Abstract
The approach for constructing the irreducible polynomials of arbitrary degree n over finite fields which is based on the number of the roots over the extension field is presented. At the same time, this paper includes a sample to illustrate the specific construction procedures. Then in terms of the relationship between the order of a polynomial over finite fields and the order of the multiplicative group of the extension field, a method which can determine whether a polynomial over finite fields is irreducible or not is proposed. By applying the Euclidean Algorithm, this judgment can be verified easily.
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References
Garefalakis, T., Kapetanakis, G.: A note on the Hansen-Mullen conjecture for self-reciprocal irreducible polynomials. Finite Fields Appl. 35(4), 61–63 (2015)
Magamba, K., Ryan, J.A.: Counting irreducible polynomials of degree r over \( F_{{q^{n} }} \) and generating Goppa Codes using the lattice of subfields of \( F_{{q^{nr} }} \). J. Discrete Math. 2014, 1–4 (2014)
Ugolini, S.: Sequences of irreducible polynomials over odd prime fields via elliptic curve endomorphisms. J. Number Theor. 152, 21–37 (2015)
Ri, W.H., Myong, G.C., Kim, R., et al.: The number of irreducible polynomials over finite fields of characteristic 2 with given trace and subtrace. Finite Fields Appl. 29, 118–131 (2014)
Kaminski, M., Xing, C.: An upper bound on the complexity of multiplication of polynomials modulo a power of an irreducible polynomial. IEEE Trans. Inf. Theor. 59(10), 6845–6850 (2013)
Fan, H.: A Chinese remainder theorem approach to bit-parallel GF(2n) polynomial basis multipliers for irreducible trinomials. IEEE Trans. Comput.65(2), 1 (2016)
Kopparty, S., Kumar, M., Saks, M.: Efficient indexing of necklaces and irreducible polynomials over finite fields. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014. LNCS, vol. 8572, pp. 726–737. Springer, Heidelberg (2014)
Nechae, A.A., Popov, V.O.: A generalization of Ore’s theorem on irreducible polynomials over a finite field. Discrete Math. Appl. 25(4), 241–243 (2015)
Pollack, P.: Irreducible polynomials with several prescribed coefficients. Finite Fields Appl. 22(7), 70–78 (2013)
Kyuregyan, M.K.: Recurrent methods for constructing irreducible polynomials over F q of odd characteristics. Finite Fields Appl. 12(3), 357–378 (2006)
Kyuregyan, M.K., Kyureghyan, G.M.: Irreducible compositions of polynomials over finite fields. Des. Codes Cryptogr. 61(3), 301–314 (2011)
Abrahamyan, S., Kyureghyan, M.: A recurrent method for constructing irreducible polynomials over finite fields. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2011. LNCS, vol. 6885, pp. 1–9. Springer, Heidelberg (2011)
Abrahamyan, S., Alizadeh, M., Kyureghyan, M.K.: Recursive constructions of irreducible polynomials over finite fields. Finite Fields Appl. 18(4), 738–745 (2012)
Kaliski, B.: Irreducible Polynomial. Springer, New York (2011)
Acknowledgments
This work was supported by the National Natural Science Foundation of China (61373150, 11501343), the Key Technologies R&D Program of Shaanxi Province (2013k0611), and the Fundamental Research Funds for the Central Universities (GK201603087).
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Song, Y., Li, Z. (2016). The Construction and Determination of Irreducible Polynomials Over Finite Fields. In: Tan, Y., Shi, Y., Li, L. (eds) Advances in Swarm Intelligence. ICSI 2016. Lecture Notes in Computer Science(), vol 9713. Springer, Cham. https://doi.org/10.1007/978-3-319-41009-8_67
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DOI: https://doi.org/10.1007/978-3-319-41009-8_67
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