Abstract

In 1978, Robert McEliece constructed the first asymmetric code-based cryptosystem using noise-immune Goppa codes; no effective key attacks has been described for it yet. By now, quite a lot of code-based cryptosystems are known; however, their cryptographic security is inferior to that of the classical McEliece cryptosystem. In connection with the development of quantum computing, code-based cryptosystems are considered as an alternative to number theoretical ones; therefore, the problem of seeking promising classes of codes to construct new secure code-based cryptosystems is relevant. For this purpose, noncommutative codes can be used, that is, ideals in group algebras \({{\mathbb{F}}_{q}}G\) over finite noncommutative groups \(G\). The security of cryptosystems based on codes induced by subgroup codes has been studied earlier. The Artin–Wedderburn theorem, which proves the existence of an isomorphism of a group algebra to the direct sum of matrix algebras, is important for studying noncommutative codes. However, the particular form of terms and the construction of the isomorphism are not specified by this theorem; thus, for each group, there remains the problem of constructing the Wedderburn representation. The complete Wedderburn decomposition for the group algebra \({{\mathbb{F}}_{q}}{{D}_{{2n}}}\) over the dihedral group \({{D}_{{2n}}}\) has been obtained by F.E. Brochero Martinez in the case when the cardinality of the field and the order of the group are relatively prime numbers. Using these results, we study codes in the group algebra \({{\mathbb{F}}_{q}}{{D}_{{2n}}}\) in this paper. The problem on the structure of all codes is solved, and the structure of codes induced by codes over cyclic subgroups of \({{D}_{{2n}}}\) is described, which is of interest for cryptographic applications.

中文翻译：

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二面体群代数中的代码

摘要

1978年，罗伯特·麦克里斯（Robert McEliece）使用了抗噪Goppa码构建了第一个基于非对称码的密码系统。尚未对此进行有效的密钥攻击。到目前为止，已经知道了很多基于代码的密码系统。但是，它们的密码安全性不如经典的McEliece密码系统。随着量子计算的发展，基于代码的密码系统被认为是数论密码系统的替代方案。因此，寻找有希望的代码类别以构建新的基于安全代码的密码系统的问题是相关的。为此，可以使用非可交换代码，即，在有限的非可交换组\（G \）上的组代数\（{{\ mathbb {F}} _ {q}} G \）中的理想情况。先前已经研究了基于由子组代码引起的代码的密码系统的安全性。Artin-Wedderburn定理证明了组代数与矩阵代数的直接和同构的同构性，对于研究非交换码非常重要。但是，该定理未指定术语的特殊形式和同构的构造。因此，对于每个组，仍然存在构造Wedderburn表示形式的问题。完整韦德伯恩分解为组代数\（{{\ mathbb {F}} _ {Q}} {{d} _ {{2N}}} \）在二面体群\（{{d} _ {{2n个}}} \）当字段的基数和组的顺序是相对质数时，FE Brochero Martinez已获得。利用这些结果，我们在本文中研究了组代数\（{{\ mathbb {F}} _ {q}} {{D} _ {{2n}}} \\）中的代码。解决了所有代码结构上的问题，并描述了在\（{{D} _ {{2n}}} \\的循环子组上由代码引起的代码结构，这对于密码学应用很有意义。