On A Cryptographic Identity In Osborn Loops

TL;DR

This paper explores new algebraic properties of Osborn loops, introduces novel identities, and discusses their potential cryptographic applications, especially focusing on non-WIP Osborn loops and their structural characteristics.

Contribution

It introduces the left and right self inverse property loop identities, shows their equivalence in Osborn loops, and proposes an Osborn cryptographic identity for cryptography.

Findings

LSIPL and RSLPL are equivalent in Osborn loops.

Power associative CC-loops are RSLPL or LSIPL.

Identified an Osborn cryptographic identity with potential cryptographic use.

Abstract

This study digs out some new algebraic properties of an Osborn loop that will help in the future to unveil the mystery behind the middle inner mappings $T_{(x)}$ of an Osborn loop. These new algebraic properties, will open our eyes more to the study of Osborn loops like CC-loops which has received a tremendious attention in this $21^\textrm{st}$ and VD-loops whose study is yet to be explored. In this study, some algebraic properties of non-WIP Osborn loops have been investigated in a broad manner. Huthnance was able to deduce some algebraic properties of Osborn loops with the WIP i.e universal weak WIPLs. So this work exempts the WIP. Two new loop identities, namely left self inverse property loop(LSIPL) identity and right self inverse property loop(RSLPL) are introduced for the first time and it is shown that in an Osborn loop, they are equivalent. A CC-loop is shown to be power associative if and only if it is a RSLPL or LSIPL. Among the few identities that have been established for Osborn loops, one of them is recognized and recommended for cryptography in a similar spirit in which the cross inverse property has been used by Keedwell following the fact that it was observed that Osborn loops that do not have the LSIP or RSIP or 3-PAPL or weaker forms of inverse property, power associativity and diassociativity to mention a few, will have cycles(even long ones). These identity is called an Osborn cryptographic identity(or just a cryptographic identity).