On The Period Factor Of Quasigroups

Quasigroup is a algebraic structures similar with group. Quasigroup theory isrelatively classic and have more problems to research in combinatorical design theory.In recent years, the application of quasigroup theory has been widespread applied incomputer science, coding theory and confidentical communication and so on. Manyencryption algorithms, message authentication algorithms and error correcting codealgorithms are based on quasigroups. Edon80is the candidate algorithm of Europesequence cipher scheme which have into the third round. The key sequence generatorof Edon80is nonlinear pseudo random sequence generator algorithm which is basedon four quasigroups of order four. International and domestic academics research theperiods growth of quasigroups by statistical experiment and find that the increasing ofthe periods of string processed is very different for different quasigroups. What’s worse,the result of the experiments are not coherent. And for higher order of quasigroups,their statistical tests are almost impossible. So, we need find a mathematical theoryto discuss the periods growth of quasigroups. In this paper, we discuss the probabilitydistribution of periods of quasigroups based on Frobenius groups.A Frobenius group is a transitive permutation group which is not regular, but inwhich only the identity has more than one fixed point. The length of the permutationfrom Frobenius group is or1. So, the periods of quasigroups based on Frobeniusgroups are more bigger in theory. In this paper, we give the constructions of quasigroupsfrom Frobenius groups and discuss the probability distribution of periods factors of thequasigroups with large periods growth.The whole paper is divided into three chapters.Chapter1In this chapter, we introduce the backgrounds and developments ofstream cipher, present the concept of quasigroups and Latin squares, and give thedevelopments of the periods of the stream cipher Edon80.Chapter2In this chapter, we first introduce the knowledge of the periods factors and the calculate formula of the distribution of period factors of quasigroups, and givethe distribution of periods factors of quasigroups of order four, then introduc the periodfactor of order five.Chapter3In this chapter, we first present the concept of Frobenius groups,give the constructions of quasigroups, and discuss the distribution of periods factorsof quasigroups based on Frobenius groups. Then introduce the period of quasigroupsbased on the density function of normal distribution and discuss the quasigroups forinformation encryption.