Curves over Finite Fields and Permutations of the Form x k

Abstract

We consider the polynomials of the form P (x) = x k − γTr(x) over Fqn for n ≥ 2. We show that P (x) is not a permutation of Fqn in the case gcd(k, q n − 1) > 1. Our proof uses an absolutely irreducible curve over Fqn and the number of rational points on it.

Permutations polynomials of the form G(X)k − L(X) and curves over finite fields

“…We are now ready to show a main result on polynomials of the form X k − L(X). It generalizes to a large extent earlier results on the case that L(X) = γ Tr(X), see for instance [6,9] and [1].…”

“…The following result relates the number of affine rational points of curves X c with the permutation property of polynomials P (X). The proof is similar to the proof of [1,Theorem 3.1]. We present it here for the sake of convenience of the reader.…”

“…have been investigated intensively with the objective to determine values of k, γ , for which P (X) is a permutation of F q n , see [6,9] and references therein. Recently, it has been shown in [1] and in [3] as a particular case that P (X) is not a permutation of F q n if gcd(k, q n −1) > 1. While finite fields arithmetic is used in [3], the approach in [1] uses absolutely irreducible curves over F q n in a different way, since the common approach, which we described above, is not applicable for these classes of polynomials as the degrees are quite large compared to the cardinality of the finite field.…”

“…Recently, it has been shown in [1] and in [3] as a particular case that P (X) is not a permutation of F q n if gcd(k, q n −1) > 1. While finite fields arithmetic is used in [3], the approach in [1] uses absolutely irreducible curves over F q n in a different way, since the common approach, which we described above, is not applicable for these classes of polynomials as the degrees are quite large compared to the cardinality of the finite field. More precisely, the method in [1] relates the multiplicative and the additive structure of F q n via an absolutely irreducible curve.…”

“…We first show that for a permutation G and a linearized polynomial L with non-trival kernel, P (X) = G(X) k − L(X) cannot be a permutation if gcd(q n − 1, k) > 1. Although this has been recently presented in [3] by using the finite fields arithmetic, we apply the method in [1] as mentioned above. We then analyse general criteria for functions of the form (1.2), where G(X) is an arbitrary linearized polynomial.…”