Periodic quasigroup string transformations

Abstract

Given a finite quasigroup (Q,*), a quasigroup string transformations e_l and d_l over the strings of elements from Q are defined as follows. e_l(a_1a_2 ... a_n) = b_1b_2 ... b_n if and only if b_i = b_{i-1}*a_i and d_l(a_1a_2 ... a_n)= b_1b_2 ... b_n if and only if b_i = a_{i-1}*a_i, for each i = 1,2, ... ,n, where l=a_0=b_0 is a fixed element of Q. A quasigroup string e- or d-transformation t is periodical if for some periodic string we have t(a_1a_2 ... a_k a_1a_2 ... a_k ... a_1a_2 ... a_k) = a_1a_2 ... a_k a_1a_2 ... a_k ... a_1a_2 ... a_k. The quasigroup string transformations are used in many fields, like: cryptography for designing different cryptographic tools, coding theory for designing error-detecting and error-correcting codes, etc. The properties of the quasigroup string transformations depend on the used quasigroups, and some quasigroups are suitable for cryptographic designs, while some others are suitable for code designs. We give a characterization of the quasigroups producing periodic string transformations, and for that aim quasigroups with period k are defined. One can use this characterization for choosing suitable quasigroups in some applications.