论有限域上代数几何编码的线性互补对

IF 0.7 3区 数学 Q2 MATHEMATICS
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引用次数: 0

摘要

线性互补双(LCD)码和线性互补对(LCP)码作为直接和掩码(DSM)中对抗侧信道攻击(SCA)和故障注入攻击(FIA)的新应用已被提出。当整个算法需要屏蔽时(在智能卡等环境中),针对 FIA 的对策可能会导致 SCA 的漏洞。这就导致了 LCD 和 LCP 问题的变种,其中 LCD 代码获得了大量结果,但 LCP 代码只获得了部分结果。鉴于薄弱结果之间的差距及其特殊重要性,本文旨在通过进一步研究特殊码族中的码的 LCP,准确地说,是有限域上代数几何码的 LCP 码的表征和构造机制来缩小这一差距。值得注意的是,我们提出了从椭圆曲线构建显式 LCP 码。此外,我们还研究了衍生 LCP 码 (C,D) 的安全参数(尤其是循环码),这些参数由最小距离 d(C) 和 d(D⊥) 给出。此外,我们还证明了对于 LCP 代数几何编码 (C,D),在我们提出的一些特定条件下,对偶编码 C⊥ 等同于 D。最后,我们研究了代数几何编码的 MDS LCP 是否存在(由于其理论意义和实用性,MDS 编码是编码理论中最重要的编码之一)。2018年,Mesnager、Tang和Qi在[11]中给出了从任意代数曲线获得LCD码的构造方案。据我们所知,这是首次对代数几何码的 LCP 进行研究。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On linear complementary pairs of algebraic geometry codes over finite fields

Linear complementary dual (LCD) codes and linear complementary pairs (LCP) of codes have been proposed for new applications as countermeasures against side-channel attacks (SCA) and fault injection attacks (FIA) in the context of direct sum masking (DSM). The countermeasure against FIA may lead to a vulnerability for SCA when the whole algorithm needs to be masked (in environments like smart cards). This led to a variant of the LCD and LCP problems, where several results were obtained intensively for LCD codes, but only partial results were derived for LCP codes. Given the gap between the thin results and their particular importance, this paper aims to reduce this by further studying the LCP of codes in special code families and, precisely, the characterization and construction mechanism of LCP codes of algebraic geometry codes over finite fields. Notably, we propose constructing explicit LCP of codes from elliptic curves. Besides, we also study the security parameters of the derived LCP of codes (C,D) (notably for cyclic codes), which are given by the minimum distances d(C) and d(D). Further, we show that for LCP algebraic geometry codes (C,D), the dual code C is equivalent to D under some specific conditions we exhibit. Finally, we investigate whether MDS LCP of algebraic geometry codes exist (MDS codes are among the most important in coding theory due to their theoretical significance and practical interests). Construction schemes for obtaining LCD codes from any algebraic curve were given in 2018 by Mesnager, Tang and Qi in [11]. To our knowledge, it is the first time LCP of algebraic geometry codes has been studied.

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来源期刊
Discrete Mathematics
Discrete Mathematics 数学-数学
CiteScore
1.50
自引率
12.50%
发文量
424
审稿时长
6 months
期刊介绍: Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory. Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.
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