Prime Fields Example

In cryptography, prime fields play major role in its mathematical problems. Below you can see a simple example of a prime field 29 which denoted by F₂₉.

The elements of F₂₉ are {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27 28}

For any integer a, a mod p shall denote the unique integer remainder r , 0 ≤ r ≤ p − 1, obtained upon dividing a by p; this operation is called reduction modulo p.

(i) Addition: 17 + 20 = 8 since 37 mod 29 = 8

(ii) Subtraction: 17 − 20 = 26 since −3 mod 29 = 26

(iii) Multiplication: 17 · 20 = 21 since 340 mod 29 = 21

(iv) Inversion: 17⁻¹ = 12 since 17 · 12 mod 29 = 1