Rachana Y. Patil

1. Modula Arithmetic Rachana Y. Patil

2. Cryptographic Theory Primality: Two nos are relatively prime if they have no factors common in them other than 1. i.e gcd(a,n) = 1 gcd (7, 78) = 1

3. Euclid’s Alorithm What is gcd of 21 and 45??? gcd(a,b) = gcd(b, a mod b)

4. Modular Arithmetic Says that 23 and 11 are equivalent ?????? 23 mod 11 = 12 Or 23≡ 11 mod 12 …..

5. Cont… a ≡ b mod n if a = b + kn for some integer k. If a > 0 and 0 < b < n then b is the remainder of the division a/n.

6. Properties of Modulo operator a ≡ b mod n if n/a-b a ≡ b mod n ═> b ≡ a mod n a ≡ b mod n and b ≡ c mod n implies a ≡ c mod n

7. Modular Arithmetic (a mod n) +(b mod n) = ( a + b ) mod n (a mod n) x (b mod n) = (a x b) mod n (a + b) ≡ (a + c) mod n thenb ≡ c mod n

8. Euler’s theorem Euler’s Toient function Ø(n) Ø(n) is the set of +ve integers less than n and relatively prime to n n = 6 What is Ø(n) ???? n = 7 Ø(n) = ????

9. Euler’s theorem….cont For any prime no Ø(n) = n-1. Suppose p and q are two prime nos. For n=pq we have Ø(n) = Ø(pq) = Ø(p) x Ø(q) = (p-1) x (q-1) n=21 p=3 and q = 7 Ø(21) = Ø(3) x Ø(7) = 2 x 6 = 12.

10. Fermat’s Theorem If p is prime and a is a +ve integer not divisible by p then a p-1≡ 1 (mod p) Let a = 3 and p = 5 a 5-1 = a 4 =34 = 81 ≡ 1 (mod 5) proved…..

11. The Theorem For every a and n which are relatively prime a Ø(n)≡ 1 (mod n) a = 3 n = 10 Ø(n) = Ø(10) = 1,3,7,9 = 4 a Ø(n) = 3 4 = 81 ≡ 1 (mod 10) hence proved

12. Modular Exponentiation xy mod n = xy mod ø(n) mod n if y = 1 mod ø(n) then xy mod n = x mod n

13. Modular exponentiation One way function used in cryptography ax mod n Can u find x where ax = b mod n??? That is the discrete logarithm problem If 3x = 15 mod (17) find x…….

14. Discrete Logarithm problem Solution….easy enough Solve 3x mod 15 = 17 x = 6 For large nos solving this is difficult!!!