V. Dimitrova, S. Markovski and A. Mileva
Periodic quasigroup string transformations

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(a₁a₂...a_n)=b₁b₂...b_n if and only if b_i=b_i-1*a_i and d_l(a₁a₂...a_n) =b₁b₂...b_n if and only if b_i = a_i-1*a_i, for each i=1,2,...,n, where l=a₀=b₀ is a fixed element of Q. A quasigroup string e- or d-transformation t is periodical if for some periodic string we have t(a₁a₂...a_ka₁a₂...a_k...a₁a₂...a_k)= a₁a₂...a_ka₁a₂...a_k...a₁a₂...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.