有限域上低c-差分一致性的幂函数新构造

中文摘要 AI 辅助

本文研究了有限域上幂函数的c-差分一致性，这是一类具有良好差分性质的重要密码函数。具体地，对于满足q-1=en且e≥3、e|n的有限域F_q，我们证明存在c∈F_q\{0,1,ε,…,ε^{e-1}}，其中ε是F_q^*中e次本原单位根，使得所构造的幂函数f(x)=x^{ln+1}（其中1≤l≤e-1且gcd(l,e)=1）在特定分圆条件成立时满足上界Δ(f,c)≤e。我们表明我们的条件是宽松的；即对于任意给定的e≥3和素数p（p∤e），可以在F_p的无穷多个扩域F_q上构造这样的幂函数。此外，基于乘法特征和的Weil界，我们证明对于足够大的q，所获得的上界是紧的，展示了我们结果的最优性。我们还分析了特殊情况c=-1并推导出简化的显式条件。特别地，我们明确刻画了e=3情况下的容许参数，并给出了具体的函数实例以供实际验证。

英文摘要

This paper investigates the $c$-differential uniformity of power functions over finite fields, an important class of cryptographic functions with favorable differential properties. Specifically, for finite fields $\mathbb{F}_q$ satisfying $q-1=en$ with $e\ge 3$ and $e\mid n$, we prove that there exists $c\in\mathbb{F}_q\setminus\{0,1,ε,\dots,ε^{e-1}\}$, where $ε$ is an $e$-th primitive root of unity in $\mathbb{F}_q^*$, the constructed power functions $f(x)=x^{ln+1}$ with $1\le l\le e-1$ and $\gcd(l,e)=1$ satisfy the upper bound $Δ(f,c)\le e$, provided that certain cyclotomic conditions hold. We show that our conditions are mild; namely, such power functions can be constructed over infinitely many extension fields $\mathbb{F}_q$ of $\mathbb{F}_p$ for any given $e\ge 3$ and prime $p$ with $p\nmid e$. Furthermore, based on the Weil bound for multiplicative character sums, we prove that the obtained upper bound is tight for sufficiently large $q$, demonstrating the optimality of our results. We also analyze the special case $c=-1$ and derive simplified explicit conditions. In particular, we explicitly characterize the admissible parameters for the case $e=3$ and present concrete function examples for practical validation.

发表机构

• School of Science, Xi’an University of Architecture and Technology（西安建筑科技大学理学院）

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