Factorization of Dickson polynomials over Finite Fields

2019/08/15 by Fabio Enrique Brochero Martínez, Martínez, F. E. Brochero, Nelcy Esperanza Arévalo Baquero +1
Computer Science · #Coding theory and cryptography #Cryptography and Residue Arithmetic #Cellular Automata and Applications

paper · pdf · doi:10.48550/arxiv.1908.05508

Abstract

Let D_n(x;a) and E_n(x;a)∈\mathbb F_q[x] be Dickson polynomials of first and second kind respectively, where \mathbb F_q is a finite field with q elements. In this article we show explicitly the irreducible factors these polynomials in the case that every prime divisor of n divides q-1. This result generalizes the results find in Chou, W.S., The Factorization of Dickson polynomials over finite fields. Finite Fields Appl. \bf3 (1997) 84-96, Fitzgerald R. W., Yucas J. L., Explicit factorization of cyclotomic and Dickson polynomials over finite fields. Arithmetic of Finite Fields. Lecture Notes in Computer Science, vol. \bf 4547, pp. 1-10. Springer, Berlin (2007), Tosun, S., Explicit factorizations of generalized Dickson polynomials of order 2^m via generalized cyclotomic polynomials over finite fields. Finite Fields Appl. \bf 38 (2016) 40-56 and Tosun, S., Explicit factors of generalized cyclotomic polynomials and generalized Dickson polynomials of order 2^m3 over finite fields. Discrete Math. 342 (2019) DOI: j.disc.2019.111618

Related

• Irreducible factorization of translates of reversed Dickson polynomials\n over finite fields
• Reversed Dickson polynomials of the (k+1)-th kind over finite fields
• Composed Products and Explicit Factors of Cyclotomic Polynomials over Finite Fields
• Factorization of composed polynomials and applications
• Closed formulas for the factorization of Xⁿ-1, the n-th cyclotomic polynomial, Xⁿ-a and f(Xⁿ) over a finite field for arbitrary positive integers n