Holomorph of Self-Distributive Quasigroup with Key Laws

Abstract

Holomorph theory has received some considerations in the past. The holomorph of an algebraic structure is a semi-direct product of it and a set of maps on it. In this paper, the study of the holomorphy of self-distributive quasigroup with key laws were carried out. Given a quasigroup (Q, ·) a subsemigroup E(Q, ·) = E(Q) of the endomorphism semigroup of Q with holomorph H(Q) = E(Q) × Q. It was shown that H(Q) is a left (right) distributive groupoid if and only if E(Q, ·) is a left (right) distributive quasigroup and the generalized left (right) distributive law in (Q, ·) hold. Also, it was shown that H(Q) obeys the left (right) key law if and only if E(Q, ·) obeys the left (right) key law and the generalized left (right) key law in (Q, ·) hold. Therefore, H(Q) is a distributive groupoid if and only if E(Q, ·) is distributive quasigroup and
the generalized distributive law in (Q, ·) hold. Also, H(Q) obeys the key laws if and only if
E(Q, ·) obeys key laws and the generalized key laws in (Q, ·) hold. The results on the holomorph
of left and right key laws were shown to be applicable to symmetric cryptography (secret key
cryptosystem).