$4$-uniform BCT permutations from generalized butterfly structure

Abstract

As a generalization of Dillon's APN permutation, butterfly structure and generalizations have been of great interest since they generate permutations with the best known differential and nonlinear properties over the field of size 2 4k+2 .Complementary to these results, we show in this paper that butterfly structure, more precisely the closed butterfly also yields permutations with the best boomerang uniformity, a new and important parameter related to boomerang-style attacks. This is the sixth known infinite family of permutations in the literature with the best known boomerang uniformity over such fields.

A complete characterization of the APN property of a class of quadrinomials

“…As a matter of fact, polynomials of the two general forms in Problem 1.1 have been studied from different perspectives. In Crypto '16,Perrin et al. investigated the only APN permutation by means of reverse-engineering and proposed the open butterfly and the closed butterfly structures [20].…”

“…The necessary condition for f (x) to be a permutation has been characterized in [14]. Furthermore, the permutation and boomerang uniformity property of a more general quadrinomial have been studied in [13,16,17].…”

Determination of a class of permutation quadrinomials

“…We note that some of the above conjectures follow easily from our results, while others require significant additional work. In particular, our proof of the conjecture from [30,Rem. 2] relies on a new polynomial identity of independent interest (Theorem 8.1).…”

“…The final two conjectures are [23,Conj. 19] and [30,Rem. 2], which describe all permutations among a certain class of functions from F q × F q to itself that are defined by a pair of bivariate polynomials.…”

Determination of a class of permutation quadrinomials

“…The final two conjectures are [26,Conj. 19] and [33,Rem. 2], which describe all permutations among a certain class of functions from 𝔽 𝑞 × 𝔽 𝑞 to itself that are defined by a pair of bivariate polynomials.…”