1, it can be deduced that if (Q, ) and (Q, ) are principal isotopes of (Q, ) and (x, u, v) = RvR[u\(xv)]LuLx, then (Q, x, v, ) 1 = (Q, u, (x, u, v), ) where (x, u, v) = (u\([(uv)/(u\(xv))]v)) for all x, u, v Q. Let Q(z, y, ) be an arbitrary principal isotope of (Q, ). We now switch to Theorem 1.8. Let a = x, b = v, c = u, d = (x, u, v) = (u\([(uv)/(u\(xv))]v)), f = z and g = y. = (x, u, v)1 = LxLuR[u\(xv)]Rv while 1 = (x, u, v) = RvR[u\(xv)]LuLx. (f b)/d = {(u[x\(zv)])/[u\(xv)] v}/{u\([(uv)/(u\(xv))]v)} and [f (a\c1)] = {u x\{z u\((u/v)[u\(xv)])}}/[u\(xv)] v. Thus, (f b)/d = [f (a\c1)] if and only if identity OSI1 0 is obeyed by (Q, , \, /). The next formulae after OSI1 0 derived by putting u = v = e into OSI1 xRx. In an Osborn loop, T(x) = LxRx, so we have the DLIP. T(x) = LxRxR 0. Consequently, 2.2 Application Of two Universal Osborn Loops Identities To Cryptography

Among the few identities that have been established for universal Osborn loops in Theorem 2.2, we would recommend two of them; OSI1.1 0 and DLIP for cryptography in a similar spirit in which the cross inverse property has been used by Keedwell . It will be recalled that CIPLs have been found appropriate for cryptography because of the fact that the left and right inverses x and x of an element x do not coincide unlike in left and right inverse property loops, hence this gave rise to what is called cycle of inverses or inverse cycles or simply cycles i.e nite sequence of elements x1, x2, , xn such that x k = xk+1 mod n. The number n is called the length of the cycle. The origin of the idea of cycles can be traced back to Artzy [1, 2] where he also found there existence in WIPLs apart form CIPLs. In his two papers, he proved some results on possibilities for the values of n and for the number m of cycles of length n for WIPLs and especially CIPLs. We call these Cycle Theorems fo

In Corollary 3.4 of Jaiyeo. la and Adenran , it was established that in a universal Osborn loop, J = J, 3-PAP, LSIP and RSIP are equivalent conditions. Furthermore, in a CC-loop, the power associativity property, 3-PAPL, x = x, LSIP and RSIPL were shown to be equivalent in Corollary 3.5. Thus, universal Osborn loops without the LSIP or RSIP will have cycles(even long ones). This exempts groups, extra loops, and Moufang loops but includes CC-loops, VD-loops and universal WIPLs. Precisely speaking, nonpower associative CC-loops will have cycles. So broadly speaking, universal 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). The next step now is to be able to identify suitably chosen identities in universal Osborn loops, 9

These identities will be called Osborn cryptographic identities(or just cryptographic identities).