Way Authentication Three-Way Authentication all use public-key signatures Cryptography and Network Security 459 One-Way Authentication 1 message ( A->B) used to establish the identity of A and that message is from A message was intended for B integrity & originality of message message must include timestamp, nonce, B's identity and is signed by A Cryptography and Network Security 460 Two-Way Authentication 2 messages (A->B, B->A) which also establishes in addition: the identity of B and that reply is from B that reply is intended for A integrity & originality of reply reply includes original nonce from A, also timestamp and nonce from B Cryptography and Network Security 461 Three-Way Authentication 3 messages (A->B, B->A, A->B) which enables above authentication without synchronized clocks has reply from A back to B containing signed copy of nonce from B means that timestamps need not be checked or relied upon Cryptography and Network Security 462 X.509 Version 3 has been recognised that additional information is needed in a certificate email/URL, policy details, usage constraints rather than explicitly naming new fields defined a general extension method extensions consist of: extension identifier criticality indicator extension value Cryptography and Network Security 463 Certificate Extensions key and policy information convey info about subject & issuer keys, plus indicators of certificate policy certificate subject and issuer attributes support alternative names, in alternative formats for certificate subject and/or issuer certificate path constraints allow constraints on use of certificates by other CA’s Cryptography and Network Security 464 Cryptography and Network Security Identification Xiang-Yang Li Cryptography and Network Security 465 Identification Identification: user authentication convince system of your identity before it can act on your behalf sometimes also require that the computer verify its identity with the user Based on three methods what you know what you have what you are Verification Validation of information supplied against a table of possible values based on users claimed identity Cryptography and Network Security 466 What you Know Passwords or Pass-phrases prompt user for a login name and password verify identity by checking that password is correct on some (older) systems, password was stored clear more often use a one-way function, whose output cannot easily be used to find the input value either takes a fixed sized input (eg 8 chars) or based on a hash function to accept a variable sized input to create the value important that passwords are selected with care to reduce risk of exhaustive search Cryptography and Network Security 467 Weakness Traditional password scheme is vulnerable to eavesdropping over an insecure network Cryptography and Network Security 468 Solutions? One-time password these are passwords used once only future values cannot be predicted from older values Password generation either generate a printed list, and keep matching list on system to be accessed or use an algorithm based on a one-way function f (eg MD5) to generate previous values in series (eg SKey) start with a secret password s, and number N , p0 = fN(s) ith password in series is pi = fN-i(s) must reset password after N uses Cryptography and Network Security 469 What you Have Magnetic Card, Magnetic Key possess item with required code value encoded Smart Card or Calculator may interact with system may require information from user could be used to actively calculate: a time dependent password a one-shot password a challenge-response verification public-key based verification Cryptography and Network Security 470 What you Are Verify identity based on your physical characteristics, known as biometrics Characteristics used include: Signature (usually dynamic) Fingerprint, hand geometry face or body profile Speech, retina pattern Tradeoff between false rejection (type I error) false acceptance (type II error) Cryptography and Network Security 471 Cryptography and Network Security Secret Sharing Xiang-Yang Li Cryptography and Network Security 472 Threshold Scheme A (t,w)-threshold scheme Sharing key K among a set of w users Any t users can recover the key Any t-1 users can not do so Schemes Shamir’s scheme Geometric techniques Matroid theory Cryptography and Network Security 473 Information Theory The secret sharing is as large as the original secret This result is based in information theory, but can be understood intuitively. Given t-1 shares, no information whatsoever can be determined about the secret. Thus, the final share must contain as much information as the secret itself. All secret sharing schemes use random bits. To distribute a one-bit secret among threshold t people, t-1 random bits are necessary. The final share contains as much information as the secret, but the other t-1 shares still provide relevant information individually. This information cannot be the secret, so it must be random. Cryptography and Network Security 474 Shamir’s Scheme Initialization phase Dealer chooses a large prime number p Dealer chooses w distinct xi from Zp Gives value xi to person pi Share distribution of key k from Zp Dealer choose t-1 random number ai Dealer computes yi=f(xi) Here f(x)=k+ajxj mod p Dealer gives share yi to person pi Cryptography and Network Security 475 Geometry View Cryptography and Network Security 476 Simple (t,t) Sharing Procedure D secretly chooses t-1 random elements yi from Zn D computes Value yt=K- yj mod n D distributes yi to person pi for all i It is secure and easy Number n can be any number Easy to recover the key Only t persons together can do so, assume yi random Cryptography and Network Security 477 Blakley's Scheme Secret is a point in an t-dimensional space Dealer gives each user a hyper-plane passing the secret point Any t users can recover the common point Cryptography and Network Security 478 Geometry View Cryptography and Network Security 479 Avoid Cheating Two major distinct weaknesses Bogus values are undetectable. Participants need not reveal their true share. Even if a bogus value was detected, it would not necessarily give any information about the true value One participant did not reveal its true value after get the true values from other one Cryptography and Network Security 480 Ben-Or/Rabin Solution Using Checking Vectors For any two participants A and B Dealer gives A (SA, YAB) Dealer gives B (BAB, CAB) Here CAB = BAB YAB+ SA mod p SA is the secret share of A A and B keep their values secret B can use (BAB, CAB) to verify the value (SA, YAB) of A Cryptography and Network Security 481 Avoid Cheating Participant B can send A bogus value after receive A’s value Solution: bit transfer Dealer gives A (SAi, YABi) Dealer gives B (BABi, CABi) Here CABi = BABi YABi+ SAi mod p SAi is the ith bit of the secret share of A Cryptography and Network Security 482 Cont. Protocol Participant A gives its value (SAi, YABi) to B B verifies: CABi = BABi YABi+ SAi mod p B then sends its value (SBi, YBAi) to A A verifies: CBAi = BBAi YBAi+ SBi mod p The protocol terminates whenever One side detects cheating, or All values transferred Cryptography and Network Security 483 Chinese Remainder Theorem Given a number m<n, and n=n1n2…nk, Numbers ni and nj are coprimes Let ai=m mod ni Number n is public Dealer delivers ai and ni to the ith participant Then all k users can recover the number m Why it is not a good secret sharing scheme? Is it computationally for any k-1 users to recover the key if n is large? Cryptography and Network Security 484 Recover method Each user pre-computes Ni=n/ni Inverse of Ni: yi=Ni mod ni Compute the product si=aiNiyi mod n Recover the secret m Each user submits si Computes s1+s2+….+sk mod n Cryptography and Network Security 485 Access Structure Threshold scheme allows any t users to recover key! Access structure allows some subsets to recover the key! Example: {{p1,p2,p4},{p1,p3,p4},{p2,p3}} among p1,p2,p3,p4,p5 able to recover the key Assume the accessing subset is minimized No subset of any accessing subset is able to recover Cryptography and Network Security 486 Monotone Circuit Assign sharing for each accessing subset p1 a1 p2 p3 p4 c1 k-c 1 a2 b1 k-a1-a2 b2 k k-b1-b2 k k k Cryptography and Network Security 487 Cont. Distribution (a1,b1) to p1 (a2,c1) to p2 (k-c1,b2) to p3 (k-a1-a2,k-b1-b2) to p4 The sharer needs know The circuit used by dealer Which shares corresponding to which wires The shared value is secret Cryptography and Network Security 488 Visual Secret Sharing There is a secret picture to be shared among n participants. The picture is divided into n transparencies (shares) such that if any m transparencies are placed together, the picture becomes visible but if fewer than m transparencies are placed together, nothing can be seen. Cryptography and Network Security 489 Visual Secret Sharing Such a scheme is constructed by viewing the secret picture as a set of black and white pixels and handling each pixel separately. The schemes are perfectly secure and easily implemented without any cryptographic computation. A further improvement allows each transparency (share) to be an innocent picture For example, a picture of a landscape or a picture of a building thus concealing the fact of secret sharing Cryptography and Network Security 490 Interactive Proof Interactive proof is a protocol between two parties in which one party, called the prover, tries to prove a certain fact to the other party, called the verifier Often takes the form of a challengeresponse protocol Cryptography and Network Security 491 cont protocol in which one or more provers try to convince another party, called the verifier, that the prover(s) possess certain true knowledge, such as the membership of a string x in a given language, often with the goal of revealing no further details about this knowledge. The prover(s) and verifier are formally defined as probabilistic Turing machines with special "interaction tapes" for exchanging messages. Cryptography and Network Security 492 Desired Properties Desired properties of interactive proofs Completeness: The verifier always accepts the proof if the prover knows the fact and both the prover and the verifier follow the protocol. Soundness: Verifier always rejects the proof if prover doesnot know the fact, and verifier follows protocol. Zero knowledge: The verifier learns nothing about the fact being proved (except that it is correct) from the prover that he could not already learn without the prover. In a zero-knowledge proof, the verifier cannot even later prove the fact to anyone else. Cryptography and Network Security 493 Typical Protocol A typical round in a zero-knowledge proof consists of a "commitment" message from the prover, followed by a challenge from the verifier, and then a response to the challenge from the prover. The protocol may be repeated for many rounds. Based on the prover's responses in all the rounds, the verifier decides whether to accept or reject the proof. Cryptography and Network Security 494 An example Ali Baba’s Cave Cryptography and Network Security 495 Cont. Alice wants to prove to Bob that she knows the secret words to open the portal at CD but does not wish to reveal the secret to Bob. In this scenario, Alice’s commitment is to go to C or D. Cryptography and Network Security 496 Proof Protocol A typical round in the proof proceeds as follows: Bob goes to A, waits there while Alice goes to C or D. Bob then asks Alice to appear from either the right side or the left side of the tunnel. If Alice does not know the secret words there is only a 50 percent chance that she will come out from the right tunnel. Bob will repeat this round as many times as he desires until he is certain that Alice knows the secret words. No matter how many times that the proof repeats, Bob does not learn the secret words. Cryptography and Network Security 497 Graph Isomorphism Problem Instance Two graphs G1=(V1,E1) and G2=(V2,E2) Question Is there a bijection f from V1 to V2, so (u,v)E1 implies that (f(u),f(v))E2 If such bijection exists, then graphs G1 and G2 are said to be isomorphic If such bijection does not exist, then graphs G1 and G2 are said to be non-isomorphic Cryptography and Network Security 498 Graph Non-isomorphism Input: graphs G1 and G2 over {1,2,…n} Prover want to prove G1 and G2 are not isomophic Assumption Prover has unbounded computational power Verifier has limited computational power Cryptography and Network Security 499 Proof Protocol Protocol (repeated for n rounds) Verifier Randomly chooses i=1 or 2 Selects a random permutation f and compute H to be the image of Gi under f, sends H to prover Prover Determines the value j such that Gj is isomorphic to H Sends j to verifier Verifier checks if j=i If equal for n rounds, then accepts the proof Cryptography and Network Security 500 Correctness and Soundness Correctness If G1 and G2 are not isomorphic, then for any round, there is only one graph of G1, G2 that could produce H under a permutation f So if the verifier knows non-isomorphism, then each round a correct j will be computed Soundness If the verifier does not know (G1 and G2 are isomorphic), then each round two answers possible, and it has half chance to get the correct i chosen by the prover. Cryptography and Network Security 501 Graph Isomorphism Input: graphs G1 and G2 over {1,2,…n} Prover want to prove G1 and G2 are isomophic Assumption Prover has unbounded computational power Verifier has limited computational power Cryptography and Network Security 502 Proof Protocol Protocol (repeated for n rounds) Prover Selects a random permutation f and compute H to be the image of G1 under f, sends H to prover Verifier Randomly chooses i=1 or 2, sends it to prover Prover Computes the permutation g such that H is the image of Gj under g, and sends g to verifier Verifier checks if H is the image of Gj under g If yes for n rounds, then accepts the proof Cryptography and Network Security 503 Correctness and Soundness Correctness If G1 and G2 are isomorphic, and the verifier knows how to find the permutation between G1 and G2, then each round a correct g will be computed Soundness If the verifier does not know (G1 and G2 are nonisomorphic or the permutation between G1 and G2), then each round prover can deceive the verifier is to guess the value i chosen by the verifier Cryptography and Network Security 504 Perfect Zero-Knowledge The graph isomorphism proof is ZKP All information seen by the verifier is the same as generated by a random simulator Define transcript of the proof as t=(G1,G2,(H1,i,g1),(H2,i,g2),….(Hn,i,gn)) Anyone can generate the transcript without knowing which permutation carries G1 to G2 Hence the verifier gains nothing by knowing the transcript (I.e., the proof history) Cryptography and Network Security 505 ZKP for Verifier Perfect Zero-knowledge for verifier Suppose we have a poly-time interactive proof system and a poly-time simulator S. Let T be all yes-instance transcripts and let F be all transcripts generated by S. For any transcript t if Pr(t occurs in T)=Pr(t occurs in F) We say the interactive proof system are perfect zeroknowledge for the verifier Cryptography and Network Security 506 Isomorphism Proof: ZKP-verifier Graph isomorphism is a perfect zero- knowledge for verifier A triple (H,i,g). There are 2n! valid triples. All triples (H,i,g) occurs equiprobable in some transcript Here, assume that both the verifier and the prover are honest Both of them randomly chooses parameters that supposed to be chosen randomly Cryptography and Network Security 507 Cheating Verifier What happened if verifier does not follow the protocol (does not choose i randomly) Transcript produced by ZKP is not same as that produced by the random simulator anymore The verifier may gain some information due to this imbalance But, there is another expected poly-time simulator to generate the same transcript Hence, the verifier still gains nothing Cryptography and Network Security 508 Perfect Zero-Knowledge Definition Suppose we have a poly-time interactive proof system, a poly-time algorithm V to generate random numbers by verifier, and a poly-time simulator S. Let T be all yes-instance transcripts (depending on V) and let F be all transcripts generated by S and V. For any transcript t if Pr(t occurs in T)=Pr(t occurs in F) We say the interactive proof system are perfect zeroknowledge Cryptography and Network Security 509 Forging Simulator Initial transcript t=(G1,G2), repeat n rounds Let old-state=state(V), repeat follows Chooses ij from {1,2} randomly Chooses gj to be a random permutation over {1,...n} Compute Hj to be the image of Gi under g Call V with input Hj, obtaining a challenge ij’ If ij=ij’, then concatenate (Hj, ij, gj) onto the end of t Else reset V by state(V)=old-state Until ij=ij’ Cryptography and Network Security 510 Perfect Zero-knowledge The graph isomorphism is perfect ZKP The expected running time of simulator is 2n For the kth round of the interactive proof system Let pk be the probability that verifier chooses i=1 Then (H,1,g) occurs in actual transcript with pk/n!, (H,2,g) occurs in actual transcript with (1-pk)/n! For simulator, when it terminates the simulation for the kth round, same probability distribution for (H,1,g) and (H,2,g) Therefore, all transcripts by simulator or actual has the same probability distribution Cryptography and Network Security 511 Quadratic Residue Fiat-Shamir Identification Question Given integer n=pq, here p, q are primes. Prover wants to prove Integer x is a quadratic residue mod n In other words, knows u so x=u2 mod n Quadratic residue is hard to solve if do not knowing the factoring of n Cryptography and Network Security 512 Proof Protocol Repeat the following for log2n times Prover Chooses random v less than n and computes y=v2 mod n. Sends y to verifier Verifier Chooses a random I from {0,1}, sends it to prover Prover Computes z=u2v mod n, sends z to verifier Verifier Checks if z2=xiy mod n Accepts the proof if equation holds all log2n rounds Cryptography and Network Security 513 Cont Correctness Show that verifier will accept the prover if indeed knows Soundness Show that verifier will detect the prover if it does not know with a good probability Zero-knowledge Show that verifier gets nothing from the protocol Cryptography and Network Security 514 Guillou Quisquater Protocol The GQ protocol is an extension of the Fiat Shamir protocol that limits the number t of rounds required. One Time Set-up: 1. A trusted authority T selects two random primes p and q and forms a modulus n = p · q. 2. T defines a public exponent v > 4 with gcd(v, (p-1)(q -1) = 1 so that T can compute s = v-1 mod (p-1) (q-1). 3. T publishes parameters n and v. Cryptography and Network Security 515 Cont. Selection of per-user parameters: 1. Each entity A has a unique identification Id(A). Everyone can calculate a value J(A) = f(Id(A)) mod n (the redundant identity). 2. T gives to each entity A the secret data secret(A) = J(A)-s, which it can calculate. Cryptography and Network Security 516 Cont. Protocol: A proves her identity to B using 1. 2. 3. 4. t rounds, each of which consists of: A selects a random secret r and sends her identity Id(A) and x = rv mod n to B. B selects a random challenge e in {1, 2, ... , v}. A computes and sends the following response to B: y = r · secret(A)e mod n. B receives y, constructs J(A) = f(Id(A)) mod n, computes z = J(A)eyv, and accepts this round if z = x mod n. In this protocol, v determines the security level. In Fiat Shamir, v = 2 and there are many rounds. A fraudulent claimant can defeat the protocol by correctly guessing the challenge e (with a 1 in v chance.) GQ seems secure, because we need to extract v-roots modulo n. Cryptography and Network Security 517 Discrete Logarithm Question: Prover wants to prove to verifier that he knows x such that y=gx mod p . Here g, y, and p are public information Prover does not want to publicize the value of x. Cryptography and Network Security 518 Proof Protocol Repeat the following for log2n times Prover Chooses random j < p-1 and computes r=gj mod p. Sends r to verifier Verifier Chooses a random i from {0,1}, sends it to prover Prover Computes h=i x +j mod p-1, sends h to verifier Verifier Checks if gh=yir mod n Accepts the proof if equation holds all log2n rounds Cryptography and Network Security 519 Cont Correctness Show that verifier will accept the prover if indeed knows Soundness Show that verifier will detect the prover if it does not know with a good probability Zero-knowledge Show that verifier gets nothing from the protocol Cryptography and Network Security 520 Bit Commitments Bit commitment Sometimes, it is desirable to give someone a piece of information, but not commit to it until a later date. It may be desirable for the piece of information to be held secret for a certain period of time. Example: stock up and down Cryptography and Network Security 521 Properties Bit commitment scheme The sender encrypts the b in some way The encrypted form of b is called blob Scheme f: (X,b)Y Properties Concealing: verifier cannot detect b from f(x,b) Binding: sender can open the blob by revealing x Hence, the sender must use random x to mask b Cryptography and Network Security 522 Methods One can choose any encryption method E Function f((x0,k),b)=Ek((x0,b)) Need supply decryption k to reveal b Assume the decryption method D is known Choose any integer n=pq, p and q are large primes Function f(x,b)=mbx2 mod n Goldwasser-Micali Scheme Here n=pq, m is not quadratic residule, m,n public mx12 mod n x22 mod n So sender can not change mind after commitment Cryptography and Network Security 523 Coin Flip Even protocols Alice has a coin flip result i or j Bob wants to guess the result Alice has a message M that is commitment If bob guesses correct, Bob should have M received Alice starts with 2 pairs of public keys (Ei,Di) and (Ej,Dj) Bob starts with a symmetric encryption S and a key k Cryptography and Network Security 524 Protocol Procedure Alice sends Ei, Ej to Bob Bob guess h and sends y=Eh(k) to Alice Alice computes p=Dj(y) and sends the encryption z of M by p using S to Bob Bob decrypts the encryption z using S and key k If the guess is correct, then Bob gets the commitment Cryptography and Network Security 525 Oblivious Transfer What is oblivious transfer Alice wants to send Bob a secret in such a way that Bob will know whether he gets it, but Alice won't. Another version is where Alice has several secrets and transfers one of them to Bob in such a way that Bob knows what he got, but Alice doesn't. This kind of transfer is said to be oblivious (to Alice). Cryptography and Network Security 526 Transfer Factoring By means of RSA, oblivious transfer of any secret amounts to oblivious transfer of the factorization of n=pq Bob chooses x and sends x2 mod n to Alice Alice (who knows p,q) computes the square roots x,x,y,-y of x2 mod n and sends one of them to Bob. Note that Alice does not know x. If Bob gets one of y or -y, he can factor n. This means that with probability 1/2, Bob gets the secret. Alice doesn't know whether Bob got one of y or -y because she doesn't know x. Cryptography and Network Security 527 Factoring If one knows x and y such that 1) x2=y2 mod n 2) 0<x,y<n, xy and x+y0 mod n Number n is the production of two primes Then n can be factored First gcd(x+y,n) is a factor of n And gcd(x-y,n) is a factor of n Cryptography and Network Security 528 Quadratic Solution Given n=p, and a is a quadratic residue Then there is two positive integers x less than n Such that x2=a mod n Given n=pq, and a is a quadratic residue Then there is four positive integers x less than n Such that x2=a mod n Cryptography and Network Security 529 Oblivious Transfer of Message Alice has a message M, bob wants to get M through oblivious transfer Alice does not know if Bob get M or not Bob knows if he gets it or not Bob gets M with probability ½ Coin flipping can be used to achieve this Cryptography and Network Security 530 Contract Signing It requires two things Commitment: after certain point, both parties are bound by the contract, until then, neither is Unforgeability: it must be possible for either party to prove the signature of the other party With Pen and Paper Two party together, face to face Sign simultaneously (or one character by one) Cryptography and Network Security 531 Remote Contract Signing Simple one Alice generate a signature, divided into SL, SR Alice randomly select two keys KL, KR Encrypt the signatures SL, SR Transfer encrypted SL,SR to Bob Obliviously transfer KL, KR to bob Bob gets one, but Alice does not know which one Bob decrypts the encrypted SL or SR Verify the decrypted signature, if invalid, stop Alice sends the ith bits of keys KL and KR to Bob Here i=1 to the length of the keys Cryptography and Network Security 532 Cont. The protocol will be conducted by Bob also What is the chance of Alice to cheat successfully? Alice can guess which key will be transferred obliviously ---(1/2 chance) Then send wrong signature for the other half or send the wrong key of the other half Bob can not detect it if Alice can guess which key Bob got How about Alice stop prematurely? One bit advance over Bob Enhanced protocol Use many pair of keys and signatures instead of one Cryptography and Network Security 533 Cryptography and Network Security Pseudo-random Number Xiang-Yang Li Cryptography and Network Security 534 Random number, Pseudorandom The outputs of pseudorandom number generators are not truly random they only approximate some of the properties of random numbers. "Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin.”--- John von Neumann Truely random numbers can be generated using hardware random number generators Cryptography and Network Security 535 Randomness Definition Chaitin-Kolmogorov randomness (also called algorithmic randomness) a string of bits is random if and only if it is shorter than any computer program that can produce that string this basically means that random strings are those that cannot be compressed. Statistical Randomness A numeric sequence is said to be statistically random when it contains no recognizable patterns or regularities; sequences such as the results of an ideal die roll, or the digits of Pi (as far as we can tell) exhibit statistical randomness. Cryptography and Network Security 536 Inherent non-randomness Because any PRNG run on a deterministic computer (contrast quantum computer) is deterministic, its output will inevitably have certain properties that a true random sequence would not exhibit. guaranteed periodicity—it is certain that if the generator uses only a fixed amount of memory then, given a sufficient number of iterations, the generator will revisit the same internal state twice, after which it will repeat forever. A generator that isn't periodic can be designed, but its memory requirements would grow as it ran. In addition, a PRNG can be started from an arbitrary starting point, or seed state, and will always produce an identical sequence from that point on. Cryptography and Network Security 537 cont In practice, many PRNGs exhibit artifacts which can cause them to fail statistically significant tests. These include, but are certainly not limited to: Shorter than expected periods for some seed states (not full period) Poor dimensional distribution Successive values are not independent Some bits may be 'more random' than others Lack of uniformity Cryptography and Network Security 538 Pseudo-random Bit Generator Several applications Key generation Some encryption algorithms, or one-time pad Let l>k be integers f: Z2k Z2l computable in poly-time Then f called (k,l)-pseudo-random bit generator The input s0 Z2k is called the seed Output f(s0) is called the pseudo-random string Function Cryptography and Network Security 539 Desired Properties Three important properties: Unbiased (uniform distribution): All values of whatever sample size is collected are equiprobable Unpredictable (independence): It is impossible to predict what the next output will be, given all the previous outputs, but not the internal "hidden" state. Irreproducible: Two of the same generators, given the same starting conditions, will produce different outputs. Cryptography and Network Security 540 Desired Properties Usually when a person says A "good" pseudo-random number generator they mean it is unbiased. A "true" PRNG they usually mean it's irreproducible A "cryptographically strong" PRNG they mean it's unpredictable Very rarely they mean it's all threes Cryptography and Network Security 541 More Properties Long period The generator should be of long period Fast computation The generator should be reasonably fast Security The generator should be secure What is security level of PRNG? Cryptography and Network Security 542 Security A PRNG suitable for cryptographic applications is called a cryptographically secure PRNG (CSPRNG). Its output should not only pass all statistical tests for randomness but satisfy some additional cryptographic requirements. Used in many aspects of cryptography require random numbers, for example: Key generation Nonces Salts in certain signature schemes, (ECDSA, RSASSA-PSS). One-time pads Cryptography and Network Security 543 CSPRNG CSPRNG requirements fall into two groups: their statistical properties are good (passing tests of randomness), they hold up well in case of attack, even when (part of) their secrets are revealed. A CSPRNG should satisfy the 'next-bit test'. Given the first l bits of a random sequence there is no polynomial-time algorithm that can predict the next bit with probability of success significantly higher than 1/2. It has been proven that a generator passing the next-bit test will pass all other polynomial-time statistical tests for randomness. should withstand state compromise extensions. That is, in the unfortunate case that part or all of the state has been revealed (or guessed correctly), it should be impossible to reconstruct the stream of random numbers prior to the incident. Also if there is an input of entropy, it should be infeasible to use knowledge of the state to predict future conditions of the state. Cryptography and Network Security 544 Example the CSPRNG being considered produces output by computing some function of the next digit of pi (ie, 3.1415...), it may well be random as pi appears to be a random sequence. However, this does not satisfy the next-bit test, and thus is not cryptographically secure. There exists an algorithm that will predict the next bit. Cryptography and Network Security 545 Design divide designs of CSPRNGs into classes: those based on block ciphers; those based upon hard mathematical problems, and special-purpose designs. Cryptography and Network Security 546 Designs based on cryptographic primitives Designs based on cryptographic primitives A secure block cipher can also be converted into a CSPRNG by running it in counter mode. This is done by choosing an arbitrary key and encrypting a zero, then encrypting a 1, then encrypting a 2, etc. The counter can also be started at an arbitrary number other than zero. Obviously, the period will be 2n for an n-bit block cipher; equally obviously, the initial values (i.e. key and 'plaintext') must not become known to an attacker lest, however good this CSPRNG construction might be otherwise, all security be lost. A cryptographically secure hash of a counter might also act as a good CSPRNG in some cases. it is necessary that the initial value of this counter is random and secret. If the counter is a bignum, then CSPRNG could have an infinite period. Cryptography and Network Security 547 DES Based Generator ANSI X9.17 PRNG (used by PGP,..) Inputs: two pseudo-random inputs one is a 64-bit representation of date and time The other is 64-bit seed values Keys: three 3DES encryptions using same keys Output: a 64-bit pseudorandom number and A 64-bit seed value for next-round use Cryptography and Network Security 548 ANSI X9.17 K1,K2 DT EDE EDE Si+1 Si EDE Ri Cryptography and Network Security 549 Linear Congruential Generator Protocol Let M be an integer and a, b less than M Let k be number of bits of M Integer l is between k+1 and M-1 Let s0 be a seed less than M Define si=asi-1+b mod M Then the ith random bit is si mod 2 It is not proved to be secure Cryptography and Network Security 550 Parameter Setting Not all a, b are good and m should be large For example, m is a large prime number For fast computation, usually m=231-1 And b is set to 0 often For this m, there are less than 100 integers a It generates all numbers less than m The generated sequences appear to be random One such a=7516807 Used in IBM 360 family of computers Cryptography and Network Security 551 RSA Generator Protocol Let p, q be two k/2 bits primes and define n=pq Integer b: gcd(b, (n))=1 Public: n, b; Private p,q A seed s0 with k bits Sequence si+1=sib mod n Then the ith random bit is si mod 2 It is proved to be secure! Cryptography and Network Security 552 BBS Generator Blum-Blum-Shub Generator Let p, q be two k/2 bits primes and define n=pq Here p=q=3 mod 4 this guarantees that each quadratic residue has one square root which is also a quadratic residue gcd(φ(p-1), φ(q-1)) should be small this makes the cycle length large. Let QR(n) be all quadratic residues modulo n Public: n; Private p,q A seed s0 with k bits from QR(n) Sequence si+1=si2 mod n Then the ith random bit is si mod 2 Cryptography and Network Security 553 Cont on BBS Provably “secure” When the primes are chosen appropriately, and O(log log n) bits of each Si are output, then in the limit as n grows large, distinguishing the output bits from random will be at least as difficult as factoring n. However, it's theoretically possible that a fast algorithm for factoring will someday be found, so BBS is not yet guaranteed to be secure. Cryptography and Network Security 554 Discrete Logarithm Generator Protocol Let p be a k-bit prime, Let be primitive element modulo p A seed s0 is any non-zero integer less than p Define si+1 = si mod p Then the ith random bit is 1 if si is larger than p/2 0 if si is less than p/2 Cryptography and Network Security 555 Standards A number of designs of CSPRNGs have been standardized. They can be found in: FIPS 186-2 ANSI X9.17-1985 Appendix C ANSI X9.31-1998 Appendix A.2.4 ANSI X9.62-1998 Annex A.4 Cryptography and Network Security 556 Network Security Cryptography and Network Security 557 Topics to be covered Applications Email security www security Malicious software Networks Wireless LAN security 802.11 IPsec Firewall Intrusions Cryptography and Network Security 558 Cryptography and Network Security Email Security Xiang-Yang Li Cryptography and Network Security 559 Electronic Mail Security Despite the refusal of VADM Poindexter and LtCol North to appear, the Board's access to other sources of information filled much of this gap. The FBI provided documents taken from the files of the National Security Advisor and relevant NSC staff members, including messages from the PROF system between VADM Poindexter and LtCol North. The PROF messages were conversations by computer, written at the time events occurred and presumed by the writers to be protected from disclosure. In this sense, they provide a first-hand, contemporaneous account of events. —The Tower Commission Report to President Reagan on the Iran-Contra Affair, 1987 Cryptography and Network Security 560 Email Security email is one of the most widely used and regarded network services currently message contents are not secure may be inspected either in transit or by suitably privileged users on destination system Cryptography and Network Security 561 Email Security Enhancements confidentiality protection from disclosure authentication of sender of message message integrity protection from modification non-repudiation of origin protection from denial by sender Cryptography and Network Security 562 Pretty Good Privacy (PGP) widely used de facto secure email developed by Phil Zimmermann selected best available crypto algs to use integrated into a single program available on Unix, PC, Macintosh and Amiga systems originally free, now have commercial versions available also Cryptography and Network Security 563 PGP Five services Authentication, confidentiality, compression, email compatibility, segmentation Functions Digital signature Message encryption Compression Email compatibility segmentation Cryptography and Network Security 564 PGP Operation – Authentication 1. sender creates a message 2. SHA-1 used to generate 160-bit hash code of message 3. hash code is encrypted with RSA using the sender's private key, and result is attached to message 4. receiver uses RSA or DSS with sender's public key to decrypt and recover hash code 5. receiver generates new hash code for message and compares with decrypted hash code, if match, message is accepted as authentic Cryptography and Network Security 565 PGP Operation – Confidentiality 1. 2. 3. 4. 5. sender generates message and random 128-bit number to be used as session key for this message only message is encrypted, using CAST-128 / IDEA/3DES with session key session key is encrypted using RSA with recipient's public key, then attached to message receiver uses RSA with its private key to decrypt and recover session key session key is used to decrypt message Cryptography and Network Security 566 PGP Operation – Confidentiality & Authentication uses both services on same message create signature & attach to message encrypt both message & signature attach RSA encrypted session key Cryptography and Network Security 567 PGP Operation – Compression by default PGP compresses message after signing but before encrypting so can store uncompressed message & signature for later verification & because compression is non deterministic uses ZIP compression algorithm Cryptography and Network Security 568 PGP Operation – Email Compatibility when using PGP will have binary data to send (encrypted message etc) however email was designed only for text hence PGP must encode raw binary data into printable ASCII characters uses radix-64 algorithm maps 3 bytes to 4 printable chars also appends a CRC PGP also segments messages if too big Cryptography and Network Security 569 PGP Operation – Summary Cryptography and Network Security 570 Segmentation & Reassembly Email systems impose maximum length 50 Kb, for example PGP provides automatic segmentation Done after all other operations Thus only one session key needed Cryptography and Network Security 571 Key management Generating unpredictable session keys Identifying keys Multiple public, private key pairs for a user Maintain keys Its own public, private keys of a PGP entity Public keys of correspondents Cryptography and Network Security 572 Session Key Generation Algorithm used: CAST-128 Input to CAST-128 A 128-bit key Two 64 bits plaintexts to be encrypted Output using cipher feedback mode Generates 2 64-bits ciphers form session key Plaintexts are from 128-bits randomized number Based on key stroke of user (timing and actual keys) Then combined with previous session key Cryptography and Network Security 573 Key Identifiers Receiver has multiple public keys How to know which private key is proper? Approach Sending the least significant 64 bits as key ID Need send the receiver’s public key ID used for encrypting the session key Need send the sender’s public key ID, whose corresponding private key used for signature Cryptography and Network Security 574 Key Rings Private key rings Timestamp, Key ID, public key, encrypted private key, user ID Public key rings Timestamp, Key ID, public key, owner trust, user ID, key legitimacy, signature, signature trust Cryptography and Network Security 575 Public Key Management A public key attributed to B may belong to C C can send messages to A forge B’s sig C can read any encrypted message to B Approach to true public key Physically get key from B Obtain B’s key from mutual trusted authority Using key legitimacy field computed from the signature trust field and number of certificates for the key Cryptography and Network Security 576 Revoking Public Key Reason It is compromised: private key is open Simply to avoid use of same key for a period Approach Owner issues key revocation certificate, signed by owner Using corresponding private key to sign the certificate Disseminate the certificate as widely and as quickly as possible Cryptography and Network Security 577 S/MIME (Secure/Multipurpose Internet Mail Extensions) security enhancement to MIME email original Internet RFC822 email was text only MIME provided support for varying content types and multi-part messages with encoding of binary data to textual form S/MIME added security enhancements have S/MIME support in various modern mail agents: MS Outlook, Netscape etc Cryptography and Network Security 578 S/MIME Functions enveloped data encrypted content and associated keys signed data encoded message + signed digest clear-signed data cleartext message + encoded signed digest signed & enveloped data nesting of signed & encrypted entities Cryptography and Network Security 579 S/MIME Cryptographic Algorithms hash functions: SHA-1 & MD5 digital signatures: DSS & RSA session key encryption: ElGamal & RSA message encryption: Triple-DES, RC2/40 and others have a procedure to decide which algorithms to use Cryptography and Network Security 580 S/MIME Certificate Processing S/MIME uses X.509 v3 certificates managed using a hybrid of a strict X.509 CA hierarchy & PGP’s web of trust each client has a list of trusted CA’s certs and own public/private key pairs & certs certificates must be signed by trusted CA’s Cryptography and Network Security 581 Certificate Authorities have several well-known CA’s Verisign one of most widely used Verisign issues several types of Digital IDs with increasing levels of checks & hence trust Class 1 2+ 3+ Identity Checks Usage name/email check web browsing/email enroll/addr check email, subs, s/w validate ID documents e-banking/service access Cryptography and Network Security 582 Email SPAM Spam is flooding the Internet with many copies of the same message, in an attempt to force the message on people who would not otherwise choose to receive it. Most spam is commercial advertising, often for dubious products, get-rich-quick schemes, or quasi-legal services. Spam costs the sender very little to send -- most of the costs are paid for by the recipient or the carriers rather than by the sender Cryptography and Network Security 583 Email Spam E-mail spam has existed since the beginning of the Internet, and has grown to about 90 billion messages a day, although about 80% is sent by fewer than 200 spammers. Botnets, virus infected computers, account for about 80% of spam. E-mail addresses are collected from chatrooms, websites, newsgroups, and viruses which harvest users address books, and are sold to other spammers Cryptography and Network Security 584 Anti-Spam Techs Some popular methods for filtering and refusing spam include e-mail filtering based on the content of the e-mail, DNS-based blackhole lists (DNSBL), greylisting, spamtraps, enforcing technical requirements, checksumming systems to detect bulk email, and by putting some sort of cost on the sender via a Proof-ofwork system or a micropayment. Each method has strengths and weaknesses and each is controversial due to its weaknesses. Cryptography and Network Security 585 Filtering Methods Bayesian spam filtering CRM114 dSPAM Markovian discrimination POPFile Policyd-weight Postfix policy-daemon before SMTP DATA Procmail is an MDA (Mail Delivery Agent) for Unix systems. Maildrop is an MDA (Mail Delivery Agent) for Unix systems. Sendmail supports libmilter for mail filtering Sieve (mail filtering language) is an RFC standard for describing mail filters SpamAssassin Anti-Spam SMTP Proxy information filtering White list#E-mail whitelists Cryptography and Network Security 586 Summary have considered: secure email PGP S/MIME Cryptography and Network Security 587 Cryptography and Network Security Security on WWW Xiang-Yang Li Cryptography and Network Security 588 Introduction Introduction Presentation of SSL The inner workings of SSL • Attacks on SSL Presentation of S-HTTP • Comparison with SSL/TLS • Attacks on S-HTTP Other aspects of Web security • TLS • IPSec, Kerberos, SET Conclusion • Cryptography and Network Security 589 Web Security Web now widely used by business, government, individuals but Internet & Web are vulnerable have a variety of threats integrity confidentiality denial of service authentication need added security mechanisms Cryptography and Network Security 590 SSL (Secure Socket Layer) transport layer security service originally developed by Netscape version 3 designed with public input subsequently became Internet standard known as TLS (Transport Layer Security) uses TCP to provide a reliable end-to-end service SSL has two layers of protocols Cryptography and Network Security 591 Location of SSL Application Layer Secure Socket Layer (SSL) Transmission Control Protocol (TCP) SSL is build on top of TCP Provides a TCP like interface In theory can be used by all type of applications in a transparent manner Internet Protocol (IP) Cryptography and Network Security 592 SSL Architecture Cryptography and Network Security 593 SSL Architecture SSL session an association between client & server created by the Handshake Protocol define a set of cryptographic parameters may be shared by multiple SSL connections SSL connection a transient, peer-to-peer, communications link associated with 1 SSL session Cryptography and Network Security 594 SSL Record Protocol confidentiality using symmetric encryption with a shared secret key defined by Handshake Protocol IDEA, RC2-40, DES-40, DES, 3DES, Fortezza, RC440, RC4-128 message is compressed before encryption message integrity using a MAC with shared secret key similar to HMAC but with different padding Cryptography and Network Security 595 SSL Change Cipher Spec Protocol one of 3 SSL specific protocols which use the SSL Record protocol a single message causes pending state to become current hence updating the cipher suite in use Cryptography and Network Security 596 SSL Alert Protocol conveys SSL-related alerts to peer entity severity warning or fatal specific alert unexpected message, bad record mac, decompression failure, handshake failure, illegal parameter close notify, no certificate, bad certificate, unsupported certificate, certificate revoked, certificate expired, certificate unknown compressed & encrypted like all SSL data Cryptography and Network Security 597 SSL Handshake Protocol allows server & client to: authenticate each other to negotiate encryption & MAC algorithms to negotiate cryptographic keys to be used comprises a series of messages in phases Establish Security Capabilities Server Authentication and Key Exchange Client Authentication and Key Exchange Finish Cryptography and Network Security 598 General purpose 1.Handshake ` 2. Data transmission Two step process: • Handshake : exchange private keys using a public key encryption algorithm • Data transmission: exchange the required data using a private key encryption Cryptography and Network Security 599 SSL Handshake Protocol Cryptography and Network Security 600 handshake ` Client Client Hello Server Server Hello Server Certificate Server Hello Done Client Key Exchange Change Cipher Specification Handshake Finished Change Cipher Specifications Handshake Finished Cryptography and Network Security 601 hello Client “Hello”: • List of supported private key encryptions + • Client random number Server “Hello”: • Selected encryption algorithm • Server Random number • Session ID Server Certificate: • Verify server’s identity ` Client Server Client Hello Server Hello Server Certificate Server Hello Done Client Key Exchange Change Cipher Specification Handshake Finished Change Cipher Specifications Handshake Finished Cryptography and Network Security 602 Key exchange Client Key Exchange: • Client Generate second random: Pre Master Key Send Pre Master Key Calculate Master Key Calculate Secret Key Calculate MAC Key • Server Calculate Master Key Calculate Secret Key Calculate MAC Key ` Client Server Client Hello Server Hello Server Certificate Server Hello Done Client Key Exchange Change Cipher Specification Handshake Finished Change Cipher Specifications Handshake Finished Cryptography and Network Security 603 Resumed based on Session Id ` Client Client Hello Server Server Hello Change Cipher Specification Handshake Finished Change Cipher Specifications Handshake Finished Cryptography and Network Security 604 Certificate authority Certificate Authority (CA) is a trusted third party that helps identify the server. How does everything work? • • • • Server sends ID, public key to CA CA creates and signs the server’s Certificate Client receives the Certificate from Server Client verifies the Certificate using the signature and the CA’s public key Cryptography and Network Security 605 MAC MAC = Message Authentication Code The initial message is split into fragments For each fragment a “fingerprint” is calculated using the MAC key The fragment, fingerprint and record header are encrypted and sent Receiver checks the “fingerprint” using MAC key to detect inconsistent messages Cryptography and Network Security 606 Attacks on SSL Certificate Injection Attack • • The list of trusted Certificate Authorities is altered Can be avoided by upgrading the OS or switching to a safer one. Man in the Middle • Cipher Spec Rollback : regresses the public key encryption algorithms Version Rollback : regression from SSL 3.0 to weaker SSL 2.0 Algorithm rollback : modify public encryption method Truncation attack : TCP FIN|RST used to terminate connection • Can be avoided by randomly delaying the computations • Can be used on servers that accept small key sizes: 40 bits for symmetric encryptions and 512 for the asymmetric one. • • • Timing attack Brute force Cryptography and Network Security 607 TLS (Transport Layer Security) IETF standard RFC 2246 similar to SSLv3 with minor differences in record format version number uses HMAC for MAC a pseudo-random function expands secrets has additional alert codes some changes in supported ciphers changes in certificate negotiations changes in use of padding Cryptography and Network Security 608 TLS TLS was developed by IETF to replace SSL version 3. • Based on SSL version 3, with some changes: • Replaced FORTEZZA key exchange option with DSS. • Include the hash method HMAC used by IPSec for authentication in IP headers. • More differentiation between sub-protocols. • TLS has mechanisms for backwards compatibility with SSL. Cryptography and Network Security 609 TLS TLS has about 30 possible cipher ‘suites’, combinations of key exchange, encryption method, and hashing method. • Key exchange includes: RSA, DSS, Kerberos • Encryption includes: IDEA(CBC), RC2, RC4, DES, 3DES, and AES • Hashing: SHA and MD5 (Note: Some of the suites are intentionally weak export versions.) Cryptography and Network Security 610 Secure Electronic Transactions (SET) open encryption & security specification to protect Internet credit card transactions developed in 1996 by Mastercard, Visa etc not a payment system rather a set of security protocols & formats secure communications amongst parties trust from use of X.509v3 certificates privacy by restricted info to those who need it Cryptography and Network Security 611 SET Components Cryptography and Network Security 612 SET Transaction 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. customer opens account customer receives a certificate merchants have their own certificates customer places an order merchant is verified order and payment are sent merchant requests payment authorization merchant confirms order merchant provides goods or service merchant requests payment Cryptography and Network Security 613 Dual Signature customer creates dual messages order information (OI) for merchant payment information (PI) for bank neither party needs details of other but must know they are linked use a dual signature for this signed concatenated hashes of OI & PI Cryptography and Network Security 614 Purchase Request – Customer Cryptography and Network Security 615 Purchase Request – Merchant Cryptography and Network Security 616 Purchase Request – Merchant 1. 2. 3. 4. verifies cardholder certificates using CA sigs verifies dual signature using customer's public signature key to ensure order has not been tampered with in transit & that it was signed using cardholder's private signature key processes order and forwards the payment information to the payment gateway for authorization (described later) sends a purchase response to cardholder Cryptography and Network Security 617 Payment Gateway Authorization 1. 2. 3. 4. 5. 6. 7. 8. verifies all certificates decrypts digital envelope of authorization block to obtain symmetric key & then decrypts authorization block verifies merchant's signature on authorization block decrypts digital envelope of payment block to obtain symmetric key & then decrypts payment block verifies dual signature on payment block verifies that transaction ID received from merchant matches that in PI received (indirectly) from customer requests & receives an authorization from issuer sends authorization response back to merchant Cryptography and Network Security 618 Payment Capture merchant sends payment gateway a payment capture request gateway checks request then causes funds to be transferred to merchants account notifies merchant using capture response Cryptography and Network Security 619 A B C D C- Secure-HTTP Presentation of S-HTTP Designed by E. Rescorla and A. Schiffman of EIT to secure HTTP connections Proposed in 1994 but never used commercially Not to be confused with HTTPS: encrypts HTTP messages at the application level Security on the WWW Cryptography and Network Security 620 A B C D C- Secure-HTTP Location of S-HTTP Secure-HTTP Message encryption and signature Application Layer: HTTP message Transmission Control Protocol (TCP) HTTP-specific message encryption Can possibly be used over a secure channel Designed to be compatible with HTTP for handling at lower layers Internet Protocol (IP) Security on the WWW Cryptography and Network Security 621 A B C D C- Secure-HTTP S-HTTP vs. SSL/TLS HTTP-specific vs. general purpose SSL (IMAPS, POPS, LDAPS…) Burden of encryption not on transmission/reception but rather on message production/unpacking Similar set of available ciphers, plus added capabilities for signing (DSS, RSA) Very general specifications, leaving a lot to implement and a potential for incompatible implementations Only one reference implementation in NCSA Mosaic Security on the WWW Cryptography and Network Security 622 A B C D C- Secure-HTTP S-HTTP vs. SSL/TLS: functionalities Security Service S-HTTP SSL Privacy Public or private cryptosystem Encryption of the complete HTTP transaction Symmetric key cryptosystem Complete communication encryption Integrity Simple MAC or signing MAC only Authentication Key management on the keys used, or digital signature During the initial public key exchange (server auth. mandatory, client auth. optional) Non-repudiation Digital signature Not provided S-HTTP can make use of key management Non-repudiation is not provided by SSL Signing is optional, but a major attraction to S-HTTP Security on the WWW Cryptography and Network Security 623 A B C D C- Secure-HTTP S-HTTP vs. SSL/TLS: proxy traversal Proxy traversal: SSL connection OR cleartext SSL tunnel External secure server SSL tunnel SSL-aware proxy Enterprise environment Proxy traversal: S-HTTP messaging Encrypted data Authentication External secure server Security on the WWW S-HTTP-aware proxy Enterprise environment Cryptography and Network Security 624 A B C D C- Secure-HTTP S-HTTP inner working Message-based encryption Superset of HTTP: “outer” envelope Specific headers added S-HTTP message S-HTTP headers HTTP payload headers: Security-Scheme, Encryption-Identity, Certificate-Info… + regular HTTP headers Request: Secure*Secure-HTTP/1.2 Response: Secure-HTTP/1.2 200 OK HTTP message body Security on the WWW Cryptography and Network Security 625 A B C D C- Secure-HTTP S-HTTP attacks Basically the same as on SSL, since the ciphers are the same Default values more secure in S-HTTP than SSL at the time of proposal (e.g. DES vs. RC4) S-HTTP generally stronger by design (more resilient to proxy compromising) More complex and wider specifications create a potential for faulty implementations No real-world use to field test the actual security of SHTTP Security on the WWW Cryptography and Network Security 626 A B C D D- Other protocols HTTP Basic Authentication HTTP has an authentication scheme as part of its original protocol. • Supported by almost all browsers and web servers. • Password and username are sent in clear text (base64 encoded) in the HTTP request message. • Obviously not secure enough for sensitive information. This scheme is being replaced by the slightly more secure HTTP Digest Authentication, which sends a MD5 hash of the password and other information. Security on the WWW Cryptography and Network Security 627 IPsec IPSec is a security layer added to a computer’s protocol stack in the kernel (Below TCP). It is invisible to the application. It is implemented by adding additional protocol numbers in the IP protocol field. • Good for implementing a VPN. • Packets can be either tunneled inside IPSec packets, or Transported with only the data portion of the original packet encrypted. • Every IPSec end machine (which could be a LAN’s router) must implement IPSec for it to work. Cryptography and Network Security 628 Summary have considered: need for web security SSL/TLS transport layer security protocols de facto standard, versatile and low-level enough to accommodate many types of payloads SET secure credit card payment protocols IPSec: true network-layer security for any applications (not just the Web) Kerberos: robust 2-way authentication framework with emphasis on security manageability Cryptography and Network Security 629 A B C D D- Conclusion Web Security • SSL/TLS: de facto standard, versatile and low-level enough to accommodate many types of payloads • S-HTTP: never took off, restricted to HTTP messages • IPSec: true network-layer security for any applications (not just the Web) • Kerberos: robust 2-way authentication framework with emphasis on security manageability Security on the WWW Cryptography and Network Security 630 Cryptography & Network Security Malicious Software XiangYang Li Cryptography and Network Security 631 Malicious Software What is the concept of defense: The parrying of a blow. What is its characteristic feature: Awaiting the blow. —On War, Carl Von Clausewitz Cryptography and Network Security 632 Viruses and Other Malicious Content computer viruses have got a lot of publicity one of a family of malicious software effects usually obvious have figured in news reports, fiction, movies (often exaggerated) getting more attention than deserve are a concern though Cryptography and Network Security 633 Malicious Software Cryptography and Network Security 634 Trapdoors secret entry point into a program allows those who know access bypassing usual security procedures have been commonly used by developers a threat when left in production programs allowing exploited by attackers very hard to block in O/S requires good s/w development & update Cryptography and Network Security 635 Logic Bomb one of oldest types of malicious software code embedded in legitimate program activated when specified conditions met eg presence/absence of some file particular date/time particular user when triggered typically damage system modify/delete files/disks Cryptography and Network Security 636 Trojan Horse program with hidden side-effects which is usually superficially attractive eg game, s/w upgrade etc when run performs some additional tasks allows attacker to indirectly gain access they do not have directly often used to propagate a virus/worm or install a backdoor or simply to destroy data Cryptography and Network Security 637 Zombie program which secretly takes over another networked computer then uses it to indirectly launch attacks often used to launch distributed denial of service (DDoS) attacks exploits known flaws in network systems Cryptography and Network Security 638 Viruses a piece of self-replicating code attached to some other code cf biological virus both propagates itself & carries a payload carries code to make copies of itself as well as code to perform some covert task Cryptography and Network Security 639 Virus Operation virus phases: dormant – waiting on trigger event propagation – replicating to programs/disks triggering – by event to execute payload execution – of payload details usually machine/OS specific exploiting features/weaknesses Cryptography and Network Security 640 Virus Structure program V := {goto main; 1234567; subroutine infect-executable := {loop: file := get-random-executable-file; if (first-line-of-file = 1234567) then goto loop else prepend V to file; } subroutine do-damage := {whatever damage is to be done} subroutine trigger-pulled := {return true if some condition holds} main: main-program := {infect-executable; if trigger-pulled then do-damage; goto next;} next: } Cryptography and Network Security 641 Types of Viruses can classify on basis of how they attack parasitic virus memory-resident virus boot sector virus stealth polymorphic virus macro virus Cryptography and Network Security 642 Macro Virus macro code attached to some data file interpreted by program using file eg Word/Excel macros esp. using auto command & command macros code is now platform independent is a major source of new viral infections blurs distinction between data and program files making task of detection much harder classic trade-off: "ease of use" vs "security" Cryptography and Network Security 643 Email Virus spread using email with attachment containing a macro virus cf Melissa triggered when user opens attachment or worse even when mail viewed by using scripting features in mail agent usually targeted at Microsoft Outlook mail agent & Word/Excel documents Cryptography and Network Security 644 Worms replicating but not infecting program typically spreads over a network cf Morris Internet Worm in 1988 led to creation of CERTs using users distributed privileges or by exploiting system vulnerabilities widely used by hackers to create zombie PC's, subsequently used for further attacks, esp DoS major issue is lack of security of permanently connected systems, esp PC's Cryptography and Network Security 645 Worm Operation worm phases like those of viruses: dormant propagation search for other systems to infect establish connection to target remote system replicate self onto remote system triggering execution Cryptography and Network Security 646 Morris Worm best known classic worm released by Robert Morris in 1988 targeted Unix systems using several propagation techniques simple password cracking of local pw file exploit bug in finger daemon exploit debug trapdoor in sendmail daemon if any attack succeeds then replicated self Cryptography and Network Security 647 Recent Worm Attacks new spate of attacks from mid-2001 Code Red exploited bug in MS IIS to penetrate & spread probes random IPs for systems running IIS had trigger time for denial-of-service attack 2nd wave infected 360000 servers in 14 hours Code Red 2 had backdoor installed to allow remote control Nimda used multiple infection mechanisms email, shares, web client, IIS, Code Red 2 backdoor Cryptography and Network Security 648 Virus Countermeasures viral attacks exploit lack of integrity control on systems to defend need to add such controls typically by one or more of: prevention - block virus infection mechanism detection - of viruses in infected system reaction - restoring system to clean state Cryptography and Network Security 649 Anti-Virus Software first-generation scanner uses virus signature to identify virus or change in length of programs second-generation uses heuristic rules to spot viral infection or uses program checksums to spot changes third-generation memory-resident programs identify virus by actions fourth-generation packages with a variety of antivirus techniques eg scanning & activity traps, access-controls Cryptography and Network Security 650 Advanced Anti-Virus Techniques generic decryption use CPU simulator to check program signature & behavior before actually running it digital immune system (IBM) general purpose emulation & virus detection any virus entering org is captured, analyzed, detection/shielding created for it, removed Cryptography and Network Security 651 Behavior-Blocking Software integrated with host O/S monitors program behavior in real-time eg file access, disk format, executable mods, system settings changes, network access for possibly malicious actions if detected can block, terminate, or seek ok has advantage over scanners but malicious code runs before detection Cryptography and Network Security 652 Summary have considered: various malicious programs trapdoor, logic bomb, trojan horse, zombie viruses worms countermeasures Cryptography and Network Security 653 Cryptography & Network Security Wireless LAN Security Road to 802.11i Xiangyang Li Cryptography and Network Security 654 Contents Introduction Problem: 802.11b Not Secure! Wired Equivalent Privacy – WEP Types of Attacks 802.11b Proposed Solutions 802.1X Wi-Fi Protected Access (WPA) 802.11i: The Solution Conclusion Cryptography and Network Security 655 Introduction Popular in offices, homes and public spaces (airport, coffee shop) Most popular: 802.11b Example: Yahoo! DSL Wireless Kit Theoretical max @ 11Mbps Operate at 2.4GHz band DSSS/FSSS modulation – similar to CDMA phones Cryptography and Network Security 656 Introduction Standards: IEEE 802.11 Series 802.11b – 11Mbps @ 2.4GHz 802.11a – 54Mbps @ 5.7GHz band 802.11g – 54Mbps @ 2.4GHz band 802.1X – security add-on 802.11i – high security Cryptography and Network Security 657 Problem: 802.11b Not Secure! “No inherent security” Wired Wireless media change was the objective Wired Equivalent Privacy (WEP) The only “security” built into 802.11 Uses RC4 Stream Cipher – in a bad way Vulnerable to several types of attacks Sometimes not even turned ON Cryptography and Network Security 658 Wired Equivalent Privacy – WEP RC4 stream cipher Designed by Ron Rivest for RSA Security Very simple Initialization Vector (IV) Shared Key The issue is in the way RC4 is used IV (24 bits) reuse and fixed key Early versions used 40-bit key 128-bit mode effectively uses 104 bits Cryptography and Network Security 659 Wired Equivalent Privacy – WEP RC4 Key Stream Encryption (source: http://mason.gmu.edu/~gharm/wireless.html) Cryptography and Network Security 660 Types of Attacks Attacks Confidentiality Integrity Availability Cryptography and Network Security 661 Types of Attacks Attacks on Confidentiality Traffic Analysis Passive Eavesdropping Very easy to do Active Eavesdropping Unauthorized Access Cryptography and Network Security 662 Types of Attacks Attacks on Confidentiality and/or Integrity Man-In-The-Middle Attacks on Integrity Session Hijacking Replay Attacks on Availability Denial of Service Cryptography and Network Security 663 802.11b Proposed Solutions Virtual Private Network Closed Network Through the use of SSID Ethernet MAC address control lists Replace RC4 with block cipher Don’t reuse IV Automatic Key Assignment Cryptography and Network Security 664 802.1X: Interim Solution Port-based authentication Not specific to wireless networks Authentication servers RADIUS Client authentication EAP Cryptography and Network Security 665 802.1X Problems 802.1X still has problems Extensible Authentication Protocol (EAP) One-way authentication Attacks Man-in-Middle Session Hijacking Cryptography and Network Security 666 802.1X Proposed Solutions Per-packet authenticity and integrity Lots of overhead Authenticity and integrity of EAPOL messages Two-way authentication Cryptography and Network Security 667 Wi-Fi Protected Access (WPA) Addresses issues with WEP Key management TKIP Key expansion Message Integrity Check Software upgrade only Compatible with 802.1X Compatible with 802.11i Cryptography and Network Security 668 802.11i Finalized: June, 2004 Robust Security Network Wi-Fi Alliance: WPA2 Improvements made Authentication enhanced Key Management created Data Transfer security enhanced Cryptography and Network Security 669 802.11i - Authentication Authentication Server Two-way authentication Prevents man-in-the-middle attacks Master Key (MK) Pairwise Master Key (PMK) Cryptography and Network Security 670 802.11i – Key Management Key Types Pairwise Transient Key Key Confirmation Key Key Encryption Key Group Transient Key Temporal Key Cryptography and Network Security 671 802.11i – Key Management Source: http://csrc.nist.gov/wireless/S10_802.11i%20Overview-jw1.pdf Cryptography and Network Security 672 802.11i – Data Transfer CCMP Long term solution – mandatory for 802.11i compliance Latest AES encryption Requires hardware upgrades WRAP Provided for early vendor support TKIP Carried over from WPA Cryptography and Network Security 673 802.11i – Additional Enhancements Pre-authentication Roaming clients Client Validation Password-to-key mappings Random number generation Cryptography and Network Security 674 Conclusion Basic 802.11b (with WEP) Massive security holes Easily attacked 802.1X Good interim solution Allows use of existing hardware Can still be attacked Cryptography and Network Security 675 Conclusion Wi-Fi Protected Access Allows use of existing hardware Compatible with 802.1X Compatible with 802.11i 802.11i May require hardware upgrades Very secure Nothing is ever guaranteed Cryptography and Network Security 676 Cryptography & Network Security IPsec XiangYang Li Cryptography and Network Security 677 IP Security If a secret piece of news is divulged by a spy before the time is ripe, he must be put to death, together with the man to whom the secret was told. —The Art of War, Sun Tzu Cryptography and Network Security 678 IP Security have considered some application specific security mechanisms eg. S/MIME, PGP, Kerberos, SSL/HTTPS however there are security concerns that cut across protocol layers would like security implemented by the network for all applications Cryptography and Network Security 679 IPSec general IP Security mechanisms provides authentication confidentiality key management applicable to use over LANs, across public & private WANs, & for the Internet Cryptography and Network Security 680 IPSec Uses Cryptography and Network Security 681 Benefits of IPSec in a firewall/router provides strong security to all traffic crossing the perimeter is resistant to bypass is below transport layer, hence transparent to applications can be transparent to end users can provide security for individual users if desired Cryptography and Network Security 682 IP Security Architecture specification is quite complex defined in numerous RFC’s incl. RFC 2401/2402/2406/2408 many others, grouped by category mandatory in IPv6, optional in IPv4 Cryptography and Network Security 683 IPSec Services Access control Connectionless integrity Data origin authentication Rejection of replayed packets a form of partial sequence integrity Confidentiality (encryption) Limited traffic flow confidentiality Cryptography and Network Security 684 Security Associations a one-way relationship between sender & receiver that affords security for traffic flow defined by 3 parameters: Security Parameters Index (SPI) IP Destination Address Security Protocol Identifier has a number of other parameters seq no, AH & EH info, lifetime etc have a database of Security Associations Cryptography and Network Security 685 Authentication Header (AH) provides support for data integrity & authentication of IP packets end system/router can authenticate user/app prevents address spoofing attacks by tracking sequence numbers based on use of a MAC HMAC-MD5-96 or HMAC-SHA-1-96 parties must share a secret key Cryptography and Network Security 686 Authentication Header Cryptography and Network Security 687 Transport & Tunnel Modes Cryptography and Network Security 688 Encapsulating Security Payload (ESP) provides message content confidentiality & limited traffic flow confidentiality can optionally provide the same authentication services as AH supports range of ciphers, modes, padding incl. DES, Triple-DES, RC5, IDEA, CAST etc CBC most common pad to meet blocksize, for traffic flow Cryptography and Network Security 689 Encapsulating Security Payload Cryptography and Network Security 690 Transport vs Tunnel Mode ESP transport mode is used to encrypt & optionally authenticate IP data data protected but header left in clear can do traffic analysis but is efficient good for ESP host to host traffic tunnel mode encrypts entire IP packet add new header for next hop good for VPNs, gateway to gateway security Cryptography and Network Security 691 Combining Security Associations SA’s can implement either AH or ESP to implement both need to combine SA’s form a security bundle have 4 cases (see next) Cryptography and Network Security 692 Combining Security Associations Cryptography and Network Security 693 Key Management handles key generation & distribution typically need 2 pairs of keys 2 per direction for AH & ESP manual key management sysadmin manually configures every system automated key management automated system for on demand creation of keys for SA’s in large systems has Oakley & ISAKMP elements Cryptography and Network Security 694 Oakley a key exchange protocol based on Diffie-Hellman key exchange adds features to address weaknesses cookies, groups (global params), nonces, DH key exchange with authentication can use arithmetic in prime fields or elliptic curve fields Cryptography and Network Security 695 ISAKMP Internet Security Association and Key Management Protocol provides framework for key management defines procedures and packet formats to establish, negotiate, modify, & delete SAs independent of key exchange protocol, encryption alg, & authentication method Cryptography and Network Security 696 ISAKMP Cryptography and Network Security 697 Summary have considered: IPSec security framework AH ESP key management & Oakley/ISAKMP Cryptography and Network Security 698 Cryptography & Network Security Firewalls XiangYang Li Cryptography and Network Security 699 Firewalls The function of a strong position is to make the forces holding it practically unassailable —On War, Carl Von Clausewitz Cryptography and Network Security 700 Introduction seen evolution of information systems now everyone want to be on the Internet and to interconnect networks has persistent security concerns can’t easily secure every system in org need "harm minimisation" a Firewall usually part of this Cryptography and Network Security 701 What is a Firewall? a choke point of control and monitoring interconnects networks with differing trust imposes restrictions on network services only authorized traffic is allowed auditing and controlling access can implement alarms for abnormal behavior is itself immune to penetration provides perimeter defence Cryptography and Network Security 702 Firewall Limitations cannot protect from attacks bypassing it eg sneaker net, utility modems, trusted organisations, trusted services (eg SSL/SSH) cannot protect against internal threats eg disgruntled employee cannot protect against transfer of all virus infected programs or files because of huge range of O/S & file types Cryptography and Network Security 703 Firewalls – Packet Filters Cryptography and Network Security 704 Firewalls – Packet Filters simplest of components foundation of any firewall system examine each IP packet (no context) and permit or deny according to rules hence restrict access to services (ports) possible default policies that not expressly permitted is prohibited that not expressly prohibited is permitted Cryptography and Network Security 705 Firewalls – Packet Filters Cryptography and Network Security 706 Attacks on Packet Filters IP address spoofing fake source address to be trusted add filters on router to block source routing attacks attacker sets a route other than default block source routed packets tiny fragment attacks split header info over several tiny packets either discard or reassemble before check Cryptography and Network Security 707 Firewalls – Stateful Packet Filters examine each IP packet in context keeps tracks of client-server sessions checks each packet validly belongs to one better able to detect bogus packets out of context Cryptography and Network Security 708 Firewalls - Application Level Gateway (or Proxy) Cryptography and Network Security 709 Firewalls - Application Level Gateway (or Proxy) use an application specific gateway / proxy has full access to protocol user requests service from proxy proxy validates request as legal then actions request and returns result to user need separate proxies for each service some services naturally support proxying others are more problematic custom services generally not supported Cryptography and Network Security 710 Firewalls - Circuit Level Gateway Cryptography and Network Security 711 Firewalls - Circuit Level Gateway relays two TCP connections imposes security by limiting which such connections are allowed once created usually relays traffic without examining contents typically used when trust internal users by allowing general outbound connections SOCKS commonly used for this Cryptography and Network Security 712 Bastion Host highly secure host system potentially exposed to "hostile" elements hence is secured to withstand this may support 2 or more net connections may be trusted to enforce trusted separation between network connections runs circuit / application level gateways or provides externally accessible services Cryptography and Network Security 713 Firewall Configurations Cryptography and Network Security 714 Firewall Configurations Cryptography and Network Security 715 Firewall Configurations Cryptography and Network Security 716 Access Control given system has identified a user determine what resources they can access general model is that of access matrix with subject - active entity (user, process) object - passive entity (file or resource) access right – way object can be accessed can decompose by columns as access control lists rows as capability tickets Cryptography and Network Security 717 Access Control Matrix Cryptography and Network Security 718 Trusted Computer Systems information security is increasingly important have varying degrees of sensitivity of information cf military info classifications: confidential, secret etc subjects (people or programs) have varying rights of access to objects (information) want to consider ways of increasing confidence in systems to enforce these rights known as multilevel security subjects have maximum & current security level objects have a fixed security level classification Cryptography and Network Security 719 Bell LaPadula (BLP) Model one of the most famous security models implemented as mandatory policies on system has two key policies: no read up (simple security property) a subject can only read/write an object if the current security level of the subject dominates (>=) the classification of the object no write down (*-property) a subject can only append/write to an object if the current security level of the subject is dominated by (<=) the classification of the object Cryptography and Network Security 720 Reference Monitor Cryptography and Network Security 721 Evaluated Computer Systems governments can evaluate IT systems against a range of standards: TCSEC, IPSEC and now Common Criteria define a number of “levels” of evaluation with increasingly stringent checking have published lists of evaluated products though aimed at government/defense use can be useful in industry also Cryptography and Network Security 722 Summary have considered: firewalls types of firewalls configurations access control trusted systems Cryptography and Network Security 723 Cryptography and Network Security Third Edition by William Stallings Lecture slides by Lawrie Brown Cryptography and Network Security 724 Intruders They agreed that Graham should set the test for Charles Mabledene. It was neither more nor less than that Dragon should get Stern's code. If he had the 'in' at Utting which he claimed to have this should be possible, only loyalty to Moscow Centre would prevent it. If he got the key to the code he would prove his loyalty to London Central beyond a doubt. —Talking to Strange Men, Ruth Rendell Cryptography and Network Security 725 Intruders significant issue for networked systems is hostile or unwanted access either via network or local can identify classes of intruders: masquerader misfeasor clandestine user varying levels of competence Cryptography and Network Security 726 Intruders clearly a growing publicized problem from “Wily Hacker” in 1986/87 to clearly escalating CERT stats may seem benign, but still cost resources may use compromised system to launch other attacks Cryptography and Network Security 727 Intrusion Techniques aim to increase privileges on system basic attack methodology target acquisition and information gathering initial access privilege escalation covering tracks key goal often is to acquire passwords so then exercise access rights of owner Cryptography and Network Security 728 Password Guessing one of the most common attacks attacker knows a login (from email/web page etc) then attempts to guess password for it try default passwords shipped with systems try all short passwords then try by searching dictionaries of common words intelligent searches try passwords associated with the user (variations on names, birthday, phone, common words/interests) before exhaustively searching all possible passwords check by login attempt or against stolen password file success depends on password chosen by user surveys show many users choose poorly Cryptography and Network Security 729 Password Capture another attack involves password capture watching over shoulder as password is entered using a trojan horse program to collect monitoring an insecure network login (eg. telnet, FTP, web, email) extracting recorded info after successful login (web history/cache, last number dialed etc) using valid login/password can impersonate user users need to be educated to use suitable precautions/countermeasures Cryptography and Network Security 730 Intrusion Detection inevitably will have security failures so need also to detect intrusions so can block if detected quickly act as deterrent collect info to improve security assume intruder will behave differently to a legitimate user but will have imperfect distinction between Cryptography and Network Security 731 Approaches to Intrusion Detection statistical anomaly detection threshold profile based rule-based detection anomaly penetration identification Cryptography and Network Security 732 Audit Records fundamental tool for intrusion detection native audit records part of all common multi-user O/S already present for use may not have info wanted in desired form detection-specific audit records created specifically to collect wanted info at cost of additional overhead on system Cryptography and Network Security 733 Statistical Anomaly Detection threshold detection count occurrences of specific event over time if exceed reasonable value assume intrusion alone is a crude & ineffective detector profile based characterize past behavior of users detect significant deviations from this profile usually multi-parameter Cryptography and Network Security 734 Audit Record Analysis foundation of statistical approaches analyze records to get metrics over time counter, gauge, interval timer, resource use use various tests on these to determine if current behavior is acceptable mean & standard deviation, multivariate, markov process, time series, operational key advantage is no prior knowledge used Cryptography and Network Security 735 Rule-Based Intrusion Detection observe events on system & apply rules to decide if activity is suspicious or not rule-based anomaly detection analyze historical audit records to identify usage patterns & auto-generate rules for them then observe current behavior & match against rules to see if conforms like statistical anomaly detection does not require prior knowledge of security flaws Cryptography and Network Security 736 Rule-Based Intrusion Detection rule-based penetration identification uses expert systems technology with rules identifying known penetration, weakness patterns, or suspicious behavior rules usually machine & O/S specific rules are generated by experts who interview & codify knowledge of security admins quality depends on how well this is done compare audit records or states against rules Cryptography and Network Security 737 Base-Rate Fallacy practically an intrusion detection system needs to detect a substantial percentage of intrusions with few false alarms if too few intrusions detected -> false security if too many false alarms -> ignore / waste time this is very hard to do existing systems seem not to have a good record Cryptography and Network Security 738 Distributed Intrusion Detection traditional focus is on single systems but typically have networked systems more effective defense has these working together to detect intrusions issues dealing with varying audit record formats integrity & confidentiality of networked data centralized or decentralized architecture Cryptography and Network Security 739 Distributed Intrusion Detection Architecture Cryptography and Network Security 740 Distributed Intrusion Detection – Agent Implementation Cryptography and Network Security 741 Honeypots decoy systems to lure attackers away from accessing critical systems to collect information of their activities to encourage attacker to stay on system so administrator can respond are filled with fabricated information instrumented to collect detailed information on attackers activities may be single or multiple networked systems Cryptography and Network Security 742 Password Management front-line defense against intruders users supply both: login – determines privileges of that user password – to identify them passwords often stored encrypted Unix uses multiple DES (variant with salt) more recent systems use crypto hash function Cryptography and Network Security 743 Managing Passwords need policies and good user education ensure every account has a default password ensure users change the default passwords to something they can remember protect password file from general access set technical policies to enforce good passwords minimum length (>6) require a mix of upper & lower case letters, numbers, punctuation block know dictionary words Cryptography and Network Security 744 Managing Passwords may reactively run password guessing tools note that good dictionaries exist for almost any language/interest group may enforce periodic changing of passwords have system monitor failed login attempts, & lockout account if see too many in a short period do need to educate users and get support balance requirements with user acceptance be aware of social engineering attacks Cryptography and Network Security 745 Proactive Password Checking most promising approach to improving password security allow users to select own password but have system verify it is acceptable simple rule enforcement (see previous slide) compare against dictionary of bad passwords use algorithmic (markov model or bloom filter) to detect poor choices Cryptography and Network Security 746 Summary have considered: problem of intrusion intrusion detection (statistical & rule-based) password management Cryptography and Network Security 747 Cryptography and Network Security 748

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# Chapter 9 slides, 3rdedition