{"ID":2852192,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.18506","arxiv_id":"2510.18506","title":"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, $c \\neq 0$, 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 $p \\nmid \\text{deg} \\left( F \\right)$.","short_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 $c = -1$, - $...","url_abs":"https://arxiv.org/abs/2510.18506","url_pdf":"https://arxiv.org/pdf/2510.18506v1","authors":"[\"Matthias Johann Steiner\"]","published":"2025-10-21T10:45:35Z","proceeding":"math.AG","tasks":"[\"math.AG\",\"cs.CR\",\"math.NT\"]","methods":"[]","has_code":false}