Chapter 8: Cryptography 06/19/23 Cryptography 1

Chapter 8: Cryptography 06/19/23 Cryptography 1

Lecture Materials A few slides are adapted from the slides copyrighted by Jim Kurose, Keith Ross Addison-Wesley, Pearson Education2010. Computer Networking: A Top Down Approach Featuring the Internet, 5th edition. 2

Friends and enemies: Alice, Bob, Trudy well-known in network security world Bob, Alice (lovers!) want to communicate “securely” Eve (or Trudy, intruder) may intercept, delete, add messages Alice channel data secure sender Bob data, control messages secure receiver data Eve 7-3

The language of cryptography Alice’s K encryption A key plaintext encryption algorithm Bob’s K decryption B key ciphertext decryption algorithm plaintext symmetric key crypto: sender, receiver keys identical public-key crypto: encryption key public, decryption key secret (private) Network Security 7-4

Symmetric Cryptosystem Scenario – Alice wants to send a message (plaintext P) to Bob. – The communication channel is insecure and can be eavesdropped – If Alice and Bob have previously agreed on a symmetric encryption scheme and a secret key K, the message can be sent encrypted (ciphertext C) Issues – What is a good symmetric encryption scheme? – What is the complexity of encrypting/decrypting? – What is the size of the ciphertext, relative to the plaintext? P K 06/19/23 P C K Cryptography 5

Basics Notation – – – – – Secret key K Encryption function EK(P) Decryption function DK(C) Plaintext length typically the same as ciphertext length Encryption and decryption are one-one mapping functions on the set of all n-bit arrays Efficiency – functions EK and DK should have efficient algorithms Consistency – Decrypting the ciphertext yields the plaintext – DK(EK(P)) P 06/19/23 Cryptography 6

Attacks Attacker may have a) b) c) d) 06/19/23 collection of ciphertexts (ciphertext only attack) collection of plaintext/ciphertext pairs (known plaintext attack) collection of plaintext/ciphertext pairs for plaintexts selected by the attacker (chosen plaintext attack) collection of plaintext/ciphertext pairs for ciphertexts selected by the attacker (chosen ciphertext attack) Encryption Algorithm Plaintext (a) Hi, Bob. Hi, Bob. Don’t Don’t invite Eve toinvite the Eve to the party! party! Love, Alice Love, Alice Ciphertext key Eve Encryption Algorithm Plaintext (b) Hi, Bob. Hi, Bob. Don’t Don’t invite Eve toinvite the Eve to the party! party! Love, Alice Love, Alice key Eve Plaintext (c) ABCDEFG ABCDEFG HIJKLMNO HIJKLMNO PQRSTUV PQRSTUV WXYZ. WXYZ. key Encryption Algorithm IJCGA, IJCGA, CAN DO CAN GOT DO HIFFA HIFFA GOT TIME. TIME. Cryptography Ciphertext Encryption Algorithm Plaintext (d) Ciphertext Eve Ciphertext 001101 110111 key Eve Eve 7

Brute-Force Attack Try all possible keys K and determine if DK(C) is a likely plaintext – Requires some knowledge of the structure of the plaintext (e.g., PDF file or email message) Key should be a sufficiently long random value to make exhaustive search attacks unfeasible 06/19/23 Cryptography Image by Michael Cote from http://commons.wikimedia.org/wiki/File:Bingo cards.jpg 8

Classical Cryptography Transposition Cipher Substitution Cipher – Simple substitution cipher (Caesar cipher) – Vigenere cipher – One-time pad Network Security 7-9

Transposition Cipher: rail fence Write plaintext in two rows Generate ciphertext in column order Example: “HELLOWORLD” HLOOL ELWRD ciphertext: HLOOLELWRD Problem: does not affect the frequency of individual symbols Network Security 7-10

Substitution Ciphers Each letter is uniquely replaced by another. There are 26! possible substitution ciphers for English language. There are more than 4.03 x 1026 such ciphers. One popular substitution “cipher” for some Internet posts is ROT13. Public domain image from http://en.wikipedia.org/wiki/File:ROT13.png 06/19/23 Cryptography 11

Frequency Analysis Letters in a natural language, like English, are not uniformly distributed. Knowledge of letter frequencies, including pairs and triples can be used in cryptologic attacks against substitution ciphers. 06/19/23 Cryptography 12

Distribution of Letters in English Frequency analysis Network Security 7-13

Simple substitution cipher substituting one thing for another – Simplest one: monoalphabetic cipher: substitute one letter for another (Caesar Cipher) ABCDEFGHIJKLMNOPQRSTUVWXYZ DEFGHIJKLMNOPQRSTUVWXYZABC Example: encrypt “I attack” Network Security 7-14

Vigenere Cipher Idea: Uses Caesar's cipher with various different shifts, in order to hide the distribution of the letters. A key defines the shift used in each letter in the text A key word is repeated as many times as required to become the same length Plain text: I a t t a c k Key: 2342342 Cipher text: K d x v d g m Network Security (key is “234”) 7-15

Problem of Vigenere Cipher Vigenere is easy to break (Kasiski, 1863): Assume we know the length of the key. We can organize the ciphertext in rows with the same length of the key. Then, every column can be seen as encrypted using Caesar's cipher. The length of the key can be found using several methods: – 1. If short, try 1, 2, 3, . . . . – 2. Find repeated strings in the ciphertext. Their distance is expected to be a multiple of the length. Compute the gcd of (most) distances. – 3. Use the index of coincidence. Network Security 7-16

Substitution Boxes Substitution can also be done on binary numbers. Such substitutions are usually described by substitution boxes, or S-boxes. 06/19/23 Cryptography 17

One-Time Pads Extended from Vigenere cipher There is one type of substitution cipher that is absolutely unbreakable. – The one-time pad was invented in 1917 by Joseph Mauborgne and Gilbert Vernam – We use a block of shift keys, (k1, k2, . . . , kn), to encrypt a plaintext, M, of length n, with each shift key being chosen uniformly at random. Since each shift is random, every ciphertext is equally likely for any plaintext. 06/19/23 Cryptography 18

Weaknesses of the One-Time Pad In spite of their perfect security, one-time pads have some weaknesses The key has to be as long as the plaintext Keys can never be reused – Repeated use of one-time pads allowed the U.S. to break some of the communications of Soviet spies during the Cold War. 06/19/23 Cryptography 19 Public domain declassified government image from https://www.cia.gov/library/center-for-the-study-of-intelligence/csi-publications/books-and-monographs/venona-soviet-espionage-and-the-american-response-1939-1957/part2.htm

Block Ciphers In a block cipher: – Plaintext and ciphertext have fixed length b (e.g., 128 bits) – A plaintext of length n is partitioned into a sequence of m blocks, P[0], , P[m 1], where n bm n b Each message is divided into a sequence of blocks and encrypted or decrypted in terms of its blocks. Requires padding with extra bits. Plaintext Blocks of plaintext 06/19/23 Cryptography 20

Padding Block ciphers require the length n of the plaintext to be a multiple of the block size b Padding the last block needs to be unambiguous (cannot just add zeroes) When the block size and plaintext length are a multiple of 8, a common padding method (PKCS5) is a sequence of identical bytes, each indicating the length (in bytes) of the padding Example for b 128 (16 bytes) – Plaintext: “Roberto” (7 bytes) – Padded plaintext: “Roberto999999999” (16 bytes), where 9 denotes the number and not the character We need to always pad the last block, which may consist only of padding 06/19/23 Cryptography 21

Block Ciphers in Practice Data Encryption Standard (DES) – Developed by IBM and adopted by NIST in 1977 – 64-bit blocks and 56-bit keys – Small key space makes exhaustive search attack feasible since late 90s Triple DES (3DES) – – – – Nested application of DES with three different keys KA, KB, and KC Effective key length is 168 bits, making exhaustive search attacks unfeasible C EKC(DKB(EKA(P))); P DKA(EKB(DKC(C))) Equivalent to DES when KA KB KC (backward compatible) Advanced Encryption Standard (AES) – Selected by NIST in 2001 through open international competition and public discussion – 128-bit blocks and several possible key lengths: 128, 192 and 256 bits – Exhaustive search attack not currently possible – AES-256 is the symmetric encryption algorithm of choice 06/19/23 Cryptography 22

Symmetric key crypto: DES DES operation initial permutation 16 identical “rounds” of function application, each using different 48 bits of key final permutation Network Security 7-23

The Advanced Encryption Standard (AES) In 1997, the U.S. National Institute for Standards and Technology (NIST) put out a public call for a replacement to DES. It narrowed down the list of submissions to five finalists, and ultimately chose an algorithm that is now known as the Advanced Encryption Standard (AES). AES is a block cipher that operates on 128-bit blocks. It is designed to be used with keys that are 128, 192, or 256 bits long, yielding ciphers known as AES-128, AES-192, and AES-256. 06/19/23 Cryptography 24

AES Round Structure The 128-bit version of the AES encryption algorithm proceeds in ten rounds. Each round performs an invertible transformation on a 128-bit array, called state. The initial state X0 is the XOR of the plaintext P with the key K: X0 P XOR K. Round i (i 1, , 10) receives state Xi-1 as input and produces state Xi. The ciphertext C is the output of the final round: C X10. 06/19/23 Cryptography 25

AES Rounds Each round is built from four basic steps: 1.SubBytes step: an S-box substitution step 2.ShiftRows step: a permutation step 3.MixColumns step: a matrix multiplication step 4.AddRoundKey step: an XOR step with a round key derived from the 128-bit encryption key 06/19/23 Cryptography 26

Block Cipher Modes A block cipher mode describes the way a block cipher encrypts and decrypts a sequence of message blocks. Electronic Code Book (ECB) Mode (is the simplest): – Block P[i] encrypted into ciphertext block C[i] EK(P[i]) – Block C[i] decrypted into plaintext block M[i] DK(C[i]) 06/19/23 Cryptography Public domain images from http://en.wikipedia.org/wiki/File:Ecb encryption.png and http://en.wikipedia.org/wiki/File:Ecb decryption.png 27

Strengths and Weaknesses of ECB Weakness: Strengths: – Is very simple – Allows for parallel encryptions of the blocks of a plaintext – Can tolerate the loss or damage of a block 06/19/23 Cryptography – Documents and images are not suitable for ECB encryption since patters in the plaintext are repeated in the ciphertext: 28

Another Example t 1 P(1) “HTTP/1.1” t 17 06/19/23 P(17) “HTTP/1.1” block cipher C(1) “k329aM02” block cipher C(17) “k329aM02” Cryptography 29

Cipher Block Chaining (CBC) Mode In Cipher Block Chaining (CBC) Mode – The previous ciphertext block is combined with the current plaintext block C[i] EK (C[i 1] P[i]) – C[ 1] V, a random block separately transmitted encrypted (known as the initialization vector) – Decryption: P[i] C[i 1] DK (C[i]) CBC Encryption: P[0] P[1] CBC Decryption: P[2] P[0] P[1] P[2] P[3] EK DK DK DK DK C[3] C[0] C[1] C[2] C[3] P[3] V V EK EK EK C[0] C[1] C[2] 06/19/23 Cryptography 30

Strengths and Weaknesses of CBC Strengths: Weaknesses: – Doesn’t show patterns in the plaintext – Is the most common mode – Is fast and relatively simple 06/19/23 Cryptography – CBC requires the reliable transmission of all the blocks sequentially – CBC is not suitable for applications that allow packet losses (e.g., music and video streaming) 31

Java AES Encryption Example Source http://java.sun.com/javase/6/docs/technotes/guides/security/crypto/CryptoSpec.html Generate an AES key KeyGenerator keygen KeyGenerator.getInstance("AES"); SecretKey aesKey keygen.generateKey(); Create a cipher object for AES in ECB mode and PKCS5 padding Cipher aesCipher; aesCipher Cipher.getInstance("AES/ECB/PKCS5Padding"); Encrypt aesCipher.init(Cipher.ENCRYPT MODE, aesKey); byte[] plaintext "My secret message".getBytes(); byte[] ciphertext aesCipher.doFinal(plaintext); Decrypt aesCipher.init(Cipher.DECRYPT MODE, aesKey); byte[] plaintext1 aesCipher.doFinal(ciphertext); 06/19/23 Cryptography 32

Hill Cipher: a cipher based on matrix multiplication Message P “ACTDOG”, use m 3 – – – – 06/19/23 Break into two blocks: “ACT”, and “DOG” 'A' is 0, 'C' is 2 and 'T' is 19, “ACT” is the vector: x Encryption key is a 3*3 matrix: K The cipher text of the first block is: c K x c ‘POH’ Cryptography 33

Hill Cipher If the first block plaintext is ‘CAT’ – – – – x c K x c ‘FIN’ The Hill cipher has achieved Shannon's diffusion, and an n-dimensional Hill cipher can diffuse fully across n symbols at once. – This and the previous slide’s examples are from Wikipedia http:// en.wikipedia.org/wiki/Hill cipher 06/19/23 Cryptography 34

Hill Cipher Decryption 06/19/23 Cryptography 35

Hill Cipher to Realize Transposition 06/19/23 Cryptography 36

Stream Cipher Key stream – Pseudo-random sequence of bits S S[0], S[1], S[2], – Can be generated on-line one bit (or byte) at the time Stream cipher – XOR the plaintext with the key stream C[i] S[i] P[i] – Suitable for plaintext of arbitrary length generated on the fly, e.g., media stream Synchronous stream cipher – Key stream obtained only from the secret key K Independent with plaintext and ciphertext – Works for high-error channels if plaintext has packets with sequence numbers – Sender and receiver must synchronize in using key stream – If a digit is corrupted in transmission, only a single digit in the plaintext is affected and the error does not propagate to other parts of the message. 06/19/23 Cryptography 37

Self-synchronizing stream cipher – Key stream obtained from the secret key and N previous ciphertexts – the receiver will automatically synchronize with the keystream generator after receiving N ciphertext digits, making it easier to recover if digits are dropped or added to the message stream. – Lost packets cause a delay of q steps before decryption resumes – Single-digit errors are limited in their effect, affecting only up to N plaintext digits. 06/19/23 Cryptography 38

Key Stream Generation RC4 – – – – Designed in 1987 by Ron Rivest for RSA Security Trade secret until 1994 Uses keys with up to 2,048 bits Simple algorithm Block cipher in counter mode (CTR) – Use a block cipher with block size b – The secret key is a pair (K,t), where K is key and t (counter) is a b-bit value – The key stream is the concatenation of ciphertexts EK (t), EK (t 1), EK (t 2), – Can use a shorter counter concatenated with a random value – Synchronous stream cipher 06/19/23 Cryptography 39

Attacks on Stream Ciphers Repetition attack – if key stream reused, attacker obtains XOR of two plaintexts (why?) 06/19/23 Cryptography 40

Public Key Encryption 06/19/23 Cryptography 41

Public Key Cryptography symmetric key crypto requires sender, receiver know shared secret key Q: how to agree on key in first place (particularly if never “met”)? public key cryptography radically different approach [DiffieHellman76, RSA78] sender, receiver do not share secret key public encryption key known to all private decryption key known only to receiver – Typical chicken and egg dilemma. Network Security 7-42

Public key cryptography Bob’s public B key K K plaintext, P encryption algorithm ciphertext C EK (P) B Network Security - Bob’s private B key decryption algorithm Plaintext, P - P DK (C) B 7-43

Facts About Numbers Prime number p: – p is an integer – p 2 – The only divisors of p are 1 and p Examples – 2, 7, 19 are primes – 3, 0, 1, 6 are not primes Prime decomposition of a positive integer n: n p1e1 pkek Example: – 200 23 52 Fundamental Theorem of Arithmetic The prime decomposition of a positive integer is unique 06/19/23 Cryptography 44

Greatest Common Divisor The greatest common divisor (GCD) of two positive integers a and b, denoted gcd(a, b), is the largest positive integer that divides both a and b The above definition is extended to arbitrary integers Examples: gcd(18, 30) 6 gcd( 21, 49) 7 gcd(0, 20) 20 Two integers a and b are said to be relatively prime if gcd(a, b) 1 Example: – Integers 15 and 28 are relatively prime 06/19/23 Cryptography 45

Modular Arithmetic Modulo operator for a positive integer n r a mod n equivalent to a r kn and r a a/n n Example: 29 mod 13 3 13 mod 13 0 1 mod 13 12 29 3 2 13 13 0 1 13 12 1 1 13 For a 0, we first add a large kn to a such that it becomes positive Modulo and GCD: gcd(a, b) gcd(b, a mod b) Example: gcd(21, 12) 3 06/19/23 gcd(12, 21 mod 12) gcd(12, 9) 3 Cryptography 46

Euclid’s GCD Algorithm Euclid’s algorithm for computing the GCD repeatedly applies the formula gcd(a, b) gcd(b, a mod b) Example Algorithm EuclidGCD(a, b) Input integers a and b Output gcd(a, b) – gcd(412, 260) 4 06/19/23 Cryptography if b 0 return a else return EuclidGCD(b, a mod b) 47

RSA: Choosing keys 1. Choose two large prime numbers p, q. (e.g., 1024 bits each) 2. Compute n pq, z (p-1)(q-1) 3. Choose e (with e n) that has no common factors with z. (e, z are “relatively prime”). 4. Choose d such that ed-1 is exactly divisible by z. (in other words: ed mod z 1 ). 5. Public key is (n,e). Private key is (n,d). - KB KB Network Security 7-48

RSA: Encryption, decryption 0. Given (n,e) and (n,d) as computed above 1. To encrypt bit pattern, m, compute e e (i.e., remainder when m is divided by n) c m mod n 2. To decrypt received bit pattern, c, compute dis divided by n) d (i.e., remainder when c m c mod n Magic happens! e n) m (m mod d mod n c Network Security 7-49

RSA example: Bob chooses p 5, q 7. Then n 35, z 24. e 5 (so e, z relatively prime). d 29 (so ed-1 exactly divisible by z). encrypt: decrypt: letter m me l 12 1524832 c 17 c d 481968572106750915091411825223071697 c me mod n 17 m cd mod n 12 letter l Computational extensive Network Security 7-50

e n) m (m mod RSA: Why is that d mod n Useful number theory result: If p,q prime and n pq, then: y y mod (p-1)(q-1) x mod n x mod n e (m mod n) d ed mod n m mod n ed mod (p-1)(q-1) m mod n (using number theory result above) 1 m mod n (since we chose ed to be divisible by (p-1)(q-1) with remainder 1 ) m Network Security 7-51

RSA: another important property The following property will be very useful later: use public key first, followed by private key use private key first, followed by public key Result is the same! Network Security 7-52

RSA Cryptosystem Setup: – n pq, with p and q primes – e relatively prime to (n) (p 1) (q 1) – d inverse of e in Z (n) Example p 7, q 17 n 7 17 119 (n) 6 16 96 e 5 d 77 ed mod z 1 Keys: – Public key: KE (n, e) – Private key: KD d Keys: public key: (119, 5) private key: 77 Encryption: – Plaintext M in Zn – C Me mod n Encryption: M 19 C 195 mod 119 66 Decryption: – M Cd mod n 06/19/23 Setup: Decryption: C 6677 mod 119 19 Cryptography 53

Complete RSA Example Setup: Encryption C M3 mod 55 – p 5, q 11 – n 5 11 55 – (n) 4 10 40 – e 3 – d 27 3 27 81 2 40 1) Decryption M C27 mod 55 Pre-compute lookup table (size of n-1, M should not be 0) Why? M C M C M C 1 1 19 39 37 53 06/19/23 2 8 20 25 38 37 3 27 21 21 39 29 4 9 22 33 40 35 5 15 23 12 41 6 6 51 24 19 42 3 7 13 25 5 43 32 8 17 26 31 44 44 9 14 27 48 45 45 Cryptography 10 10 28 7 46 41 11 11 29 24 47 38 12 23 30 50 48 42 13 52 31 36 49 4 14 49 32 43 50 40 15 20 33 22 51 46 16 26 34 34 52 28 17 18 35 30 53 47 54 18 2 36 16 54 54

Security Security of RSA based on difficulty of factoring of n pq – Widely believed – Best known algorithm takes exponential time RSA Security factoring challenge (discontinued) In 1999, 512-bit challenge factored in 4 months using 35.7 CPU-years – – – – 06/19/23 In 2005, a team of researchers factored the RSA-640 challenge number using 30 2.2GHz CPU years In 2004, the prize for factoring RSA2048 was 200,000 Current practice is 2,048-bit keys Estimated resources needed to factor a number within one year 160 175-400 MHz SGI and Sun 8 250 MHz SGI Origin 120 300-450 MHz Pentium II 4 500 MHz Digital/Compaq Cryptography 55

Algorithmic Issues The implementation of the RSA cryptosystem requires various algorithms Overall Setup –Representation of integers of arbitrarily large size and arithmetic operations on them Encryption –Modular power Decryption –Modular power 06/19/23 Cryptography –Generation of random numbers with a given number of bits (to generate candidates p and q) –Primality testing (to check that candidates p and q are prime) –Computation of the GCD (to verify that e and (n) are relatively prime) –Computation of the multiplicative inverse (to compute d from e) 56

Modular Power The repeated squaring algorithm speeds up the computation of a modular power ap mod n Write the exponent p in binary p pb 1 pb 2 p1 p0 Start with Q1 apb 1 mod n Repeatedly compute Qi ((Qi 1)2 mod n)apb i mod n We obtain Qb ap mod n The repeated squaring algorithm performs O (log p) arithmetic operations 06/19/23 Example Cryptography –318 mod 19 (18 010) –Q1 31 mod 19 3 –Q2 32 mod 19)30 mod 19 9 –Q3 92 mod 19)30 mod 19 81 mod 19 5 –Q4 52 mod 19)31 mod 19 (25 mod 19)3 mod 19 18 mod 19 18 –Q5 182 mod 19)30 mod 19 (324 mod 19) mod 19 (17 19 1) mod 19 1 57

Cryptographic Hash Functions 06/19/23 Cryptography 58

Hash Functions A hash function h maps a plaintext x to a fixedlength value x h(P) called hash value or digest of P – Usually x is much smaller in size compared to P. – A collision is a pair of plaintexts P and Q that map to the same hash value, h(P) h(Q) – Collisions are unavoidable – For efficiency, the computation of the hash function should take time proportional to the length of the input plaintext 06/19/23 Cryptography 59

Simplex Example of Hash Functions Parity bit: map a binary bit stream to ‘1’ or ‘0’ – Hash value space is only 2. Repeated addition of n-byte chunks without considering carry-on bits – Hash value space is 28n 06/19/23 Cryptography 60

Cryptographic Hash Functions A cryptographic hash function satisfies additional properties – Preimage resistance (aka one-way) Given a hash value x, it is hard to find a plaintext P such that h(P) x – Second preimage resistance (aka weak collision resistance) Given a plaintext P, it is hard to find a plaintext Q such that h(Q) h(P) – Collision resistance (aka strong collision resistance) It is hard to find a pair of plaintexts P and Q such that h(Q) h(P) Collision resistance implies second preimage resistance Hash values of at least 256 bits recommended to defend against brute-force attacks 06/19/23 Cryptography 61

Hash Function Algorithms MD5 hash function widely used (RFC 1321) – computes 128-bit message digest in 4-step process. – arbitrary 128-bit string x, appears difficult to construct msg m whose MD5 hash is equal to x. SHA-1 is also used. – US standard [NIST, FIPS PUB 180-1] – 160-bit message digest There are many hash functions, but most of them do not satisfy cryptographic hash function requirements – example: checksum Network Security 7-62

Message-Digest Algorithm 5 (MD5) Developed by Ron Rivest in 1991 Uses 128-bit hash values Still widely used in legacy applications although considered insecure Various severe vulnerabilities discovered Chosen-prefix collisions attacks found by Marc Stevens, Arjen Lenstra and Benne de Weger – Start with two arbitrary plaintexts P and Q – One can compute suffixes S1 and S2 such that P S1 and Q S2 collide under MD5 by making 250 hash evaluations – Using this approach, a pair of different executable files or PDF documents with the same MD5 hash can be computed 06/19/23 Cryptography 63

Secure Hash Algorithm (SHA) Developed by NSA and approved as a federal standard by NIST SHA-0 and SHA-1 (1993) – – – – 160-bits Considered insecure Still found in legacy applications Vulnerabilities less severe than those of MD5 SHA-2 family (2002) – 256 bits (SHA-256) or 512 bits (SHA-512) – Still considered secure despite published attack techniques Public competition for SHA-3 announced in 2007 06/19/23 Cryptography 64

Iterated Hash Function A compression function works on input values of fixed length An iterated hash function extends a compression function to inputs of arbitrary length – padding, initialization vector, and chain of compression functions – inherits collision resistance of compression function MD5 and SHA are iterated hash functions IV P1 P2 P3 P4 digest SHA-1 MD5 msec Hashing Time 0.06 0.05 0.04 0.03 0.02 0.01 0 0 06/19/23 100 200 300 400 500 600 Input Size (Bytes) Cryptography 700 800 900 1000 65

Birthday Attack The brute-force birthday attack aims at finding a collision for a hash function h – Randomly generate a sequence of plaintexts X1, X2, X3, – For each Xi compute yi h(Xi) and test whether yi yj for some j i – Stop as soon as a collision has been found If there are m possible hash values, the probability that the i-th plaintext does not collide with any of the previous i 1 plaintexts is 1 (i 1)/m The probability Fk that the attack fails (no collisions) after k plaintexts is Fk (1 1/m) (1 2/m) (1 3/m) (1 k 1)/m) Using the standard approximation 1 x e x Fk e (1/m 2/m 3/m (k 1)/m) e k(k 1)/2m The attack succeeds/fails with probability ½ when Fk ½ , that is, e k(k 1)/2m ½ k 1.17 m½ We conclude that a hash function with b-bit values provides about b/2 bits of security 06/19/23 Cryptography 66