Cryptographic Classification Of Quasigroups Of Order4

With the development of information science and technology, the safe and efcienttransmission of information has become a popular research subject cryptography. As animportant content to combination design, the theory of quasigroups have been widelyused in cryptography, such as the hash function Edon80is one of the third roundcandidates of the European stream cipher scheme. Edon80is a stream cipher designthat had advanced to the third and last phase of the eSTREAM project. The core ofthe cipher consists quasigroup string e-transformations and it employs four quasigroupsof order4. There are576quasigroups of order4in total, but not every Latin squareis suitable for information encryption. For pseudo random key sequence generators ofEdon80, the increasing of the periods of string processed is very diferent. For someof them it is linear, for others it is exponential. The increasing of periods is the mostimportant index in pseudo random key sequence generators algorithm of stream cipher.By numerous computer experiments, Gligoroski et al. shows that384quasigroups aresuitable and64of them are very suitable. What’s worse, the result of the experimentis not very accurate. So it is very necessary to find a way to determine whether a Latinsquare is suitable for pseudo random sequence algorithm from mathematics theory.In this paper, we analyze the quasigroups of order4from the view of displace-ment, every twenty-four Latin squares have the same of column permutations sets.According to the characteristics of the column permutations sets of quasigroups, wedivided the576quasigroups order of four into24column isomorphism classes, thenwe divided the quasigroups order of four into six column conjugate classes on basicof the conjugate of the permutations of quasigroups and prove the quasigroups in thesame column conjugate classes have the same period factor. Select one quasigroupfrom each column conjugate class, we calculate the mathematical expectation of eachcolumn conjugate classes. According to the characteristics of the mathematical expec-tation of each column conjugate classes, we analyze the quasigroups whether they are suitable for information encryption.The detailed work is as follow:(1) Introduce some relevant concepts of quasigroup and Latin square and the linkbetween them, given the calculation methods of e-transformation of quasigroup strings,and the concept of period factor and the calculation methods of the quasigroups ofperiod factors.(2) According to the characteristics of the column permutations sets of quasi-groups, we divided the576quasigroups order of four into24column isomorphismclasses, then we divided the quasigroups order of four into six column conjugate classeson basic of the conjugate of the permutations of quasigroups and prove the quasigroupsin the same column conjugate classes have the same period factor.(3) According to the probability distribution characteristics of period factors, wecalculate the mathematical expectation of each column conjugate classes. According tothe characteristics of the mathematical expectation of each column conjugate classes,we analyze the quasigroups whether they are suitable for information encryption.