A note on the differential spectrum of a class of locally APN functions
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909451116609536 |
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| author | Yan, Haode Ren, Ketong |
| author_facet | Yan, Haode Ren, Ketong |
| contents | Let $\gf_{p^n}$ denote the finite field containing $p^n$ elements, where $n$ is a positive integer and $p$ is a prime. The function $f_u(x)=x^{\frac{p^n+3}{2}}+ux^2$ over $\gf_{p^n}[x]$ with $u\in\gf_{p^n}\setminus\{0,\pm1\}$ was recently studied by Budaghyan and Pal in \cite{Budaghyan2024ArithmetizationorientedAP}, whose differential uniformity is at most $5$ when $p^n\equiv3~(mod~4)$. In this paper, we study the differential uniformity and the differential spectrum of $f_u$ for $u=\pm1$. We first give some properties of the differential spectrum of any cryptographic function. Moreover, by solving some systems of equations over finite fields, we express the differential spectrum of $f_{\pm1}$ in terms of the quadratic character sums. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_04233 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note on the differential spectrum of a class of locally APN functions Yan, Haode Ren, Ketong Information Theory Cryptography and Security Let $\gf_{p^n}$ denote the finite field containing $p^n$ elements, where $n$ is a positive integer and $p$ is a prime. The function $f_u(x)=x^{\frac{p^n+3}{2}}+ux^2$ over $\gf_{p^n}[x]$ with $u\in\gf_{p^n}\setminus\{0,\pm1\}$ was recently studied by Budaghyan and Pal in \cite{Budaghyan2024ArithmetizationorientedAP}, whose differential uniformity is at most $5$ when $p^n\equiv3~(mod~4)$. In this paper, we study the differential uniformity and the differential spectrum of $f_u$ for $u=\pm1$. We first give some properties of the differential spectrum of any cryptographic function. Moreover, by solving some systems of equations over finite fields, we express the differential spectrum of $f_{\pm1}$ in terms of the quadratic character sums. |
| title | A note on the differential spectrum of a class of locally APN functions |
| topic | Information Theory Cryptography and Security |
| url | https://arxiv.org/abs/2501.04233 |