## EXTENDED EUCLIDEAN ALGORITHM OF AUNU BINARY POLYNOMIALS OF CARDINALITY ELEVEN

#### Authors:

S.I. Abubakar,Zaid Ibrahim,A. A. Ibrahim,Sadiq Shehu,A. Rufa’i,#### DOI NO:

https://doi.org/10.26782/jmcms.2022.01.00001#### Abstract:

*Binary polynomials representation of Aunu permutation patterns has been used to perform arithmetic operations on words and sub-words of the polynomials using addition, multiplication, and division modulo two. The polynomials were also found to form some mathematical structures such as group, ring, and field. This paper presents the extension of our earlier work as it reports the Aunu binary polynomials of cardinality eleven and how to find their greatest common divisor (gcd) using the extended Euclidean algorithm. The polynomials are pairly permuted and the results found showed that one polynomial is a factor of the other polynomial or one polynomial is relatively prime to the other and some gave different results. This important feature is of combinatorial significance and can be investigated further to formulate some theoretic axioms for this class of Aunu permutation pattern. Binary polynomials have important applications in coding theory, circuit design, and the construction of cryptographic primitives.*

#### Keywords:

Algorithm,Aunu,Cryptography,Euclidean,Extended,Galois field,Greatest common divisor,Permutation,Patterns,Polynomials,Binary,#### Refference:

I. A. A Ibrahim (2007). An Enumeration Scheme and Algebraic properties of a Special (132)-avoiding Class of permutation Pattern. Trends in Applied sciences Research Academic Journals Inc. USA. 2(4) 334-340.

II. Aminu Alhaji Ibrahim and Sa’idu Isah Abubakar (2016). Aunu Integer Sequence as Non-Associative Structure and Their Graph Theoretic Properties. Advances in Pure Mathematics, (6), 409-419 http://www.scirp.org/journal/apm http://dx.doi.org/10.4236/apm.2016.66028

III. Aminu Alhaji Ibrahim, Saidu Isah Abubakar (2016). Non-Associative Property of 123-Avoiding Class of Aunu Permutation Patterns Advances in Pure Mathematics, 2016, 6, 51-57 http://www.scirp.org/journal/apm http://dx.doi.org/10.4236/apm.2016.62006.

IV. Abubakar S.I, Shehu S., Ibrahim Z. Ibrahim A.A (2014). Some polynomials representation using the 123-avoiding class of the Aunu permutation patterns of cardinality five using binary codes. International Journal of Scientific and Engineering Research 5(8), 1-4.

V. Abubakar S.I, Ibrahim Z. Ibrahim A.A (2014). Binary polynomials representation using the 123-avoiding class of the Aunu permutation patterns of cardinality seven. A paper presented at the 1st National Conference organized by Faculty of Science, Sokoto State University in conjunction with The Algebra Group Usmanu Danfodiyo University, Sokoto held at Sokoto State University from 17th-20th March, 2014.

VI. Benvenuto, C. J. (2012). Galois field in cryptography. University of Washington, 1(1), 1-11.

VII. Daniel Panario (June 2006). A Minicourse in Finite Fields and Applications, School of Mathematics and Statistics, Carleton University.

VIII. De Piccoli, A., Visconti, A., & Rizzo, O. G. (2018). Polynomial multiplication over binary finite fields: new upper bounds. Journal of Cryptographic Engineering, 1-14.

IX. Homma, N., Saito, K., Aoki, T. (2014). Toward formal design of practical cryptographic hardware based on Galois field arithmetic. IEEE Transactions on Computers 63(10), 2604-2613.

XI. Paul Pollack (2008). Prime Polynomials over Finite Fields; A PhD Thesis, Darmouth College.

XII. Sheueling Chang Shantz (2001). From Eculid’s GCD to Montgomery Multiplication to the Great Divide” sun Microsystems laboratories MSLI TR-2001-95.

XIII. Shparlinski, I. (2013). Finite Fields: Theory and Computation: The meeting point of number theory, computer science, coding theory and cryptography 477.

XIV. Stein J. (1961). Computational problems associated with Racah algebra. Journal Computational Physics, 1.