A Degree Bound for the c-Boomerang Uniformity
Fuente:
arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918164778975232 |
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| author | Steiner, Matthias Johann |
| author_facet | Steiner, Matthias Johann |
| contents | 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 $p \nmid \text{deg} \left( F \right)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_18506 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Degree Bound for the c-Boomerang Uniformity Steiner, Matthias Johann Algebraic Geometry Cryptography and Security Number Theory 94A60, 11T06, 14G50, 14H50 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 $p \nmid \text{deg} \left( F \right)$. |
| title | A Degree Bound for the c-Boomerang Uniformity |
| topic | Algebraic Geometry Cryptography and Security Number Theory 94A60, 11T06, 14G50, 14H50 |
| url | https://arxiv.org/abs/2510.18506 |