Construction of APN permutations via Walsh zero spaces

Abstract

A Walsh zero space (WZ space) for \(f:{\mathbb {F}_{2^n}}\rightarrow {\mathbb {F}_{2^n}}\) is an n-dimensional vector subspace of \({\mathbb {F}_{2^n}}\times {\mathbb {F}_{2^n}}\) 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 \({\mathbb {F}_{2^n}}\) 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.