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Understanding Cryptographic Theory and Modular Arithmetic in Primality

This text delves into the concepts of modular arithmetic and cryptographic theory, particularly focusing on primality. It explains key concepts such as relative primality, the Greatest Common Divisor (GCD) through Euclid’s algorithm, and modular operations. The discussion also covers Euler's theorem, the Euler's totient function Ø(n), Fermat’s theorem, and modular exponentiation, highlighting their significance in cryptography and number theory. Examples and applications, including the discrete logarithm problem, illustrate the practical use of these theories in securing data.

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Understanding Cryptographic Theory and Modular Arithmetic in Primality

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  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!!!

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