On the Application of Keedwell Cross Inverse Quasigroup to Cryptography

Jaíyéọlá Tèmítọpé Gbọláhàn

Jaíyéọlá Tèmítọpé Gbọláhàn

Abstract

In 1999, A. D. Keedwell found cross inverse property quasigroups (CIPQs) applicable to J. H. Ellis's original schema for a public key encryption. The present study devices a mechanism of changing the use of the Keedwell CIPQs against attack on a system(as required by the author). This is done as follows. The holomorphic structure of automorphic inverse property quasigroups (loops) [AIPQs (AIPLs) ] and cross inverse property quasigroups (loops) [ 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 ). By constructing two isotopic quasigroups(loops) and such that their automorphism groups are not trivial, it is shown that is a AIPQ or CIPQ (AIPL or CIPL) if and only if is a AIPQ or CIPQ (AIPL or CIPL). Explanations are given on how these CIPQs can be incorporated into the encryption scheme of Keedwell for higher security using a computer.

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