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
Euclids 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 gt 0 and 0 lt b lt 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 -gt 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 then b c mod n

8
Eulers theorem

• Eulers 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
Eulers theorem.cont

• For any prime no Ø(n) n-1.
• Suppose p and q are two prime nos.
• For npq we have
• Ø(n) Ø(pq) Ø(p) x Ø(q) (p-1) x (q-1)
• n21 p3 and q 7
• Ø(21) Ø(3) x Ø(7) 2 x 6 12.

10
Fermats 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!!!