New Results about the Boomerang Uniformity of Permutation Polynomials

TL;DR

This paper introduces new theoretical and experimental results on the boomerang uniformity of permutation polynomials, including a simplified computation method, a Walsh transform characterization, and new classes of low uniformity permutations.

Contribution

It provides a simpler technique for computing boomerang uniformity, a Walsh transform-based characterization, and new classes of permutation polynomials with low boomerang uniformity.

Findings

Simplified method for computing BCT and boomerang uniformity.

Walsh transform characterization of functions with given boomerang uniformity.

Discovery of new 4-uniform BCT permutation polynomials, including the first binomial.

Abstract

In EUROCRYPT 2018, Cid et al. \cite{BCT2018} introduced a new concept on the cryptographic property of S-boxes: Boomerang Connectivity Table (BCT for short) for evaluating the subtleties of boomerang-style attacks. Very recently, BCT and the boomerang uniformity, the maximum value in BCT, were further studied by Boura and Canteaut \cite{BC2018}. Aiming at providing new insights, we show some new results about BCT and the boomerang uniformity of permutations in terms of theory and experiment in this paper. Firstly, we present an equivalent technique to compute BCT and the boomerang uniformity, which seems to be much simpler than the original definition from \cite{BCT2018}. Secondly, thanks to Carlet's idea \cite{Carlet2018}, we give a characterization of functions $f$ from $\mathbb{F}_{2}^n$ to itself with boomerang uniformity $\delta_{f}$ by means of the Walsh transform. Thirdly, by our method, we consider boomerang uniformities of some specific permutations, mainly the ones with low differential uniformity. Finally, we obtain another class of $4$-uniform BCT permutation polynomials over $\mathbb{F}_{2^n}$, which is the first binomial.