**ON MIDDLE UNIVERSAL $m$-INVERSE QUASIGROUPS AND THEIR APPLICATIONS TO CRYPTOGRAPHY**

*JAIYEOLA Temitope Gbolahan*

**Abstract: **This study presents a special type of middle isotopism under which
$m$-inverse quasigroups are isotopic invariant. A sufficient
condition for an $m$-inverse quasigroup that is specially isotopic
to a quasigroup to be isomorphic to the quasigroup isotope is
established. It is shown that under this special type of middle
isotopism, if $n$ is a positive even integer, then, a quasigroup is
an $m$-inverse quasigroup with an inverse cycle of length $nm$ if
and only if its quasigroup isotope is an $m$-inverse quasigroup with
an inverse cycle of length $nm$. But when $n$ is an odd positive
integer. Then, if a quasigroup is an $m$-inverse quasigroup with an
inverse cycle of length $nm$, its quasigroup isotope is an
$m$-inverse quasigroup with an inverse cycle of length $nm$ if and
only if the two quasigroups are isomorphic. Hence, they are
isomorphic $m$-inverse quasigroups. Explanations and procedures are
given on how these results can be used to apply $m$-inverse
quasigroups to cryptography, double cryptography and triple
cryptography.

**Category / Keywords: **$m$-inverse quasigroups, ${\cal T}_m$ condition,length of inverse cycles, cryptography

**Publication Info: **Submitted for Publication

**Date: **received 4 Jun 2008, last revised 4 Jun 2008

**Contact author: **tjayeola at oauife edu ng

**Available format(s): **PDF | BibTeX Citation

**Note: **m-inverse quasigroups are generalizations of weak and cross inverse loops which are useful in cryptography.

**Version: **20080610:130013 (All versions of this report)

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