Triprojective almost perfect nonlinear permutations and functions

TL;DR

This paper introduces a broad family of APN permutations and functions with triprojective structure for finite vector spaces, expanding the understanding of nonlinear functions in cryptography.

Contribution

It presents new APN permutations for odd dimensions divisible by three and non-bijective APN functions for even dimensions, with a novel triprojective structure.

Findings

Provides APN permutations for all odd dimensions divisible by three.

Introduces non-bijective APN functions for even dimensions.

Defines functions with triprojective structure induced by GL(3, 2^m).

Abstract

We give a large family of almost perfect nonlinear (APN) permutations of finite vector spaces of every odd dimension divisible by three. We also give APN functions that are not bijective on even dimensions and related highly nonlinear functions. The functions we provide admit a so-called triprojective structure induced by the general linear group $\mathrm{GL}(3,2^m)$.