On Some Applications of Transformation Group Theory

Abstract

In this article we give several results and problems on the permutation groups, having an interest for cryptography. (2000): 05E20, 20B05, 20F05. Mathematics Subject ClassificationsKey words: permutation groups, generator systems of groups, primitive groups, length of group, width of group of inertia.Finite groups, and especially permutation groups and the groups of linear transformations of finite vector spaces, are widely applied in various areas of discrete mathematics. They especially play an important role in cryptography. As a matter of fact, any method of encryption of information includes (in a clear or implied way) the algorithm of obtaining of permutations of some alphabet depending on the particular key information. Moreover, a set of applied permutations has to meet a number of requirements. It is desirable, in particular, that permutations be numerous so that they could provide good intermixing of the information, be easily realized and restored with difficulty. The intermixing properties of permutation systems are formalized in a probability-theoretic language, and are provided with their combinatorial-algebraic properties. Thus, such properties of permutation groups as transitivity, primitiveness, multiple transitivity, factorability, etc., play a special role. The permutations and other transformations of finite sets are employed in cryptography not only for encryption of information, but also for various supplementary purposes. The extensive material on permutation groups is contained in the book [15] and review [16]. Here we shall only look into some problems of a group theory having particular value for cryptography, and give several results as examples.Below, symmetrical and alternating permutation groups of the set will be designated accordingly as S( ), A( ), or S n , A n when = {1, . . . , n}. The case, when = {0, 1, . . . , 2 m − 1}, plays a special role in applications. In this caseThe abstract of the talk at the international conference "Algebra and its applications" (Krasnoyarsk, August 5-9, 2002).