On Differential and Boomerang Properties of a Class of Binomials over Finite Fields of Odd Characteristic
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| Format: | Preprint |
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2025
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| _version_ | 1866911356059385856 |
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| author | Koo, Namhun Kwon, Soonhak |
| author_facet | Koo, Namhun Kwon, Soonhak |
| contents | In this paper, we investigate the differential and boomerang properties of a class of binomial $F_{r,u}(x) = x^r(1 + uχ(x))$ over the finite field $\mathbb{F}_{p^n}$, where $r = \frac{p^n+1}{4}$, $p^n \equiv 3 \pmod{4}$, and $χ(x) = x^{\frac{p^n -1}{2}}$ is the quadratic character in $\mathbb{F}_{p^n}$. We show that $F_{r,\pm1}$ is locally-PN with boomerang uniformity $0$ when $p^n \equiv 3 \pmod{8}$. To the best of our knowledge, it is the second known non-PN function class with boomerang uniformity $0$, and the first such example over odd characteristic fields with $p > 3$. Moreover, we show that $F_{r,\pm1}$ is locally-APN with boomerang uniformity at most $2$ when $p^n \equiv 7 \pmod{8}$. We also provide complete classifications of the differential and boomerang spectra of $F_{r,\pm1}$. Furthermore, we thoroughly investigate the differential uniformity of $F_{r,u}$ for $u\in \mathbb{F}_{p^n}^* \setminus \{\pm1\}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_11486 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Differential and Boomerang Properties of a Class of Binomials over Finite Fields of Odd Characteristic Koo, Namhun Kwon, Soonhak Information Theory Cryptography and Security Number Theory In this paper, we investigate the differential and boomerang properties of a class of binomial $F_{r,u}(x) = x^r(1 + uχ(x))$ over the finite field $\mathbb{F}_{p^n}$, where $r = \frac{p^n+1}{4}$, $p^n \equiv 3 \pmod{4}$, and $χ(x) = x^{\frac{p^n -1}{2}}$ is the quadratic character in $\mathbb{F}_{p^n}$. We show that $F_{r,\pm1}$ is locally-PN with boomerang uniformity $0$ when $p^n \equiv 3 \pmod{8}$. To the best of our knowledge, it is the second known non-PN function class with boomerang uniformity $0$, and the first such example over odd characteristic fields with $p > 3$. Moreover, we show that $F_{r,\pm1}$ is locally-APN with boomerang uniformity at most $2$ when $p^n \equiv 7 \pmod{8}$. We also provide complete classifications of the differential and boomerang spectra of $F_{r,\pm1}$. Furthermore, we thoroughly investigate the differential uniformity of $F_{r,u}$ for $u\in \mathbb{F}_{p^n}^* \setminus \{\pm1\}$. |
| title | On Differential and Boomerang Properties of a Class of Binomials over Finite Fields of Odd Characteristic |
| topic | Information Theory Cryptography and Security Number Theory |
| url | https://arxiv.org/abs/2506.11486 |