Inversion in Finite Fields and Rings

Synonyms

Inversion in Galois fields

Related Concepts

Euclidean Algorithm; Finite Field

Definition

Inversion is the computation of the element A ⁻¹ such that

$$A{A}^{-1} = 1$$

where A is a nonzero element of a finite field or ring, and “1” is the neutral element of the algebraic structure. In finite fields, or Galois fields, the inverse exists for all nonzero elements. In finite rings, not all elements have an inverse.

Theory

One distinguishes between inversion in a finite ring and in a finite field (or Galois field). In the case of inversion in a finite integer ring or polynomial ring, the extended Euclidean algorithm can be used. Let u be the element whose inverse is to be computed and v the modulus. Note that u and v must be relatively prime in order for the inverse to exist. The extended Euclidean algorithm computes the coefficients s and t such that us +vt = gcd(u, v) = 1. The parameter s is the inverse of u modulo v. In the case of finite integer rings, using the binary Euclidean...