# On the Application of Keedwell Cross Inverse Quasigroup to Cryptography

### Abstract

In 1999, A. D. Keedwell found cross inverse property quasigroups (CIPQs) applicable to J. H. Ellis's original schema for a public key encryption. The present study devices a mechanism of changing the use of the Keedwell CIPQs against attack on a system(as required by the author). This is done as follows. The holomorphic structure of automorphic inverse property quasigroups (loops) [AIPQs (AIPLs) ] and cross inverse property quasigroups (loops) [ CIPQs (CIPLs) ] are investigated. Necessary and sufficient conditions for the holomorph of a quasigroup(loop) to be an AIPQ (AIPL) or CIPQ (CIPL) are established. It is shown that if the holomorph of a quasigroup(loop) is a AIPQ(AIPL) or CIPQ (CIPL), then the holomorph is isomorphic to the quasigroup(loop). Hence, the holomorph of a quasigroup(loop) is an AIPQ (AIPL) or CIPQ (CIPL) if and only if its automorphism group is trivial and the quasigroup(loop) is a AIPQ(AIPL) or CIPQ (CIPL). Furthermore, it is discovered that if the holomorph of a quasigroup(loop) is a CIPQ (CIPL), then the quasigroup (loop) is a flexible unipotent CIPQ(flexible CIPL of exponent ). By constructing two isotopic quasigroups(loops) and such that their automorphism groups are not trivial, it is shown that is a AIPQ or CIPQ (AIPL or CIPL) if and only if is a AIPQ or CIPQ (AIPL or CIPL). Explanations are given on how these CIPQs can be incorporated into the encryption scheme of Keedwell for higher security using a computer.### References

J. O. Adeniran (2005): On holomorphic theory of a class of left Bol loops, Al.I.Cuza 51(1), pp. 23--28.

J. O. Adeniran, Y. T. Oyebo and D. Mohammed (2011): On certain isotopic maps of central loops, Proyecciones Journal of Mathematics, 30( 3), pp. 303-318.

R. Artzy (1955): On loops with special property, Proc. Amer. Math. Soc. 6, pp. 448--453.

R. Artzy (1959): Crossed inverse and related loops, Trans. Amer. Math. Soc. 91(3), pp. 480--492.

R. Artzy (1959): On Automorphic-Inverse Properties in Loops, Proc. Amer. Math. Soc. 10(4), pp. 588--591.

R. Artzy (1978): Inverse-Cycles in Weak-Inverse Loops, Proc. Amer. Math. Soc. 68(2), pp. 132--134.

V. D. Belousov and B. F. Curkan (1969): Crossed inverse quasigroups(CI-quasigroups), Izv. Vyss. Ucebn; Zaved. Matematika 82, pp. 21--27.

R. H. Bruck (1944): Contributions to the theory of loops, Trans. Amer. Math. Soc. 55, pp. 245--354.

R. H. Bruck and L. J. Paige (1956): Loops whose inner mappings are automorphisms, The annuals of Mathematics, 63(2), pp. 308--323.

V. O. Chiboka and A. R. T. Solarin (1991): Holomorphs of conjugacy closed loops, Scientific Annals of Al.I.Cuza. Univ. 37(3), pp. 277--284.

W. Diffie and M. E. Hellman, (1976): New directions in cryptography, IEE Trans. Inform. Theory IT-22, pp. 644--654.

J. H. Ellis, (1970): The possibility of secure non-secret digtal encryption, CESG Report.

J. H. Ellis (1987): The story of non-secret digital encryption, http://www.cesg.gov.uk/ellisdox.ps.

J. H. Ellis (1987): The history of Non-Secret Encryption, http://www.cesg.gov.uk/about/nsecret.htm.

E. D. Huthnance Jr.(1968): A theory of generalised Moufang loops, Ph.D. thesis, Georgia Institute of Technology.

T. G. Jaiyé lá (2008): An holomorphic study of Smarandache automorphic and cross inverse property loops, Proceedings of the 4 International Conference on Number Theory and Smarandache Problems, Scientia Magna Journal, 4(1), pp. 102--108.

A. D. Keedwell (1999): Crossed-inverse quasigroups with long inverse cycles and applications to cryptography, Australas. J. Combin. 20, pp. 241–-250.

A. D. Keedwell and V. A. Shcherbacov (2002): On m-inverse loops and quasigroups with a long inverse cycle, Australas. J. Combin. 26, pp. 99–-119.

A. D. Keedwell and V. A. Shcherbacov (2003): Construction and properties of -inverse quasigroups I, Discrete Math. 266, pp. 275–-291.

A. D. Keedwell and V. A. Shcherbacov (2004): Construction and properties of -inverse quasigroups II, Discrete Math. 288, pp. 61-–71.

J. M. Osborn (1961): Loops with the weak inverse property, Pac. J. Math. 10,pp. 295--304.

D. A. Robinson (1964): Bol loops, Ph.D. thesis, University of Wisconsin, Madison, Wisconsin.

D. A. Robinson (1971): Holomorphic theory of extra loops, Publ. Math. Debrecen 18, pp. 59--64.