A Degree Bound for the c-Boomerang Uniformity

• URL: http://arxiv.org/abs/2510.18506v1
• Date: Tue, 21 Oct 2025 10:45:35 GMT
• Title: A Degree Bound for the c-Boomerang Uniformity
• Authors: Matthias Johann Steiner,
• Abstract summary: We prove that the $c$-Boomerang, $c neq 0$, of $F$ is bounded by - $d2$ if $c2 neq 1$, - $d cdot (d - 1)$ if $c = -1$, - $d cdot (d - 2)$ if $c = 1$.
• Score: 2.538209532048867
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $p \nmid \text{deg} \left( F \right)$.

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