Factorization of Polynomials Over Finite Fields

Abstract

If $f(x)$ is a polynomial over $GF(q)$, we observe (as has Berlekamp) that if $h{(x)^q} \equiv h(x)(\bmod f(x))$, then $f(x) = \prod {{{_a}_{ \in GF(q)}}\gcd (f(x),h(x) - a)}$. The object of this paper is to give an explicit construction of enough such $h$’s so that the repeated application of this result will succeed in separating all irreducible factors of $f$. The $h$’s chosen are loosely defined by ${h_i}(x) \equiv {x^i} + {x^{iq}} + {x^{i{q^2}}} + \cdots (\bmod f(x))$. A detailed example over $GF(2)$ is given, and a table of the factors of the cyclotomic polynomials ${\Phi _n}(x)(\bmod p){\text { for }}p = 2,n \leqq 250;p = 3,n \leqq 100;p = 5,7,n \leqq 50$, is included.