Applications of Finite Fields

A Note on Construction of Irreducible Polynomials Over Finite Fields With Characteristic 2

“…Theorem 3 (see [4], Theorem 3.5). The trinomial x p − x − b, b ∈ F q where q is a prime power p m , is irreducible over F q if and only if T r Fq/Fp (b) = 0.…”

“…For any polynomial f (x) over F q whose constant term is nonzero, the reciprocal of f (x) is a polynomial f * (x) = x m f ( 1 x ) of degree m over F q and the roots of f * (x) are the reciprocals of the roots of f (x). For more information on terminology and notations we refer [3], [4].…”

Prescribing coefficients of invariant irreducible polynomials

“…For the general case (i.e. for arbitrary r), see [17,Theorem 3.2.5] or [2,Theorem 3.9]. In our case, since r stands for the order of α ∈ F * q , it is clear that gcd(r, q) = 1, r | (q − 1) and 4 gcd(r, q n + 1), hence the irreducibility of P (X r ) depends solely on the choice of p 0 .…”

Factoring Polynomials over Finite Fields Using Differential Equations and Normal Bases

“…The potential bottleneck in the above procedure for determining gi, ... , gm is the calculation of the rm gcd's in (2). For random polynomials / over finite fields and r and d = deg(/) of reasonable size, this problem is not so serious since the average order of magnitude of the number m of distinct monic irreducible factors of / is small, namely logo?…”