Differential uniformity and costacyclic code from some power mapping

TL;DR

This work analyzes the differential behavior of the power mapping $f(x)=x^{d}$ over $\\mathbb{F}_{p^n}$ with $d=p^{2l}-p^{l}+1$ and $n=4l$, obtaining the complete differential spectrum and nontrivial $c$-differential uniformity bounds. It then determines the value distribution of a related exponential sum and constructs a class of six-weight $eta^{-1}$-constacyclic codes with explicitly known weight distributions, thereby linking differential properties, exponential sums, and coding theory. The results rely on counting rational points of associated curves and intricate case analyses of differential equations, and they complement existing cross-correlation analyses in the literature. The findings have implications for cryptographic function design and coding theory by providing exact spectral and weight-distribution data for specific power mappings over finite fields.

Abstract

In this paper, we study the differential properties of $x^d$ over $\mathbb{F}_{p^n}$ with $d=p^{2l}-p^{l}+1$. By studying the differential equation of $x^d$ and the number of rational points on some curves over finite fields, we completely determine differential spectrum of $x^{d}$. Then we investigate the $c$-differential uniformity of $x^{d}$. We also calculate the value distribution of a class of exponential sum related to $x^d$. In addition, we obtain a class of six-weight consta-cyclic codes, whose weight distribution is explicitly determined. Part of our results is a complement of the works shown in [\ref{H1}, \ref{H2}] which mainly focus on cross correlations.

Theorems & Definitions (28)

• Definition 1
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• Lemma 1
• Lemma 2
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• Lemma 4
• proof
• Theorem 1
• proof
• Theorem 2