The present study further strengthens the use of the Keedwell CIPQ against attack on a system by the use of the Smarandache Keedwell CIPQ for cryptography in a similar spirit in which the cross inverse property has been used by Keedwell. This is done as follows. By constructing two S-isotopic S-quasigroups(loops) $U$ and $V$ such that their Smarandache automorphism groups are not trivial, it is shown that $U$ is a SCIPQ(SCIPL) if and only if $V$ is a SCIPQ(SCIPL). Explanations and procedures are given on how these SCIPQs can be used to double encrypt information.
A loop is a weak inverse property loop(WIPL) if and only if it obeys the identity
A loop(quasigroup) is a cross inverse property loop(quasigroup)[CIPL(CIPQ)] if and only if it obeys the identity
A loop(quasigroup) is an automorphic inverse property loop(quasigroup)[AIPL(AIPQ)] if and only if it obeys the identity As observed by Osborn [17], a loop is a WIPL and an AIPL if and only if it is a CIPL. The past efforts of Artzy [1,4,3,2], Belousov and Tzurkan [5] and recent studies of Keedwell [12], Keedwell and Shcherbacov [13,14,15] are of great significance in the study of WIPLs, AIPLs, CIPQs and CIPLs, their generalizations(i.e m-inverse loops and quasigroups, (r,s,t)-inverse quasigroups) and applications to cryptography.
Interestingly, Huthnance [7] showed that if (L, •) is a loop with holomorph (H, •), (L, •) is a WIPL if and only if (H, •) is a WIPL. But the holomorphic structure of AIPL and a CIPL has just been revealed by Jaíyéo . lá [11].
In the quest for the application of CIPQs with long inverse cycles to cryptography, Keedwell [12] constructed the following CIPQ which we shall specifically call Keedwell CIPQ. The author also gave examples and detailed explanation and procedures of the use of this CIPQ for cryptography. Cross inverse property quasigroups have been found appropriate for cryptography because of the fact that the left and right inverses x λ and x ρ 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 finite sequence of elements
The number n is called the length of the cycle. The origin of the idea of cycles can be traced back to Artzy [1,4] 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” for now.
In application, it is assumed that the message to be transmitted can be represented as single element x of a quasigroup (L, •) and that this is enciphered by multiplying by another element y of L so that the encoded message is yx. At the receiving end, the message is deciphered by multiplying by the right inverse y ρ of y. If a left(right) inverse quasigroup is used and the left(right) inverse of x is x λ (x ρ ), then the left(right) inverse of x λ (x ρ ) is necessarily x. But if a CIPQ is used, this is not necessary the situation. This fact makes an attack on the system more difficult in the case of CIPQs.
The study of Smarandache loops was initiated by W. B. Vasantha Kandasamy in 2002. In her book [19], she defined a Smarandache loop(S-loop) as a loop with at least a subloop which forms a subgroup under the binary operation of the loop. In [9], the present author defined a Smarandache quasigroup(S-quasigroup) to be a quasigroup with at least a non-trivial associative subquasigroup called a Smarandache subsemigroup (S-subsemigroup). Examples of Smarandache quasigroups are given in Muktibodh [16]. In her book, she introduced over 75 Smarandache concepts on loops. In her first paper [20], on the study of Smarandache notions in algebraic structures, she introduced Smarandache : left(right) alternative loops, Bol loops, Moufang loops, and Bruck loops. But in [8], the present author introduced Smarandache : inverse property loops(IPL) and weak inverse property loops(WIPL).
A quasigroup(loop) is called a Smarandache “certain” quasigroup(loop) if it has at least a non-trivial subquasigroup(subloop) with the “certain” property and the latter is referred to as the Smarandache “certain” subquasigroup(subloop). For example, a loop is called a Smarandache CIPL(SCIPL) if it has at least a non-trivial subloop that is a CIPL and the latter is referred to as the Smarandache CIP-subloop. By an “initial S-quasigroup” L with an “initial S-subquasigroup” L ′ , we mean L and L ′ are pure quasigroups, i.e. they do not obey a “certain” property(not of any variety).
If L is a S-groupoid with a S-subsemigroup H, then the set SSY M(L, •) = SSY M(L) of all bijections A in L such that A : H → H forms a group called the Smarandache permutation(symmetric) group of the S-groupoid. In fact, SSY M(L) ≤ SY M(L). Definition 1.1 Let (L, •) and (G, •) be two distinct groupoids that are isotopic under a triple (U, V, W ). Now, if (L, •) and (G, •) are S-groupoids with S-subsemigroups L ′ and G ′ respectively such that A :
δ, γ ∈ SAUM(V ) such that their Smarandache automorphism groups are non-trivial and are conjugates in SSY M(L) i.e there exists a ψ ∈ SSY M(L) such that for any γ ∈ SAUM(V ), γ = ψ -1 αψ where α ∈ SAUM(U). Then, U is a SCIPQ(SCIPL) if and only if V is a SCIPQ(SCIPL).
Let U be an SCIPQ(SCIPL), then since H S (U) has a subquasigroup(subloop) that is isomorphic to a S-CIP-subquasigroup(subloop) of U and that subquasigroup(subloop) is isomorphic to a S-subquasigroup(subloop) of H S
This content is AI-processed based on open access ArXiv data.