Determination of a class of permutation quadrinomials

Abstract

We determine all permutation polynomials over 𝔽 𝑞 2 of the form 𝑋 𝑟 𝐴(𝑋 𝑞−1 ) where, for some 𝑄 that is a power of the characteristic of 𝔽 𝑞 , we have 𝑟 ≡ 𝑄 + 1 (mod 𝑞 + 1) and all terms of 𝐴(𝑋) have degrees in {0, 1, 𝑄, 𝑄 + 1}. We use this classification to resolve eight conjectures and open problems from the literature, and we list 77 recent results from the literature that follow immediately from the simplest special cases of our result. Our proof makes a novel use of geometric techniques in a situation where they previously did not seem applicable, namely to understand the arithmetic of high-degree rational functions over small finite fields, despite the fact that in this situation the Weil bounds do not provide useful information.

Recursive construction of normal polynomials over finite fields

“…Follows directly from Lemma 2.1 and Lemma 3.1 Theorem 3.2 generalises Corollary 3 from [10]. Following lemma is an observation made by Zieve in [3]. For the sake of convenience we state the proof of the same.…”

“…In what follows here on, we shall use the notations and terminologies of [3]. Let us also recall a few results which aid the proof of the results stated in Section 3 of this paper.…”

“…A lot of work have already been commenced in the direction of finding permutation trinomials over F q 2 , which can be found in [8]. Quadrinomials, which form permutations of F q 2 have also been studied by several authors in [3] and [7], where characteristic of the field can be both odd and even. The goal of this paper is to give a new approach that aids the construction of PP's of F q 2 with fewer terms.…”

“…The idea is to use the self conjugate reciprocal (SCR) polynomials for the same. The term SCR was first introduced in [3] by Zieve. In this paper we build up a method to find conditions such that an SCR polynomial has no roots in the set of (q + 1)-th roots of unity. A significant step of our method is to take the composition of our SCR polynomial with a degree one rational function which induces a bijection from F q to µ q+1 (set of (q + 1)-th roots of unity).…”