On permutation quadrinomials and $4$-uniform BCT

TL;DR

This paper investigates a specific class of quadrinomials over binary fields, identifying conditions for permutation behavior with optimal boomerang uniformity, enhancing understanding of cryptographic resistance.

Contribution

It characterizes when certain quadrinomials over binary fields are permutations with optimal boomerang uniformity, extending prior research in cryptographic function analysis.

Findings

Identified conditions for quadrinomials to be permutations

Established criteria for achieving 4-uniform BCT

Extended previous results on boomerang uniformity

Abstract

We study a class of general quadrinomials over the field of size $2^{2m}$ with odd $m$ and characterize conditions under which they are permutations with the best boomerang uniformity, a new and important parameter related to boomerang-style attacks. This vastly extends previous results from several recent papers.