On the Application of Keedwell Cross Inverse Quasigroup to Cryptography

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.