Prime and Relatively Prime Numbers Divisors: We say that b  0 divides a if a = mb for some m, where a, b and m are integers. b divides a if there is no remainder on division. The notation b|a is commonly used to mean that b divides a. If b|a, we say that b is a divisor of a. YSL Information Security -- Public-Key Cryptography

Prime and Relatively Prime Numbers (cont’d) If a|1, then a =  1. If a|b and b|a, then a =  b. Any b  0 divides 0. If b|g and b|h, then b|(mg + nh) for arbitrary integers m and n. YSL Information Security -- Public-Key Cryptography

Prime and Relatively Prime Numbers (cont’d) YSL Information Security -- Public-Key Cryptography

Prime and Relatively Prime Numbers (cont’d) Table 7.1 Primes under 2000 YSL Information Security -- Public-Key Cryptography

Prime and Relatively Prime Numbers (cont’d) The above statement is referred to as the prime number theorem, which was proven in 1896 by Hadaward and Poussin. YSL Information Security -- Public-Key Cryptography

Prime and Relatively Prime Numbers (cont’d) YSL Information Security -- Public-Key Cryptography

Prime and Relatively Prime Numbers (cont’d) Whether there exists a simple formula to generate prime numbers? An ancient Chinese mathematician conjectured that if n divides 2n - 2 then n is prime. For n = 3, 3 divides 6 and n is prime. However, For n = 341 = 11  31, n dives 2341 - 2. Mersenne suggested that if p is prime then Mp = 2p - 1 is prime. This type of primes are referred to as Mersenne primes. Unfortunately, for p = 11, M11 = 211 -1 = 2047 = 23  89. YSL Information Security -- Public-Key Cryptography

Prime and Relatively Prime Numbers (cont’d) Fermat conjectured that if Fn = 22n + 1, where n is a non-negative integer, then Fn is prime. When n is less than or equal to 4, F0 = 3, F1 = 5, F2 = 17, F3 = 257 and F4 = 65537 are all primes. However, F5 = 4294967297 = 641  6700417 is not a prime bumber. n2 - 79n + 1601 is valid only for n < 80. There are an infinite number of primes of the form 4n + 1 or 4n + 3. There is no simple way so far to gererate prime numbers. YSL Information Security -- Public-Key Cryptography

Prime and Relatively Prime Numbers (cont’d) Factorization of an integer as a product of prime numbers Example: 91 = 7  13; 11011 = 7  112  13. Useful for checking divisibility and relative primality to be discussed later. Factorization is in gereral difficult. YSL Information Security -- Public-Key Cryptography

Prime and Relatively Prime Numbers (cont’d) Define notation gcd(a,b) to mean the greatest common divisor of a and b. The positive integer c is said to be the gcd of a and b if c|a and c|b any divisor of a and b is a dividor of c. Equivalently, gcd(a,b) = max[k, such that k|a and k|b] gcd(a,b) = gcd(-a,b) = gcd(a,-b) = gcd(-a,-b) =gcd(|a|,|b|) YSL Information Security -- Public-Key Cryptography

Prime and Relatively Prime Numbers (cont’d) gcd(a,0) = |a|. Factorization is one possible but in general inefficient way to calculate gcd. Whereas, Euclid‘s algorithm (to be discussed later) is more efficient. Relative primality the integers a and b are relatively prime if they have no prime factors in common or equivalently, their only common factor is 1 or equivalently, gcd(a,b) = 1 YSL Information Security -- Public-Key Cryptography

Information Security -- Public-Key Cryptography Modular Arithmetic YSL Information Security -- Public-Key Cryptography

Modular Arithmetic (cont’d) Examples: a = 11; n = 7; 11 = 1  7 + 4; r = 4. a = -11; n = 7; -11 = (-2)  7 + 3; r = 3. If a is an integer and n is a positive integer, define a mod n to be the remainder when a is divided by n. Then, a = a/n  n + (a mod n); Example: 11 mod 7 = 4; -11 mod 7 = 3. YSL Information Security -- Public-Key Cryptography

Modular Arithmetic (cont’d) YSL Information Security -- Public-Key Cryptography

Modular Arithmetic (cont’d) Properties of modular arithmetic operations Proof of Property 1: Define (a mod n) = ra and (b mod n) = rb. Then a = ra + jn and b = rb + kn for some integers j and k. Then, (a+b) mod n = (ra + jn + rb + kn) mod n = (ra + rb + (j + k)n) mod n = (ra + rb) mod n = [(a mod n) + (b mod n)] mod n YSL Information Security -- Public-Key Cryptography

Modular Arithmetic (cont’d)  Examples for the above three properties YSL Information Security -- Public-Key Cryptography

Modular Arithmetic (cont’d) Properties of modular arithmetic Let Zn = {0,1,2,…,(n-1)} be the set of residues modulo n. YSL Information Security -- Public-Key Cryptography

Modular Arithmetic (cont’d) Properties of modular arithmetic (cont’d) if (a + b)  (a + c) mod n, then b  c mod n (due to the existence of an additive inverse) if (a  b)  (a  c) mod n, then b  c mod n (only if a is relatively prime to n; due to the possible absence of a multiplicative inverse) e.g. 6  3 = 18  2 mod 8 and 6  7 = 42  2 mod 8 but 3  7 mod 8 (6 is not relatively prime to 8) If n is prime then the property of multiplicative inverse holds (from a ring to a field). YSL Information Security -- Public-Key Cryptography

Modular Arithmetic (cont’d) Properties of modular arithmetic (cont’d) YSL Information Security -- Public-Key Cryptography

Fermat’s and Euler’s Theorems YSL Information Security -- Public-Key Cryptography

Fermat’s and Euler’s Theorems (cont’d) Fermat’s theorem (cont’d) alternative form if p is prime and a is any positive integer, then ap  a mod p example: p = 5, a = 3, 35 = 243  3 mod 5 YSL Information Security -- Public-Key Cryptography

Fermat’s and Euler’s Theorems (cont’d) Euler’s totient function YSL Information Security -- Public-Key Cryptography

Fermat’s and Euler’s Theorems (cont’d) YSL Information Security -- Public-Key Cryptography

Fermat’s and Euler’s Theorems (cont’d) Euler’s totient function (cont’d) if n is the product of two primes p and q φ(n) = pq – [(q – 1)+(p –1) + 1] = pq – (p + q) + 1 = (p – 1)  (q – 1) = φ (p)  φ (q) YSL Information Security -- Public-Key Cryptography

Fermat’s and Euler’s Theorems (cont’d) YSL Information Security -- Public-Key Cryptography

Fermat’s and Euler’s Theorems (cont’d) Euler’s totient function (cont’d) YSL Information Security -- Public-Key Cryptography

Information Security -- Public-Key Cryptography Testing for Primality If p is an odd prime, then the equation x2  1 (mod p) has only two solutions, 1 and -1. YSL Information Security -- Public-Key Cryptography

Testing for Primality (cont’d) YSL Information Security -- Public-Key Cryptography

Testing for Primality (cont’d) Probabilistic primality test YSL Information Security -- Public-Key Cryptography

Information Security -- Public-Key Cryptography Euclid’s Algorithm YSL Information Security -- Public-Key Cryptography

Euclid’s Algorithm (cont’d) YSL Information Security -- Public-Key Cryptography

Euclid’s Algorithm (cont’d) YSL Information Security -- Public-Key Cryptography

Euclid’s Algorithm (cont’d) YSL Information Security -- Public-Key Cryptography

Extended Euclid’s Algorithm YSL Information Security -- Public-Key Cryptography

Chinese Remainder Theorem YSL Information Security -- Public-Key Cryptography

Chinese Remainder Theorem (cont’d) YSL Information Security -- Public-Key Cryptography

Information Security -- Public-Key Cryptography Discrete Logarithms YSL Information Security -- Public-Key Cryptography

Discrete Logarithms (cont’d) YSL Information Security -- Public-Key Cryptography