APN trinomials and hexanomials

TL;DR

This paper introduces new APN trinomials and hexanomials over finite fields, characterizes their properties, and provides methods to construct and count elements satisfying specific conditions, advancing understanding of cryptographically relevant functions.

Contribution

It presents a new family of APN trinomials with specific properties and characterizes, constructs, and counts elements related to hexanomials with certain differential uniformity conditions.

Findings

New APN trinomial family with permutation properties

Complete characterization and counting of elements satisfying subgroup conditions

Enhanced understanding of hexanomials' differential uniformity

Abstract

In this paper we give a new family of APN trinomials of the form $X^{2^k+1} + (\mathsf{tr}^{n}_{m}(X))^{2^k+1}$ on $\mathbb{F}_{2^n}$ where $\mathsf{gcd}(k,n)=1$ and $n = 2m = 4t$, and prove its important properties. The family satisfies for all $n = 4t$ an interesting property of the Kim function which is, up to equivalence, the only known APN function equivalent to a permutation on $\mathbb{F}_{2^{2m}}$. As another contribution of the paper, we consider a family of hexanomials $g_{C,k}$ which was shown to be differentially $2^{\mathsf{gcd}(m,k)}$-uniform by Budaghyan and Carlet (2008) when a quadrinomial $P_{C,k}$ has no roots in a specific subgroup. In this paper, for all $(m,k)$ pairs, we characterize, construct and count all $C \in \mathbb{F}_{2^n}$ satisfying the condition. Bracken, Tan and Tan (2014) and Qu, Tan and Li (2014) constructed some elements $C$ satisfying the condition when $m \equiv 2 \textrm{or} 4 \pmod{6}$ and $m \equiv 0 \pmod{6}$ respectively, both requiring $\mathsf{gcd}(m,k) = 1$. Bluher (2013) proved that such $C$ exists if and only if $k \ne m$ without characterizing, constructing or counting those $C$. To prove the results, we effectively use a Trace-$0$/Trace-$1$ (relative to the subfield $\mathbb{F}_{2^m}$) decomposition of $\mathbb{F}_{2^n}$.