1. 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.

2. 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.

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

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

5. 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.

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

7. 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 2 341 - 2. 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 = 2 11 -1 = 2047 = 23  89.

8. 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 = 65537 are all primes. However, F 5 = 4294967297 = 641  6700417 is not a prime bumber. n 2 - 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.

9. 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; 11011 = 7  11 2  13. Useful for checking divisibility and relative primality to be discussed later. Factorization is in gereral difficult.

10. 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|)

11. 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

12. YSLInformation Security -- Public-Key Cryptography12 Modular Arithmetic

13. 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.

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

15. 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

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

17. 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.

18. 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).

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

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

21. 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

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

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

24. 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)

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

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

27. 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.

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

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

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

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

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

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

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

35. YSLInformation Security -- Public-Key Cryptography35 Chinese Remainder Theorem

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

37. YSLInformation Security -- Public-Key Cryptography37 Discrete Logarithms

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