有限域$\mathbb{F}_{q^2}$上幂函数$x^{q+2}$的深入研究：差分谱、回旋镖谱与Walsh谱及其在编码理论中的应用
An in-depth study of the power function $x^{q+2}$ over the finite field $\mathbb{F}_{q^2}$: the differential, boomerang, and Walsh spectra, with an application to coding theory
AI总结：
该研究提出新方法分析有限域$\mathbb{F}_{q^2}$上幂函数$x^{q+2}$的差分谱，确定了多数情形下的回旋镖均匀性，还得到$p=3$时其Walsh谱仅取4个值，并推导了对应三元循环码的重量分布。
摘要：
设$q = p^m$，其中$p$为奇素数，$m$为正整数。本文研究包含$q^2$个元素的有限域$\mathbb{F}_{q^2}$。我们首先提出一种确定$\mathbb{F}_{q^2}$上幂函数$f(x) = x^{q+2}$差分谱的替代方法，该方法融入了若干关键简化技巧。这一方法为Man、Xia、Li与Helleseth发表于《Finite Fields and Their Applications》第84卷（2022年，文章编号102100）的结果提供了新的证明，不仅完整确定了$f$的差分谱，还为其回旋镖均匀性的分析提供了便利。\n具体而言，我们确定了$q \equiv 1$或$3 \pmod{6}$情形下$f$的回旋镖均匀性，仅排除$p = 5$且$m$为偶数的场景。此外，当$p = 3$时，我们研究了$f$的Walsh谱的值分布，证明其仅取4个不同值。利用这一结果，我们推导了一类具有4个汉明重量的三元循环码的重量分布。本文整合了代数数论与有限域理论中的精细数学技巧，运用指数和等多种工具探索有限域上函数的密码学分析，这些方法可用于研究更广泛函数类的差分/回旋镖均匀性。
英文摘要：
Let $q = p^m$, where $p$ is an odd prime number and $m$ is a positive integer. In this paper, we examine the finite field $\mathbb{F}_{q^2}$, which consists of $q^2$ elements. We first present an alternative method to determine the differential spectrum of the power function $f(x) = x^{q+2}$ on $\mathbb{F}_{q^2}$, incorporating several key simplifications. This methodology provides a new proof of the results established by Man, Xia, Li, and Helleseth in Finite Fields and Their Applications 84 (2022), 102100, which not only completely determine the differential spectrum of $f$ but also facilitate the analysis of its boomerang uniformity. Specifically, we determine the boomerang uniformity of $f$ for the cases where $q \equiv 1$ or $3$ (mod $6$), with the exception of the scenario where $p = 5$ and $m$ is even. Furthermore, for $p = 3$, we investigate the value distribution of the Walsh spectrum of $f$, demonstrating that it takes on only four distinct values. Using this result, we derive the weight distribution of a ternary cyclic code with four Hamming weights. The article integrates refined mathematical techniques from algebraic number theory and the theory of finite fields, employing several ingredients, such as exponential sums, to explore the cryptographic analysis of functions over finite fields. They can be used to explore the differential/boomerang uniformity across a wider range of functions.
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