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.