Construction of APN permutations via Walsh zero spaces.

Cryptography and Communications

A Walsh zero space (WZ space) for f : F_{2n}
→
F_{2n}
is an n-dimensional vector subspace of
F_{2n}
x
F_{2n}
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^{2i+1} on F_{2n} 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.