Investigation of the permutation and linear codes from the Welch APN function

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-07-27 DOI:10.1007/s10623-024-01461-3

Tor Helleseth, Chunlei Li, Yongbo Xia

引用次数: 0

Abstract

Dobbertin in 1999 proved that the Welch power function \(x^{2^m+3}\) was almost perferct nonlinear (APN) over the finite field \(\mathbb {F}_{2^{2m+1}}\), where m is a positive integer. In his proof, Dobbertin showed that the APNness of \(x^{2^m+3}\) essentially relied on the bijectivity of the polynomial \(g(x)=x^{2^{m+1}+1}+x^3+x\) over \(\mathbb {F}_{2^{2m+1}}\). In this paper, we first determine the differential and Walsh spectra of the permutation polynomial g(x), revealing its favourable cryptograhphic properties. We then explore four families of binary linear codes related to the Welch APN power functions. For two cyclic codes among them, we propose algebraic decoding algorithms that significantly outperform existing methods in terms of decoding complexity.

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韦尔奇 APN 函数中的置换和线性编码研究

Dobbertin 在 1999 年证明了韦尔奇幂函数 \(x^{2^m+3}\)在有限域 \(\mathbb{F}_{2^{2m+1}}\)上几乎是非线性的（APN），其中 m 是正整数。在他的证明中，Dobbertin 证明了 \(x^{2^m+3}\) 的 APN 性本质上依赖于 \(\mathbb {F}_{2^{2m+1}} 上多项式 \(g(x)=x^{2^{m+1}+1}+x^3+x\) 的双射性。）在本文中，我们首先确定了置换多项式 g(x) 的微分和沃尔什谱，揭示了其有利的加密特性。然后，我们探讨了与韦尔奇 APN 幂函数相关的四个二进制线性编码系列。对于其中的两种循环码，我们提出了代数解码算法，在解码复杂度方面明显优于现有方法。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.