A Double Cryptography Using The Keedwell Cross Inverse Quasigroup

TL;DR

This paper explores the algebraic structure of holomorphs of quasigroups and loops, establishing conditions for their properties and demonstrating how these structures can be used for double encryption in cryptography.

Contribution

It provides new necessary and sufficient conditions for holomorphs of quasigroups to be AIPQ or CIPQ, and shows how these can be applied to enhance cryptographic security.

Findings

Holomorphs are AIPQ or CIPQ iff automorphism group is trivial.

Holomorphs of CIPQ are flexible unipotent of exponent 2.

Constructed isotopic quasigroups demonstrate conditions for AIPQ and CIPQ properties.

Abstract

The present study further strenghtens the use of the Keedwell CIPQ against attack on a system. This is done as follows. The holomorphic structure of AIPQs(AIPLs) and CIPQs(CIPLs) are investigated. Necessary and sufficient conditions for the holomorph of a quasigroup(loop) to be an AIPQ(AIPL) or CIPQ(CIPL) are established. It is shown that if the holomorph of a quasigroup(loop) is a AIPQ(AIPL) or CIPQ(CIPL), then the holomorph is isomorphic to the quasigroup(loop). Hence, the holomorph of a quasigroup(loop) is an AIPQ(AIPL) or CIPQ(CIPL) if and only if its automorphism group is trivial and the quasigroup(loop) is a AIPQ(AIPL) or CIPQ(CIPL). Furthermore, it is discovered that if the holomorph of a quasigroup(loop) is a CIPQ(CIPL), then the quasigroup(loop) is a flexible unipotent CIPQ(flexible CIPL of exponent 2). By constructing two isotopic quasigroups(loops) $U$ and $V$ such that their automorphism groups are not trivial, it is shown that $U$ is a AIPQ or CIPQ(AIPL or CIPL) if and only if $V$ is a AIPQ or CIPQ(AIPL or CIPL). Explanations and procedures are given on how these CIPQs can be used to double encrypt information.