A Degree Bound For The c-Boomerang Uniformity Of Permutation Monomials

TL;DR

This paper establishes a quadratic upper bound on the c-boomerang uniformity for permutation monomials over finite fields and applies this result to analyze the security of certain cryptographic permutation systems.

Contribution

It introduces a new degree bound for the c-boomerang uniformity of permutation monomials and extends this to generalized triangular dynamical systems.

Findings

c-boomerang uniformity for permutation monomials is bounded by d^2

The bound applies to all permutation monomials with d > 1 and p ∤ d

The result aids in estimating security parameters of cryptographic permutations

Abstract

Let $\mathbb{F}_q$ be a finite field of characteristic $p$. In this paper we prove that the $c$-Boomerang Uniformity, $c \neq 0$, for all permutation monomials $x^d$, where $d > 1$ and $p \nmid d$, is bounded by $d^2$. Further, we utilize this bound to estimate the $c$-boomerang uniformity of a large class of Generalized Triangular Dynamical Systems, a polynomial-based approach to describe cryptographic permutations, including the well-known Substitution-Permutation Network.