A complete characterization of the APN property of a class of quadrinomials

Abstract

In this paper, by the Hasse-Weil bound, we determine the necessary and sufficient condition on coefficientsan APN function over F 2 n . Our result resolves the first half of an open problem by Carlet in International Workshop on the Arithmetic of Finite Fields, 83-107, 2014.

Kim-type APN functions are affine equivalent to Gold functions

“…Proof. This is proved at the bottom of page 2 in [9]. The proof given there applies to the general case when f (x) also contains the term a 2 x q+2 .…”

“…We will call functions of this form Kim-type functions in the present paper. We extend the result of [9] by proving that if a Kim-type function f is APN and m ≥ 4, then f is affine equivalent to one of two Gold functions…”

“…We start with an observation which is not stated as a numbered item in [9] but we want to give it a label for our future reference, and we also state it in a slightly different form.…”

“…Most recently, Li, Li, Helleseth and Qu [9] solved the first part of Problem 1.1 by proving Theorem 1.4 stated below, in which they completely characterize APN functions on F 2 n of the form f (x) = x 3q + a 1 x 2q+1 + a 2 x q+2 + a 3 x 3 , where q = 2 m and m = n/2, m ≥ 4. We will call functions of this form Kim-type functions in the present paper.…”

“…The motivation for solving this problem is given by possibly applying a suitable CCZ transformation to the resulting APN functions to obtain APN permutations, in analogy to the approach of [2]. There have been several lines of attack to approach Problem 1.1 and similar problems; we refer to the Introduction section of [9] for a detailed account of them.…”