An in-depth study of the power function $x^{q+2}$ over the finite field $mathbb{F}_{q^2}$: the differential, boomerang, and Walsh spectra, with an application to coding theory

2024-07-08

AI Summary

研究了$x^{q+2}$在$\mathbb{F}_{q^2}$上的特性，通过新方法简化分析，重证并深化Man等人的发现，尤其关注$q\mod 6=1,3$时的性质，揭示$p=3$时函数的频率分布，推导出具有四个汉明权重的三元循环码的权重分布，利用指数和等工具探究密码学性质。

Technology Category

• Knowledge Representation and Reasoning: Qualitative Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

Let $q = p^m$, where $p$ is an odd prime number and $m$ is a positive integer. In this paper, we examine the finite field $mathbb{F}_{q^2}$, which consists of $q^2$ elements. We first present an alternative method to determine the differential spectrum of the power function $f(x) = x^{q+2}$ on $mathbb{F}_{q^2}$, incorporating several key simplifications. This methodology provides a new proof of the results established by Man, Xia, Li, and Helleseth in Finite Fields and Their Applications 84 (2022), 102100, which not only completely determine the differential spectrum of $f$ but also facilitate the analysis of its boomerang uniformity. Specifically, we determine the boomerang uniformity of $f$ for the cases where $q equiv 1$ or $3$ (mod $6$), with the exception of the scenario where $p = 5$ and $m$ is even. Furthermore, for $p = 3$, we investigate the value distribution of the Walsh spectrum of $f$, demonstrating that it takes on only four distinct values. Using this result, we derive the weight distribution of a ternary cyclic code with four Hamming weights. The article integrates refined mathematical techniques from algebraic number theory and the theory of finite fields, employing several ingredients, such as exponential sums, to explore the cryptographic analysis of functions over finite fields. They can be used to explore the differential/boomerang uniformity across a wider range of functions.

Problem

• Finite Fields
• Coding Theory
• Information Transmission

Innovation

• Finite Fields Analysis
• Cyclic Codes Weight Distribution
• Cryptographic Applications