Groupoids orthogonal to quasigroups and their application
by
Smile Markovski
Institute of Informatics, Faculty of Sciences, Skopje, Macedonia
Coauthors: Ana Sokolova (II, Faculty of Sciences, Skopje, Macedonia), Lidija Gorachinova (PF, Shtip, Macedonia)

Given a groupoid (Q, *), a groupoid (Q, o ) is said to be orthogonal to (Q, *) iff the mapping (x, y) --> (x*y, x o y) is injective. Taking (Q, *) to be a quasigroup and defining x o y = x*(x*y) it is known that the groupoid (Q, o ) is orthogonal to (Q, *). We focus on finite quasigroups satisfying identity x*(x*y)=y or x*(x*y)=x*y or x*(x*y)=y*x and symmetric ones, and then their orthogonal complement is a left-zero groupoid or a quasigroup, respectively. Several constructions of such quasigroups are given. In many cases we have seen that if a quasigroup posesses an orthogonal complement which is left(right) zero groupoid or quasigroup itself then the so called quasigroup string processing gives as a result "fractal" image. This gives us a tool for avoiding certain quasigroups in application of quasigroups in pseudo random number generation and security protocols.

Date received: May 22, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caee-58.