Differential Spectrum of Some Power Functions With Low Differential Uniformity

TL;DR

This paper calculates the differential spectrum of specific power functions over finite fields, revealing their differential uniformity and identifying new functions with low differential uniformity, which are valuable for cryptographic applications.

Contribution

It derives the differential spectrum of certain power functions over finite fields and identifies new functions with low differential uniformity, advancing cryptographic function design.

Findings

Differential spectrum of $x^{(p^k+1)/2}$ in $ ext{GF}(p^n)$ calculated.

Differential spectrum of $x^{(p^n+1)/(p^k+1)+(p^n-1)/2}$ in $ ext{GF}(p^n)$ derived for specific primes.

New power functions with low differential uniformity identified.

Abstract

In this paper, for an odd prime $p$, the differential spectrum of the power function $x^{\frac{p^k+1}{2}}$ in $\mathbb{F}_{p^n}$ is calculated. For an odd prime $p$ such that $p\equiv 3\bmod 4$ and odd $n$ with $k|n$, the differential spectrum of the power function $x^{\frac{p^n+1}{p^k+1}+\frac{p^n-1}{2}}$ in $\mathbb{F}_{p^n}$ is also derived. From their differential spectrums, the differential uniformities of these two power functions are determined. We also find some new power functions having low differential uniformity.