论文标题
使用组代数对某些类别代码的有效描述
Efficient Description of some Classes of Codes using Group Algebras
论文作者
论文摘要
循环矩阵是在编码理论和密码学中广泛使用的重要工具。循环矩阵是一个方形矩阵，其行是第一行的循环移位。这样的矩阵可以有效地存储在内存中，因为它已由其第一行指定。 $ n \ times n $循环矩阵的戒指可以用商环$ \ mathbb {f} [x]/（x^n-1）$识别。因此，环$ \ mathbb {f} [x]/（x^n-1）$的强代代数结构可用于研究所有$ n \ times n $循环矩阵收集的属性。环$ \ mathbb {f} [x]/（x^n-1）$是组代数的特殊情况，并且可以用单个列指定的方形矩阵来表示任何有限尺寸组代数的元素。在本文中，我们研究了这种表示形式，并证明它是保留$ \ Mathbb {f} $ - 代数的同态同态的注入性锤锤，并在基础群体为Abelian的情况下对其进行了分类。 我们的工作是出于概括自行车加密系统的愿望（NIST竞争中的竞争者，以获得新的非对称加密后量子标准）。组代数可用于设计类似的密码系统，或者更普遍地用于构建线性代码的低密度或中等密度平均检查矩阵。
Circulant matrices are an important tool widely used in coding theory and cryptography. A circulant matrix is a square matrix whose rows are the cyclic shifts of the first row. Such a matrix can be efficiently stored in memory because it is fully specified by its first row. The ring of $n \times n$ circulant matrices can be identified with the quotient ring $\mathbb{F}[x]/(x^n-1)$. In consequence, the strong algebraic structure of the ring $\mathbb{F}[x]/(x^n-1)$ can be used to study properties of the collection of all $n\times n$ circulant matrices. The ring $\mathbb{F}[x]/(x^n-1)$ is a special case of a group algebra and elements of any finite dimensional group algebra can be represented with square matrices which are specified by a single column. In this paper we study this representation and prove that it is an injective Hamming weight preserving homomorphism of $\mathbb{F}$-algebras and classify it in the case where the underlying group is abelian. Our work is motivated by the desire to generalize the BIKE cryptosystem (a contender in the NIST competition to get a new post-quantum standard for asymmetric cryptography). Group algebras can be used to design similar cryptosystems or, more generally, to construct low density or moderate density parity-check matrices for linear codes.
