YSLInformation Security -- Public-Key Cryptography1 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.

YSLInformation Security -- Public-Key Cryptography2 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.

YSLInformation Security -- Public-Key Cryptography3 Prime and Relatively Prime Numbers (cont’d)

YSLInformation Security -- Public-Key Cryptography4 Prime and Relatively Prime Numbers (cont’d) Table 7.1 Primes under 2000

YSLInformation Security -- Public-Key Cryptography5 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.

YSLInformation Security -- Public-Key Cryptography6 Prime and Relatively Prime Numbers (cont’d)

YSLInformation Security -- Public-Key Cryptography7 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 2 n - 2 then n is prime. For n = 3, 3 divides 6 and n is prime. However, For n = 341 = 11  31, n dives Mersenne suggested that if p is prime then M p = 2 p - 1 is prime. This type of primes are referred to as Mersenne primes. Unfortunately, for p = 11, M 11 = = 2047 = 23  89.

YSLInformation Security -- Public-Key Cryptography8 Prime and Relatively Prime Numbers (cont’d) Fermat conjectured that if F n = 2 2 n + 1, where n is a non-negative integer, then F n is prime. When n is less than or equal to 4, F 0 = 3, F 1 = 5, F 2 = 17, F 3 = 257 and F 4 = are all primes. However, F 5 = = 641  is not a prime bumber. n n 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.

YSLInformation Security -- Public-Key Cryptography9 Prime and Relatively Prime Numbers (cont’d) Factorization of an integer as a product of prime numbers Example: 91 = 7  13; = 7  11 2  13. Useful for checking divisibility and relative primality to be discussed later. Factorization is in gereral difficult.

YSLInformation Security -- Public-Key Cryptography10 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|)

YSLInformation Security -- Public-Key Cryptography11 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

YSLInformation Security -- Public-Key Cryptography12 Modular Arithmetic

YSLInformation Security -- Public-Key Cryptography13 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.

YSLInformation Security -- Public-Key Cryptography14 Modular Arithmetic (cont’d)

YSLInformation Security -- Public-Key Cryptography15 Modular Arithmetic (cont’d) Properties of modular arithmetic operations Proof of Property 1: Define (a mod n) = r a and (b mod n) = r b. Then a = r a + jn and b = r b + kn for some integers j and k. Then, (a+b) mod n = (r a + jn + r b + kn) mod n = (r a + r b + (j + k)n) mod n = (r a + r b ) mod n = [(a mod n) + (b mod n)] mod n

YSLInformation Security -- Public-Key Cryptography16 Modular Arithmetic (cont’d)  Examples for the above three properties

YSLInformation Security -- Public-Key Cryptography17 Modular Arithmetic (cont’d) Properties of modular arithmetic –Let Z n = {0,1,2,…,(n-1)} be the set of residues modulo n.

YSLInformation Security -- Public-Key Cryptography18 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).

YSLInformation Security -- Public-Key Cryptography19 Modular Arithmetic (cont’d) Properties of modular arithmetic (cont’d)

YSLInformation Security -- Public-Key Cryptography20 Fermat’s and Euler’s Theorems Fermat’s theorem

YSLInformation Security -- Public-Key Cryptography21 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 a p  a mod p example: p = 5, a = 3, 3 5 = 243  3 mod 5

YSLInformation Security -- Public-Key Cryptography22 Fermat’s and Euler’s Theorems (cont’d) Euler’s totient function

YSLInformation Security -- Public-Key Cryptography23 Fermat’s and Euler’s Theorems (cont’d)

YSLInformation Security -- Public-Key Cryptography24 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)

YSLInformation Security -- Public-Key Cryptography25 Fermat’s and Euler’s Theorems (cont’d) Euler’s theorem

YSLInformation Security -- Public-Key Cryptography26 Fermat’s and Euler’s Theorems (cont’d) Euler’s totient function (cont’d)

YSLInformation Security -- Public-Key Cryptography27 Testing for Primality If p is an odd prime, then the equation x 2  1 (mod p) has only two solutions, 1 and -1.

YSLInformation Security -- Public-Key Cryptography28 Testing for Primality (cont’d)

YSLInformation Security -- Public-Key Cryptography29 Testing for Primality (cont’d) Probabilistic primality test

YSLInformation Security -- Public-Key Cryptography30 Euclid’s Algorithm

YSLInformation Security -- Public-Key Cryptography31 Euclid’s Algorithm (cont’d)

YSLInformation Security -- Public-Key Cryptography32 Euclid’s Algorithm (cont’d)

YSLInformation Security -- Public-Key Cryptography33 Euclid’s Algorithm (cont’d)

YSLInformation Security -- Public-Key Cryptography34 Extended Euclid’s Algorithm

YSLInformation Security -- Public-Key Cryptography35 Chinese Remainder Theorem

YSLInformation Security -- Public-Key Cryptography36 Chinese Remainder Theorem (cont’d)

YSLInformation Security -- Public-Key Cryptography37 Discrete Logarithms

YSLInformation Security -- Public-Key Cryptography38 Discrete Logarithms (cont’d)