On the construction of certain odd degree irreducible polynomials over finite fields

• Published in: Designs, codes, and cryptography Vol. 92; no. 12; pp. 4085 - 4097
• Main Authors: Çil, Melek, Kırlar, Barış Bülent
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2024; Springer Nature B.V
• Subjects: Algorithms; Coding and Information Theory; Computer Science; Cryptology; Discrete Mathematics in Computer Science; Fields (mathematics); Polynomials; Finite fields; 11T06; 11T71; 12E20; 12E10; Irreducible polynomials; Hilbert Theorem 90
• ISSN: 0925-1022; 1573-7586
• DOI: 10.1007/s10623-024-01479-7

For an odd prime power q , let F q 2 = F q ( α ) , α 2 = t ∈ F q be the quadratic extension of the finite field F q . In this paper, we consider the irreducible polynomials F ( x ) = x k - c 1 x k - 1 + c 2 x k - 2 - ⋯ - c 2 q x 2 + c 1 q x - 1 over F q 2 , where k is an odd integer and the coefficients c i are in the form c i = a i + b i α with at least one b i ≠ 0 . For a given such irreducible polynomial F ( x ) over F q 2 , we provide an algorithm to construct an irreducible polynomial G ( x ) = x k - A 1 x k - 1 + A 2 x k - 2 - ⋯ - A k - 2 x 2 + A k - 1 x - A k over F q , where the A i ’s are explicitly given in terms of the c i ’s. This gives a bijective correspondence between irreducible polynomials over F q 2 and F q . This fact generalizes many recent results on this subject in the literature.