Prime and Relatively Prime Numbers

1. 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. Information Security -- Public-Key Cryptography

2. 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. Information Security -- Public-Key Cryptography

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

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

5. 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. Information Security -- Public-Key Cryptography

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

7. 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. Information Security -- Public-Key Cryptography

8. 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. Information Security -- Public-Key Cryptography

9. 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. Information Security -- Public-Key Cryptography

10. 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|) Information Security -- Public-Key Cryptography

11. 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 Information Security -- Public-Key Cryptography

12. Modular Arithmetic Information Security -- Public-Key Cryptography

13. 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. Information Security -- Public-Key Cryptography

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

15. 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 Information Security -- Public-Key Cryptography

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

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

18. 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). Information Security -- Public-Key Cryptography

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

20. Fermat’s and Euler’s Theorems • Fermat’s theorem Information Security -- Public-Key Cryptography

21. 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 Information Security -- Public-Key Cryptography

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

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

24. 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) Information Security -- Public-Key Cryptography

25. Fermat’s and Euler’s Theorems (cont’d) • Euler’s theorem Information Security -- Public-Key Cryptography

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

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

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

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

30. Euclid’s Algorithm Information Security -- Public-Key Cryptography

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

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

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

34. Extended Euclid’s Algorithm Information Security -- Public-Key Cryptography

35. Chinese Remainder Theorem Information Security -- Public-Key Cryptography

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

37. Discrete Logarithms Information Security -- Public-Key Cryptography

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