On Middle Universal Weak and Cross Inverse Property Loops with Equal Length of Inverse Cycles

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. It is shown that under this special type of middle isotopism, whenever $n$ is a positive even integer, a finite WIPL has an inverse cycle of length $n$ if and only if its isotope is a finite WIPL with an inverse cycle of length $n$. But, when $n$ is an odd positive integer and a loop (or its isotope) is a finite WIPL with only $e$ and inverse cycles of length $n$, then its isotope (or the loop) is a finite WIPL with only $e$ and inverse cycles of length $n$ if and only if they are isomorphic. Hence, both are isomorphic CIPLs. Explanations and procedures are given on how these results can be used to apply CIPLs to cryptography.