Paper 2008/256 Original: eprint.iacr.org/2008/256 ↗

ON MIDDLE UNIVERSAL WEAK AND CROSS INVERSE PROPERTY LOOPS WITH EQUAL LENGHT OF INVERES CYCLES

Abstract

This study presents a special type of middle isotopism under which the weak inverse property(WIP) is isotopic invariant in loops. A sufficient condition for a WIPL that is specially isotopic to a loop to be isomorphic to the loop isotope is established. Cross inverse property loops(CIPLs) need not satisfy this sufficient condition. It is shown that under this special type of middle isotopism, if $n$ is a positive even integer, then a WIPL has an inverse cycle of length $n$ if and only if its isotope is a WIPL with an inverse cycle of length $n$. But, when $n$ is an odd positive integer. If a loop or its isotope is a WIPL with only $e$ and inverse cycles of length $n$, its isotope or the loop is a WIPL with only $e$ and inverse cycles of length $n$ if and only if they are isomorphic. So, that both are isomorphic CIPLs. Explanations and procedures are given on how these results can be used to apply CIPLs to cryptography.