Power functions with low differential uniformity improve cryptographic security
New Construction of Power Functions with Low c-Differential Uniformity over Finite Fields
Information Theory
Summary
Cryptographic security often relies on special mathematical functions that resist certain types of attacks. This paper shows how to build new power functions over finite fields that have very low c-differential uniformity, a measure tied to their strength against attacks. The authors prove that their constructions can be done in many cases and that their results are optimal for large fields. They also provide specific examples and simpler criteria for special cases to help practical use.
What this means in practice
- •For cryptographic engineers: Design block ciphers that resist differential cryptanalysis using these low c-differential uniformity power functions for stronger security.
- •For security protocol developers: Incorporate these functions into secure communication protocols to enhance resistance against algebraic attacks based on differential properties.
Authors
Zhiye Yang, Yan Wang, Keqin Feng
Abstract
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.