Abstract

Finite quasi-groups without sub-quasi-groups are considered. It is shown that polynomially complete quasi-groups with this property are quasi-primal. The case in which the automorphism groups act transitively on these quasi-groups is considered. Quasi-groups of prime-power order defined on an arithmetic vector space over a finite field are also studied. Necessary conditions for a multiplication in this space given in coordinate form to determine a quasi-group are found. The case of a vector space over the two-element field is considered in more detail. A criterion for a multiplication given in coordinate form by Boolean functions to determine a quasi-group is obtained. Under certain assumptions, quasi-groups of order 4 determined by Boolean functions are described up to isotopy. Polynomially complete quasi-groups are important in that the problem of solving polynomial equations is NP-complete in such quasi-groups. This property suggests using them for protecting information, because cryptographic transformations are based on quasi-group operations. In this context, an important role is played by quasi-groups containing no sub-quasi-groups.

中文翻译：

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没有子拟群的有限拟群的自同构

摘要

考虑没有子拟群的有限拟群。结果表明，具有该性质的多项式完全拟群是拟原始的。考虑自同构群在这些拟群上传递作用的情况。还研究了有限域上在算术矢量空间上定义的素数次幂的拟群。找到在此空间中以坐标形式给出的用于确定准群的乘法的必要条件。更详细地考虑了在两个元素字段上的向量空间的情况。获得了由布尔函数以坐标形式给出的用于确定准群的乘法的标准。在某些假设下，由布尔函数确定的4阶拟群被描述为同位素。多项式完全拟群是重要的，因为在这种拟群中求解多项式方程的问题是NP-完全的。此属性建议使用它们来保护信息，因为密码转换基于准组操作。在这种情况下，不包含子准群体的准群体扮演着重要角色。