What is Fermat’s Little Theorem in Information Security?

Information SecuritySafe & SecurityData Structure

Fermat's little theorem is a fundamental theorem in elementary number theory, which provides compute powers of integers modulo prime numbers. It is a specific case of Euler's theorem, and is essential in applications of elementary number theory, such as primality testing and public-key cryptography. This is referred to as Fermat’s little theorem.

Fermat’s theorem is also called a Fermat’s little theorem defines that is P is prime and ‘a’ is a positive integer not divisible by P then −

aP−1 ≡ 1 mod P

Second condition says that if P is a prime and a is an integer then aP ≡ 1 mod P.

Proof − Zp is the set of integer {0, 1…P-1} when multiplied by a modulo P, the result includes all the element of Zp in some sequence, moreover a x 0 ≡ 0 mod P. Thus, the (P-1) numbers {a mod P, 2a mod P,…((P-1) a mod P)} are only the number {1, 2,…(P-1)} in some order.

Multiplying the numbers in both steps and taking the result mod P gives

         a x 2a x …. x ((P-1)a)= [(a mod P) x (2a mod P) x …. x ((P-1) a mod P)] mod P

                                           =[1 x 2 x…x (P-1)] mod P

                                           = (P-1)! mod P

But

a x 2a x…x ((P-1)a) = (P-1)!aP−1

   (P-1)! aP−1 ≡ (𝑃 − 1)! mod P

             aP−1 ≡ 1 mod P

Consider the set of positive integers less than p: {1, 2... p1} and multiply each element by a, modulo p, to receive the set X = {a mod p, 2a mod p . . . (p1) a mod p}. None of the elements of X is similar to zero because p does not divide a.

Furthermore no two of the integers in X are same. To see this, consider that (ja ≡ p) where 1 ≤ p1. Because p, it can delete a from both sides of the equation resulting in − j≡ p).

This final similarity is inaccessible due to j and k are both positive integers less than p. Therefore, it is understand that the (p1) elements of X are all positive integers, with no two elements same.

Numerical − Fermat’s theorem states that if p is prime and a is a positive integer not divisible by P, then aP−1 ≡ 1(mod p).

Therefore, 310 ≡ 1(mod 11).

Therefore, 3201 = (310)20 x 3 ≡ 3 (mod 11).

Fermats little theorem sometimes is beneficial for quickly discovering a solution to some exponentiations. The following examples show the concept.

Example1 − Find the result of 610 mod 11.

Solution

Here, we have 610 mod 11 = 1. This is the first version of Fermat’s little theorem where p = 11.

Example2 − Find the result of 312 mod 11.

Solution

Therefore the exponent (12) and the modulus (11) are not equal. With substitution this can be defined utilizing Fermat’s little theorem.

312 mod11 = (311x3)mod11 = (311mod11)(3 mod 11) = (3x3)mod11 = 9

raja
Updated on 15-Mar-2022 11:45:24

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