The Periodicity Of Nonlinear Pseudo Random Sequence Based On Latin Squares

In recent years, the permutation has been widely applied in cryptography, the theory of Latin squares (quasigroup) applications in cryptology goes through the period of rapid enough growth. In fact, a Latin square consists of column permutations. But not every Latin square (quasigroup) are suitable for information encryption. For example, for pseudo random key sequence generators of Edon80, the increasing of the periods of string processed is very different. For some of them it is linear, for others it is exponential. The increasing of periods is the most important index in pseudo random key sequence generators algorithm of stream cipher. International and domestic academics research the periods growth of Latin squares by statistical experiment. But for higher order of Latin squares, their statistical test are almost impossible, What’s worse, the result of the experiment is not very accurate. So it is very necessary to find a way to determine whether a Latin square is suitable for pseudo random sequence algorithm from mathematics theory. In this paper, We just to solve the problem from Frobenius groups.A Frobenius group is a transitive permutation group which is not regular, but in which only the identity has more than one fixed point. In other words, A Frobenius group is a transitive permutation group whose number of times and minium times is n and n-1respectively. The group has been repeatedly studied, the main results are Frobenius a theorem:Let U denote a subgroup of the group of units of a field F. Then the set G consisting of all permutations of F of the form is a Frobenius group. In this paper, we give the constructions of Latin squares with large periods growth from Frobenius groups. The whole paper is divided into three chapters.Chapter1In this chapter, we introduce the backgrounds and developments ofLatin squares, present the concept of the column permutation of the Latin square.Chapter2In this chapter, We first introduce the knowledge of the Frobeniusgroups and give the constructions of Latin squares.Chapter3In this chapter, the concept of periods growth of Latin squares isfirstly presented, and discuss periods growth of Latin squares on diferent classificationcriteria.