Triprojective almost perfect nonlinear permutations and functions

2026-05-17

AI Summary

This work addresses the long-standing challenge of constructing almost perfect nonlinear (APN) functions in high-dimensional finite vector spaces by introducing a novel approach based on a tri-projective algebraic structure induced by the general linear group GL(3, 2^m). The proposed method yields, for the first time, a systematic construction of APN permutations that covers all odd dimensions divisible by three, and further extends to even dimensions to produce highly nonlinear, non-bijective APN functions. By achieving this, the study not only fills a critical theoretical gap in the existence and construction of APN permutations in odd dimensions but also significantly enriches the design paradigms for highly nonlinear functions in cryptographic applications.

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 $\mathrm{GL}(3,2^m)$.

Problem

• almost perfect nonlinear
• permutations
• triprojective
• finite vector spaces
• nonlinear functions

Innovation

• almost perfect nonlinear
• triprojective structure
• APN permutations
• finite vector spaces
• GL(3,2^m)