Classification of quasigroups by image patterns

Abstract

Given a finite quasigroup (Q, ∗), we define a quasigroup string transformation e over the strings of elements from Q by e(a1a2 . . . an) = b1b2 . . . bn if and only if bi = bi−1 ∗ ai for each i = 1, 2, . . . , n, where b0 is a fixed element of Q, and ai are elements from Q. These kind of quasigroup string transformations are used for designing several cryptographic primitives and error-correcting codes. Not all quasigroups are suitable for that kind of designs. The set of quasigroups of given order can be separated in two disjoint classes, the class of so called fractal quasigroups and the class of non-fractal quasigroups. The classification is obtained by presenting several consecutive sequences generated by e−transformations and their presentation in matrix form, used to produce suitable image pattern. We note that the fractal quasigroups are usually not suitable for designing cryptographic primitives.