On the differential spectrum of a class of power functions over finite fields

TL;DR

This paper investigates the differential spectrum of a specific class of power functions over finite fields, confirming a conjecture and providing a complete characterization relevant to cryptographic security.

Contribution

It proves a conjecture regarding the differential uniformity of a particular power function over finite fields and determines its differential spectrum completely.

Findings

Confirmed the conjecture on differential uniformity

Determined the complete differential spectrum of the function

Enhanced understanding of cryptographically strong power functions

Abstract

Differential uniformity is a significant concept in cryptography as it quantifies the degree of security of S-boxes respect to differential attacks. Power functions of the form $F(x)=x^d$ with low differential uniformity have been extensively studied in the past decades due to their strong resistance to differential attacks and low implementation cost in hardware. In this paper, we give an affirmative answer to a recent conjecture proposed by Budaghyan, Calderini, Carlet, Davidova and Kaleyski about the differential uniformity of $F(x)=x^d$ over $\mathbb{F}_{2^{4n}}$, where $n$ is a positive integer and $d=2^{3n}+2^{2n}+2^{n}-1$, and we completely determine its differential spectrum.