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

IF 1.4 2区数学 Q3 COMPUTER SCIENCE, THEORY & METHODS

Designs, Codes and Cryptography 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

Designs, Codes and Cryptography 工程技术-计算机：理论方法

CiteScore

2.80

自引率

12.50%

发文量

157

审稿时长

16.5 months

期刊介绍： Designs, Codes and Cryptography is an archival peer-reviewed technical journal publishing original research papers in the designated areas. There is a great deal of activity in design theory, coding theory and cryptography, including a substantial amount of research which brings together more than one of the subjects. While many journals exist for each of the individual areas, few encourage the interaction of the disciplines. The journal was founded to meet the needs of mathematicians, engineers and computer scientists working in these areas, whose interests extend beyond the bounds of any one of the individual disciplines. The journal provides a forum for high quality research in its three areas, with papers touching more than one of the areas especially welcome. The journal also considers high quality submissions in the closely related areas of finite fields and finite geometries, which provide important tools for both the construction and the actual application of designs, codes and cryptographic systems. In particular, it includes (mostly theoretical) papers on computational aspects of finite fields. It also considers topics in sequence design, which frequently admit equivalent formulations in the journal’s main areas. Designs, Codes and Cryptography is mathematically oriented, emphasizing the algebraic and geometric aspects of the areas it covers. The journal considers high quality papers of both a theoretical and a practical nature, provided they contain a substantial amount of mathematics.