Paper 2026/2800

Testing CCZ-equivalence to permutations via projections of Walsh zero spaces, with applications to APN functions

Abstract

Whether almost perfect nonlinear (APN) permutations exist in even dimension $n\ge 8$ is a long-standing open question, the big APN problem. The only example known in even dimension, on six bits, was found from a quadratic APN function that is not itself a permutation, by Carlet--Charpin--Zinoviev (CCZ) equivalence. It is therefore natural to test APN functions for CCZ-equivalence to a permutation. This amounts to finding two complementary $n$-dimensional subspaces among the zeros of the Walsh transform, which lie in $\mathbb{F}_2^{2n}$. A known necessary condition only uses their projections to $\mathbb{F}_2^n$ and cannot decide every case. We give an exact test for functions whose Walsh supports are affine, including all quadratic functions. It reconstructs the subspaces from their projections, through searches for subspaces of $\mathbb{F}_2^n$ and for constrained bilinear maps. On the benchmarked functions in dimension twelve, it lists them $39$ to $55$ times faster than the generic search of the commonly used sboxU library. We then study the functions $x^3+a^{-1}\mathrm{Tr}(a^3x^9)$, which Göloğlu and Pavlů proved not to be CCZ-equivalent to permutations when $4$ divides $n$. With their result, our computations show that this holds for every even $n\le 24$. In dimension $18$ the necessary condition is satisfied, and the new test decides the case. We conjecture this for every even dimension. We also compile $25$ classes of quadratic APN functions in dimension ten from published instances, with equivalence certificates. No function in these classes or in their switching closure is CCZ-equivalent to a permutation. All code and data are public.