A Generalisation of Dillon's APN Permutation With the Best Known Differential and Nonlinear Properties for All Fields of Size $2^{4k+2}$

Abstract

The existence of Almost Perfect Nonlinear (APN) permutations operating on an even number of variables was a long-standing open problem, until an example with six variables was exhibited by Dillon et al. in 2009. However it is still unknown whether this example can be generalised to any even number of inputs. In a recent work, Perrin et al. described an infinite family of permutations, named butterflies, operating on (4 + 2) variables and with differential uniformity at most 4, which contains the Dillon APN permutation. In this paper, we generalise this family, and we completely solve the two open problems raised by Perrin et al. Indeed we prove that all functions in this larger family have the best known nonlinearity. We also show that this family does not contain any APN permutation besides the Dillon permutation, implying that all other functions have differential uniformity exactly four.

Determining the Walsh spectra of Taniguchi's and related APN-functions

“…For the Zhou-Pott function we will also render the APN condition more precisely using the Hasse-Weil bound. Finally we point out that our method also works well for the butterfly functions in [4].…”

“…We close this section pointing out that our approach is also applicable to the butterfly functions investigated in [4]. For an odd integer m, let F : is CCZ-equivalent to the only known APN-permutation in an even number of variables in [3].…”

“…In the next sections we will apply Corollary 2.6 to some classes of quadratic functions from F 2 m ×F 2 m to F 2 m ×F 2 m , more precisely, to all of their component functions simultaneously. Among those are the APN-functions in [7,22], the recently introduced Taniguchi APN-functions, and the differentially 4-uniform butterfly functions, [4].…”

“…For an odd integer m, let F : is CCZ-equivalent to the only known APN-permutation in an even number of variables in [3]. We refer to [4] for details, where the functions (3.13) were thoroughly investigated. Amongst others it is shown that F is differentially 4-uniform if and only if β = (1 + α) 3 , but the above mentioned case is the only one for which F is APN.…”

“…Since it seems to be hard to find APN-permutations of V n when n is even, one often considers also differentially 4-uniform functions, see e.g. [4].…”