# Inverse Properties in Neutrosophic Triplet Loop and Their Application to Cryptography
## 3 Application to Cryptography  
[PERSON] [45], [PERSON] [46; 47; 48; 49], [PERSON] [50; 51; 52; 53; 54; 55], and [PERSON] and [PERSON] [56] are of great significance in the study of quasigroups and loop with the WIP, AIP, CIP, their generalizations (i.e., \(m\)-inverse loops and quasigroups, (r,s,t)-inverse quasigroups) and applications to cryptography.  
Figure 1: Inverse property neutrosophic triplet loop (NTL) Hasse diagrams. AAIP: antiautomorphic inverse property; AIP: automorphic inverse property; CIP: cross inverse property; LCIP: left cross inverse property; LIP: left inverse property; LWIP: left weak inverse property; RCIP: right cross inverse property; RIP: right inverse property; RWIP: right weak inverse property; SAIP: semi-automorphic inverse property; WIP: weak inverse property.  
Cross inverse property quasigroups have been found appropriate for cryptography because they give rise to what is called 'cycle of inverses' or 'inverse cycles' or simply 'cycles'.  
After [PERSON] [57] studied the universality of Osborn loops; a class of loop which includes universal WIP loops, some of the identities established in Jaiyeola and Adeniran [58] were singled out and christened 'cryptographic identities', and their applications to cryptography have been reported in [PERSON] [59; 60], Jaiyeola and Adeniran [61].  
Going by Lemma 1, Lemma 3, and Theorem 6, a CIPNTL, IPNTL, LWIPNTL, or RWIPNTL \(X\) obeys the property \(anti\big{(}anti(x)\big{)}=x\) for any \(x\in X\). Additionally, by Lemma 4, a commutative NTL \(X\) with RIP or LIP or RCIP or LCIP also has the property \(anti\big{(}anti(x)\big{)}=x\) for any \(x\in X\). Hence, long inverse cycles which naturally arise in CIP quasigroup will not be feasible for such NTLs. However, for an RCIPNTL, LCIPNTL, RIPNTL, or LRIPNTL \(X\) that is non-commutative, long inverse cycles will be feasible (this makes an attack on the system more difficult). Thus, such a non-commutative NTL which is not a CIPNTL, IPNTL, RWIPNTL, or RWIPNTL will be appropriate for cryptography. The procedure for applying any of them is described below.  
**RCIPNTL**: Assume that the message to be transmitted can be represented as a single element \(x\in X\). Then, this is enciphered by pre-multiplying by another element \(y\in X\) so that the cipher text is \(yx\in X\). At the receiving end, the cipher text is deciphered by post-multiplying by \(anti(y)\in X\) to get the plain text.
**LCIPNTL**: Assume that the message to be transmitted can be represented as a single element \(x\in X\). Then, this is enciphered by post-multiplying by another element \(y\in X\) so that the cipher text is \(xy\in X\). At the receiving end, the cipher text is deciphered by pre-multiplying by \(anti(y)\in X\) to get the plain text.
**RIPNTL**: Assume that the message to be transmitted can be represented as a single element \(x\in X\). Then, this is enciphered by post-multiplying by another element \(y\in X\) so that the cipher text is \(xy\in X\). At the receiving end, the cipher text is deciphered by post-multiplying by \(anti(y)\in X\) to get the plain text.
**LIPNTL**: Assume that the message to be transmitted can be represented as a single element \(x\in X\). Then, this is enciphered by pre-multiplying by another element \(y\in X\) so that the cipher text is \(yx\in X\). At the receiving end, the cipher text is deciphered by pre-multiplying by \(anti(y)\in X\) to get the plain text.  
Note that these four procedures can alternatively be carried out using Theorem 2.  
**Author Contributions:** [PERSON] has introduced many properties of the neutrosophic triplet loop and he proved them. He also presented an application of the neutrosophic triplet loops into cryptography. [PERSON] has co-introduced the neutrosophic triplet set and group and their properties. He defined the neutrosophic triplet loop and several of its properties.  
**Conflicts of Interest:** The authors declare no conflict of interest.