On A Cryptographic Identity In Osborn Loops

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).Comment: 10 pages, submitted for publicatio

On Multiplication Groups of Quasigroups

Quasigroups are algebraic structures in which divisibility is always defined. In this thesis we investigate quasigroups using a group-theoretic approach. We first construct a family of quasigroups which behave in a group-like fashion. We then focus on the multiplication groups of quasigroups, which have first appeared in the work of A. A. Albert. These permutation groups allow us to study quasigroups using group theory. We also explore how certain natural operations on quasigroups affect the associated multiplication groups. Along the way we take the time and special care to pose specific questions that may lead to further work in the near future

International Journal of Mathematical Combinatorics, Vol.2

The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 460 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences