# Public Key Encryption

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1 Public Key Encryption 3/13/2012 Cryptography 1 Facts About Numbers Prime number p: p is an integer p 2 The only divisors of p are 1 and p s 2, 7, 19 are primes -3, 0, 1, 6 are not primes Prime decomposition of a positive integer n: n = p 1 e 1 p k e k : 200 = Fundamental Theorem of Arithmetic The prime decomposition of a positive integer is unique 3/13/2012 Cryptography 2 Symmetric Cryptography 1

2 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 s: gcd(18, 30) = 6 gcd(0, 20) = 20 gcd(-21, 49) = 7 Two integers a and b are said to be relatively prime if : gcd(a, b) = 1 Integers 15 and 28 are relatively prime 3/13/2012 Cryptography 3 Modular Arithmetic Modulo operator for a positive integer n, computes a remainder r: r = a mod n if there exists r and k s.t. 0 r < n and a = r + kn which is equivalent to: r = a - a/n n : 29 mod 13 = 3 13 mod 13 = 0-1 mod 13 = = = = Modulo and GCD: gcd(a, b) = gcd(b, a mod b) : gcd(21, 12) = 3 gcd(12, 21 mod 12) = gcd(12, 9) = 3 3/13/2012 Cryptography 4 Symmetric Cryptography 2

3 Euclid s GCD Algorithm Euclid s algorithm for computing the GCD repeatedly applies the formula gcd(a, b) = gcd(b, a mod b) gcd(412, 260) = 4 Algorithm EuclidGCD(a, b) Input integers a and b Output gcd(a, b) if b = 0 return a else return EuclidGCD(b, a mod b) a b /13/2012 Cryptography 5 Analysis of Running Time Let a i and b i be the arguments of the i-th recursive call of algorithm EuclidGCD We have a i + 2 = b i + 1 = a i mod a i + 1 < a i + 1 Sequence a 1, a 2,, a n decreases exponentially, namely a i + 2 ½ a i for i > 1 Case 1 a i + 1 ½ a i a i + 2 < a i + 1 ½ a i Case 2 a i + 1 > ½ a i a i + 2 = a i mod a i + 1 = a i - a i + 1 ½ a i Thus, the maximum number of recursive calls of algorithm EuclidGCD(a, b) is log min(a, b) => Running time is linear in the number of bits describing a and b 3/13/2012 Cryptography 6 Symmetric Cryptography 3

4 Multiplicative Inverses (1) The residues modulo a positive integer n are the set Z n = {0, 1, 2,, (n - 1)} Let x and y be two elements of Z n such that xy mod n = 1 We say that y is the multiplicative inverse of x in Z n and we write y = x -1 : Multiplicative inverses of the residues modulo 11 x x /13/2012 Cryptography 7 Multiplicative Inverses (2) Theorem An element x of Z n has a multiplicative inverse if and only if x and n are relatively prime The elements of Z 10 with a multiplicative inverse are 1, 3, 7, 9 x x Corollary If is p is prime, every nonzero residue in Z p has a multiplicative inverse 3/13/2012 Cryptography 8 Symmetric Cryptography 4

5 Computing Multiplicative Inverses: Extended Euclidean Algorithm Euclid GCD algorithm on input (a,b) computes n = gcd(a,b) A simple variant of EuclidGCD algorithm, called Extended Euclid algorithm, on inputs (a,b) outputs also n=gcd(a,b) together with integers z,k s.t. n = z*a + k*b Note that if x is co-prime to n then ExtEucGCD(a,b,c) outputs (n,z,k) s.t. n=1, because gcd(x,n)=1, and therefore z,k satisfy: 1 = z*x + k*b Therefore z = x -1 mod b, i.e. z is a multiplicative inverse x modulo b. Therefore computing multiplicative inverses is as easy as computing GCD. 3/13/2012 Cryptography 9 Modular Powers: We observe interesting cycles Let p be a prime A sequence of successive powers of any element x in Z p exhibit repeating subsequences, i.e. cycles (p = 7) x x 2 x 3 x 4 x 5 x Notice: The sizes of the cycles, 1,3,6,3,6,2, are all divisors of p-1 3/13/2012 Cryptography 10 Symmetric Cryptography 5

6 Fermat s Little Theorem Theorem Let p be a prime. For each nonzero residue x of Z p, we have x p - 1 mod p = 1 (p = 5): 1 4 mod 5 = mod 5 = 16 mod 5 = mod 5 = 81 mod 5 = mod 5 = 256 mod 5 = 1 Corollary Let p be a prime. For each nonzero residue x of Z p, the multiplicative inverse of x is x p - 2 mod p Proof x(x p - 2 mod p) mod p = xx p - 2 mod p = x p - 1 mod p = 1 3/13/2012 Cryptography 11 Euler s Theorem The multiplicative group for Z n, denoted with Z* n, is the subset of elements of Z n relatively prime with n The totient function of n, denoted with f(n), is the size of Z* n Z* 10 = { 1, 3, 7, 9 } f(10) = 4 If p is prime, we have Z* p = {1, 2,, (p - 1)} f(p) = p - 1 Euler s Theorem For each element x of Z* n, we have x f(n) mod n = 1 (n = 10) 3 f(10) mod 10 = 3 4 mod 10 = 81 mod 10 = 1 7 f(10) mod 10 = 7 4 mod 10 = 2401 mod 10 = 1 9 f(10) mod 10 = 9 4 mod 10 = 6561 mod 10 = 1 3/13/2012 Cryptography 12 Symmetric Cryptography 6

7 RSA Cryptosystem Setup: n = pq, with p and q primes e relatively prime to f(n) = (p - 1) (q - 1) d inverse of e in Z f(n) Keys: Public key: K E = (n, e) Private key: K D = d Encryption: Plaintext M in Z n C = M e mod n Decryption: M = C d mod n Setup: p = 7, q = 17 n = 7 17 = 119 f(n) = 6 16 = 96 e = 5 d = 77 Keys: public key: (119, 5) private key: 77 Encryption: M = 19 C = 19 5 mod 119 = 66 Decryption: C = mod 119 = 19 3/13/2012 Cryptography 13 Correctness We show the correctness of the RSA cryptosystem for the case when the plaintext M does not divide n Namely, we show that (M e ) d mod n = M Since ed mod f(n) = 1, there is an integer k such that ed = kf(n) + 1 Since M does not divide n, by Euler s theorem we have M f(n) mod n = 1 Thus, we obtain (M e ) d mod n = M ed mod n = M kf(n) + 1 mod n = MM kf(n) mod n = M (M f(n) ) k mod n = M (M f(n) mod n) k mod n = M (1) k mod n = M mod n = M Proof of correctness can be extended to the case when the plaintext M divides n 3/13/2012 Cryptography 14 Symmetric Cryptography 7

8 Complete RSA Example Setup: p = 5, q = 11 n = 5 11 = 55 f(n) = 4 10 = 40 e = 3 d = 27 (3 27 = 81 = ) Encryption C = M 3 mod 55 Decryption M = C 27 mod 55 M C M C M C /13/2012 Cryptography 15 Security of RSA based on difficulty of factoring 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 MHz SGI and Sun MHz SGI Origin MHz Pentium II MHz Digital/Compaq Security In 2005, a team of researchers factored the RSA-640 challenge number using GHz CPU years In 2004, the prize for factoring RSA was \$200,000 Current practice is 2,048-bit keys Estimated resources needed to factor a number within one year Length (bits) 3/13/2012 Cryptography 16 PCs Memory MB ,000 4GB 1, GB 1, TB Symmetric Cryptography 8

9 Algorithmic Issues The implementation of the RSA cryptosystem requires various algorithms Overall Representation of integers of arbitrarily large size and arithmetic operations on them Encryption Modular exponentiation Decryption Modular exponentiation Setup 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 f(n) are relatively prime) Computation of the multiplicative inverse (to compute d from e) 3/13/2012 Cryptography 17 Computing Modular Exponentiation: Efficient Algorithm The repeated squaring algorithm speeds up the computation of a modular power a p mod n Write the exponent p in binary p = p b - 1 p b - 2 p 1 p 0 Start with Q 1 = a p b - 1 mod n Repeatedly compute Q i = ((Q i - 1 ) 2 mod n)a p b - i mod n We obtain Q b = a p mod n The repeated squaring algorithm performs O (log p) arithmetic operations 3 18 mod 19 (18 = 10010) Q 1 = 3 1 mod 19 = 3 Q 2 = (3 2 mod 19)3 0 mod 19 = 9 Q 3 = (9 2 mod 19)3 0 mod 19 = 81 mod 19 = 5 Q 4 = (5 2 mod 19)3 1 mod 19 = (25 mod 19)3 mod 19 = 18 mod 19 = 18 Q 5 = (18 2 mod 19)3 0 mod 19 = (324 mod 19) mod 19 = mod 19 = 1 p 5 - i p 5 - i Q i /13/2012 Cryptography 18 Symmetric Cryptography 9

10 Modular Inverse Theorem Given positive integers a and b, let d be the smallest positive integer such that d = ia + jb for some integers i and j. We have d = gcd(a,b) a = 21 b = 15 d = 3 i = 3, j = -4 3 = (-4) 15 = = 3 Given positive integers a and b, the extended Euclid s algorithm computes a triplet (d,i,j) such that d = gcd(a,b) d = ia + jb To test the existence of and compute the inverse of x Z n, we execute the extended Euclid s algorithm on the input pair (x,n) Let (d,i,j) be the triplet returned d = ix + jn Case 1: d = 1 i is the inverse of x in Z n Case 2: d > 1 x has no inverse in Z n 3/13/2012 Cryptography 19 Pseudoprimality Testing The number of primes less than or equal to n is about n / ln n Thus, we expect to find a prime among O(b) randomly generated numbers with b bits each Testing whether a number is prime (primality testing) is a difficult problem, though polynomial-time algorithms exist An integer n 2 is said to be a base-x pseudoprime if x n - 1 mod n = 1 (Fermat s little theorem) Composite base-x pseudoprimes are rare: A random 100-bit integer is a composite base-2 pseudoprime with probability less than The smallest composite base-2 pseudoprime is 341 Base-x pseudoprimality testing for an integer n: Check whether x n - 1 mod n = 1 Can be performed efficiently with the repeated squaring algorithm 3/13/2012 Cryptography 20 Symmetric Cryptography 10

11 Randomized Primality Testing Compositeness witness function witness(x, n) with error probability q for a random variable x Case 1: n is prime witness(x, n) = false always Case 2: n is composite witness(x, n) = true in most cases, false with small probability q < 1 Algorithm RandPrimeTest tests whether n is prime by repeatedly evaluating witness(x, n) A variation of base- x pseudoprimality provides a suitable compositeness witness function for randomized primality testing (Rabin-Miller algorithm) Algorithm RandPrimeTest(n, k) Input integer n,confidence parameter k and composite witness function witness(x,n) with error probability q Output an indication of whether n is composite or prime with probability 2 -k t k/log 2 (1/q) for i 1 to t x random() if witness(x, n) = true return n is composite return n is prime 3/13/2012 Cryptography 21 Symmetric Cryptography 11

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