The limited deficit method and the problem of constructing orthomorphisms and almost orthomorphisms of Abelian groups

IF 0.3 Q4 MATHEMATICS, APPLIED Discrete Mathematics and Applications Pub Date : 2021-10-01 DOI:10.1515/dma-2021-0030
A. V. Menyachikhin
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Abstract

Abstract The limited deficit method is described, which allows constructing new orthomorphisms (almost orthomorphisms) of groups with the use of those already known. A class of transformations is described under which the set of all orthomorphisms (almost orthomorphisms) remains invariant. It is conjectured that the set of all orthomorphisms (almost orthomorphisms) is generated by transformations implemented by the limited deficit method. This conjecture is verified for all Abelian groups of order at most 12. The spectral-linear method and the spectral-differential method of design of permutations over the additive group of the field 𝔽2m (m = 4, …, 8) are used to construct orthomorphisms with sufficiently high values of the most important cryptographic parameters.
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有限亏缺法与构造阿贝尔群的正态和几乎正态问题
摘要描述了利用已知群的正态构造群的新正态(几乎正态)的有限亏缺方法。描述了一类变换,在该类变换下,所有正态(几乎正态)的集合保持不变。我们推测所有正态(几乎正态)集合是由有限亏缺法实现的变换生成的。这个猜想在所有不超过12阶的阿贝尔群上都得到了验证。利用谱-线性法和谱-微分法设计域𝔽2m (m = 4,…,8)加性群上的置换,构造了具有足够高值的最重要密码参数的正态。
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来源期刊
CiteScore
0.60
自引率
20.00%
发文量
29
期刊介绍: The aim of this journal is to provide the latest information on the development of discrete mathematics in the former USSR to a world-wide readership. The journal will contain papers from the Russian-language journal Diskretnaya Matematika, the only journal of the Russian Academy of Sciences devoted to this field of mathematics. Discrete Mathematics and Applications will cover various subjects in the fields such as combinatorial analysis, graph theory, functional systems theory, cryptology, coding, probabilistic problems of discrete mathematics, algorithms and their complexity, combinatorial and computational problems of number theory and of algebra.
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