TARU PUBLICATIONS
Journal of Discrete Mathematical Sciences and Cryptography cover
Hybrid ·Peer-reviewed·ISSN (Online): 2169-0065·ISSN (Print): 0972-0529

Monthly Journal: Publishes theoretical and applied research in all areas of Discrete Mathematical Sciences, Cryptography, Combinatorics, Elliptic Curves and Information Security.

Issues up to 2022 co-published with and available at:Taylor & Francis Online
submissions@tarupublications.com
Open Access Research Article

Some new constructions of irreducible polynomials using composition of polynomials over finite fields

, * , ,

* Corresponding author · click or hover a name for details

pp. 2383–2396Vol. 28Issue 6September 2025DOI: 10.47974/JDMSC-2308 Crossmark XML
Received:
17 Sep 2024
Published Online:
26 Jul 2025
Article type:
Research Article
Language:
EN
Article no.:
JDMSC-2308
Pages:
2383–2396

Abstract

Construction of irreducible polynomials hold significant importance in coding theory, complexity theory and cryptography. Irreducible polynomials also helps us to generate non-zero elements of finite fields. In the present work, starting from a specified n degree irreducible polynomial, the methods to construct irreducible polynomials with a degree 6n over finite field F3s are explored. Additionally, a method for generating irreducible polynomial of degree p2n over finite field Fps by polynomial composition under specific constraints related to coefficients and the degree of initial irreducible polynomial is given.

Keywords

Subject Classifications

11T0612E0512E20

References

[1] S. Abrahamyan, M. Alizadeh and M. K. Kyuregyan, “Recursive constructions of irreducible polynomials over finite fields,” Finite Fields and its Applications, vol. 18, no. 4, pp. 738-745 (2012).
[2] M. Alan and B. Duman, “A note on construction of irreducible polynomials over finite fields with characteristic 2,” International Journal of Pure and Applied Mathematics, vol. 115, no. 3, pp. 529-532 (2017).
[3] M. Alizadeh, “Constructing methods for irreducible polynomials,” Mathematical Problems of Computer Sciences, vol. 35, pp. 26-32 (2011).
[4] M. Alizadeh, “Some algorithms for normality testing irreducible polynomials and computing complexity of the normal polynomials over finite fields,” Applied Mathematical Sciences, vol. 6, no. 40, pp. 1997-2003 (2012).
[5] M. Alizadeh, S. Abrahamyan, S. Mehrabi and M. K. Kyuregyan, “Constructing of N-polynomials over finite fields,” International Journal of Algebra, vol. 5, no. 29, pp. 1437-1442 (2011).
[6] M. Alizadeh, M. R. Darafsheh and S. Mehrabi, “On the k-normal elements and polynomials over finite fields,” Italian Journal of Pure and Applied Mathematics, vol. 39, pp. 451-464 (2018).
[7] M. Alizadeh and S. Mehrabi, “Construction of self-reciprocal normal polynomials over finite fields of even characteristic,” Turkish Journal of Mathematics, vol. 39, no. 2, pp. 259-267 (2015).
[8] S. D. Cohen, “On irreducible polynomials of certain types in finite fields,” In Mathematical Proceedings of the Cambridge Philosophical Society, vol. 66, no. 2, pp. 335-344 (1969).
[9] S. Huczynska, G. L. Mullen, D. Panario, and D. Thomson, “Existence and properties of k-normal elements over finite fields,” Finite Fields and Their Applications, vol. 24, pp. 170-183 (2013).
[10] X. D. Hou, “A criterion for the normality of polynomials over finite fields based on their coefficients,” Finite Fields and Their Applications, vol. 93, pp. 102313 (2024).
[11] R. Kim and H. S. Son, “Recursive constructions of k-normal polynomials using some rational transformations over finite fields,” Journal of Algebra and its Applications, vol. 19, no. 11, pp. 2050210 (2020).
[12] M. K. Kyuregyan, “Recurrent methods for constructing irreducible polynomials over GF(2s),” Finite Fields and its Applications, vol. 8, no. 1, pp. 52-68 (2002).
[13] M. K., Kyuregyan, “Recursive constructions of N-polynomials over GF (2s),” Discrete Applied Mathematics, vol. 156, no. 9, pp. 1554-1559 (2008).
[14] M. K. Kyuregyan, “Iterated constructions of irreducible polynomials over finite fields with linearly independent roots’’, Finite fields and their applications, vol. 10, no. 3, pp. 323-341 (2004).
[15] R. Lidl and H. Niederreiter, Introduction to finite fields and their applications, Cambridge University Press, Cambridge (1986).
[16] A. J. Menezes, I. F. Blake, X. Gao, R. C. Mullin, S. A. Vanstone and T. Yaghoobian, Applications of Finite Fields, Kluwer Academic Publishers, Boston, Dordrecht, Lancaster (1993).
[17] H. Meyn, “Explicit n-polynomials of 2-power degree over finite fields,” I Designs, Codes and Cryptography, vol. 6, no. 2, pp. 107-116 (1995).
[18] G. L. Mullen and D. Panario, Handbook of Finite Fields, Discrete Mathematics and its Applications, CRC Press, Taylor & Francis Group (2013).
[19] P. L. Sharma, Ashima and A. K. Sharma, “Recursive construction of normal polynomials over finite fields’’, Journal of Discrete Mathematical Sciences and Cryptography, vol. 25, no. 8, pp. 2645-2660 (2021).
[20] P. L. Sharma and Ashima, “Construction of irreducible polynomials over finite fields,” Asian-European Journal of Mathematics, vol. 15, no. 7, pp. 2250130 (2022).
[21] P. L. Sharma, S. Sharma and N. Dhiman, “Construction of infinite sequences of irreducible polynomials using Kloosterman sums,” Bulletin of Pure and Applied Sciences, vol. 33, no. 2, pp. 161-168 (2014).
[22] P. L. Sharma, S. Sharma and M. Rehan, “Construction of infinite sequences of irreducible polynomials over F2,” International Journal of Mathematical Sciences and Engineering Applications, vol. 9, no. 3, pp. 19-35 (2015).
[23] P. L. Sharma, S. Sharma and M. Rehan, “On construction of irreducible polynomials over F3,” Journal of Discrete Mathematical Sciences and Cryptography, vol. 18, no. 4, pp. 335-347 (2015).
[24] Å . Schwarz, “Irreducible polynomials over finite fields with linearly independent roots,” Mathematica Slovaca, vol. 38 no. 2, pp. 147-158 (1988).
[25] R. R. Varshamov, “A general method of synthesizing irreducible polynomials over Galois Fields,” Soviet Math. Dokl, vol. 29, pp. 334-336 (1984). 
[26] T. Baumbaugh and F. Manganiello, “Matroidal root structure of skew polynomials over finite fields,” Journal of Discrete Mathematical Sciences and Cryptography, vol. 22, no. 3, pp. 377-389 (2019).

Views: 261Downloads: 74Citations: 0