Note on Irreducible Polynomials over Finite Field

Abstract

In this note we extend an irreducibility criterion of polynomial over finite fields. Weprove the irreducibility of the polynomial P(Y ) = Yn + λn−1Y n−1 + λn−2Y n−2 + · · · + λ1Y + λ0, such that λ0 6= 0, deg λn−2 = 2 deg λn−1 + l deg λi, for all i 6= n − 2 and odd integer l.

Note on Irreducible Polynomials over $\mathbb{F}_q[X]$

“…This result was the starting point for many researches and the exploration of new criterions, see [1,3]. For older results, see [4,5].…”

Construction of Substitution Box (S-Box) Based on Irreducible Polynomials on Gf(2^8)

“…The first step in constructing the S-box is to find the inverse multiplication of each element in the 𝐺𝐹 (2 8 ). Mathematically, an inverse calculation of an element of an irreducible polynomial can be done using the Extended Euclid Algorithm [18].…”