Cryptographically suitable quasigroups via functional equations

Abstract

The use of quasigroups in cryptography is increasingly popular. One method to find quasigroups suitable for cryptographic purposes is to use identity sieves, i.e. to find appropriate identities and check candidate quasigroups against them. We propose the use of functional equation approach to this problem. Namely, every identity can be considered as a functional equation and solutions to these equations as models of given identities. The identity i.e. functional equation can be transformed into related generalized functional equation which is suitable for algebraic treatment. A new method of solution of quadratic and parastrophically uncancelablle equations is given, using trees and dichotomies (a special equivalence relations). General solution is given by closed formulas. The quasigroups obtained can be further filtered using much simpler conditions.