A Degree Bound for the c-Boomerang Uniformity

TL;DR

This paper establishes upper bounds for the c-boomerang uniformity of polynomials over finite fields, providing tight examples and proving irreducibility conditions for related bivariate polynomials.

Contribution

It introduces a degree-based bound for the c-boomerang uniformity of polynomials over finite fields and proves irreducibility results crucial for the analysis.

Findings

Bound of d^2 for c ≠ ±1

Bound of d(d-1) for c = -1

Bound of d(d-2) for c = 1

Abstract

Let $\mathbb{F}_q$ be a finite field, and let $F \in \mathbb{F}_q [X]$ be a polynomial with $d = \text{deg} \left( F \right)$ such that $\gcd \left( d, q \right) = 1$. In this paper we prove that the $c$-Boomerang uniformity, $c \neq 0$, of $F$ is bounded by - $d^2$ if $c^2 \neq 1$, - $d \cdot (d - 1)$ if $c = -1$, - $d \cdot (d - 2)$ if $c = 1$. For all cases of $c$, we present tight examples for $F \in \mathbb{F}_q [X]$. Additionally, for the proof of $c = 1$ we establish that the bivariate polynomial $F (x) - F (y) + a \in k [x, y]$, where $k$ is a field of characteristic $p$ and $a \in k \setminus \{ 0 \}$, is absolutely irreducible if $p \nmid \text{deg} \left( F \right)$.