J.Appl. Math. & Computing Vol. 19(2005), No. 1 - 2, pp. 57 - 75 CLASSIFICATION OF QUASIGROUPS BY RANDOM WALK ON TORUS SMILE MARKOVSKI * , DANILO GLIGOROSKI AND JASEN MARKOVSKI Abstract. Quasigroups are algebraic structures closely related to Latin squares which have many different applications. There are several classifi- cations of quasigroups based on their algebraic properties. In this paper we propose another classification based on the properties of strings obtained by specific quasigroup transformations. More precisely, in our research we identified some quasigroup transformations which can be applied to arbi- trary strings to produce pseudo random sequences. We performed tests for randomness of the obtained pseudo-random sequences by random walks on torus. The randomness tests provided an empirical classification of quasi- groups. AMS Mathematics Subject Classification : 20N05, 11K45, 62P99. Key words and phrases : Random walk, quasigroup transformation, χ 2 - test. 1. Introduction Finite quasigroups have applications in many theories like cryptography, cod- ing theory, design theory etc. As building elements in nonlinear Boolean func- tions, they can play important role in cryptographic algorithms with comparable properties as those in S-boxes used in popular cryptographic algorithms [1, 2]. By quasigroups, effective error correcting codes can be defined with properties com- parable as those of Turbo Codes, LDPC, Periodic Boundary Cellular Automata and others [3, 4]. Thus, gaining knowledge about classes of finite quasigroups is very important for successful application of quasigroups in various fields of mathematics and computer science. Received January 17, 2005. * Corresponding author. Part of this work was carried out during the tenure of an ERCIM fellowship of the second author - D. Gligoroski visiting Q2S - Centre for Quantifiable Quality of Service in Communication Systems at Norwegian University of Science and Technology - Trondheim, Norway. c  2005 Korean Society for Computational & Applied Mathematics and Korean SIGCAM. 57