# Construction of APN permutations via Walsh zero spaces

A Walsh zero space (WZ space) for f:F_2^n→ F_2^n is an n-dimensional vector subspace of F_2^n× 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 (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.

READ FULL TEXT