A Degree Bound for the c-Boomerang Uniformity

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, $c\ne 0$, of $F$ is bounded by - $d^2$ if $c^2\ne 1$, - $d\cdot (d-1)$ 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 $p\nmid \text{deg}\left(F\right)$.