Construction of APN permutations via Walsh zero spaces

TL;DR

This paper introduces new theoretical methods to construct Walsh zero spaces for Gold APN functions and demonstrates how these can be used to generate APN permutations with specific cryptographic properties.

Contribution

It provides novel, computer-free constructions of Walsh zero spaces for Gold APN functions and applies these to create APN permutations with unique equivalence properties.

Findings

Constructed Walsh zero spaces for Gold APN functions

Developed methods to generate APN permutations with specific CCZ but not EA equivalence

Demonstrated applications in cryptographic permutation design

Abstract

A Walsh zero space (WZ space) for $f:F_{2^n}\rightarrow F_{2^n}$ is an $n$-dimensional vector subspace of $F_{2^n}\times F_{2^n}$ whose all nonzero elements are Walsh zeros of $f$. We provide several theoretical and computer-free constructions of WZ spaces for Gold APN functions $f(x)=x^{2^i+1}$ on $F_{2^n}$ where $n$ is odd and $\gcd(i,n)=1$. We also provide several constructions of trivially intersecting pairs of such spaces. We illustrate applications of our constructions that include constructing APN permutations that are CCZ equivalent to $f$ but not extended affine equivalent to $f$ or its compositional inverse.