Recurrent methods for constructing irreducible polynomials over \(\text{GF}(2^s)\)
type: Item
instance of: scholarly article
title: Recurrent methods for constructing irreducible polynomials over \(\text{GF}(2^s)\)
zbMATH Open document ID: 1028.11073
published in: Finite Fields and their Applications
publication date: 2002-07-04T00:00:00Z
review text: The author provides a method to construct irreducible polynomials over finite fields. Starting with a polynomial \(f\in \text{GF}(q)[x]\) of degree \(n\) irreducible over \(\text{GF}(q)\) he shows how to obtain an irreducible polynomial of degree \(np\), where \(p\) is the characteristic of the field. He gives several such constructions depending on the initial polynomial \(f\). All the polynomials that can be obtained by this method have composite degree leading to composite finite fields. For binary fields a recursive application is possible for some polynomials. Therefore, the construction leads to irreducible polynomials of degree \(2^mn\) over \(\text{GF}(2^s)\). The conditions are satisfied for linear polynomials \(f(x)=x-a, a\in \text{GF}(2^s)^*\) leading to irreducible polynomials of degree \(2^m\). The method is then applied and extended to lead to self-dual irreducible polynomials. The third construction starts with primitive polynomials of degrees \(n_1,\ldots ,n_\sigma\) and an irreducible polynomial of degree \(n\) over \(\text{GF}(2)\) leading to an irreducible polynomial of degree \(n\prod_{i=1}^\sigma (2^{n_i}-1)\). Some results are extensions of [\textit{R. R. Varshamov}, Sov. Phys., Dokl. 9, 426--428 (1964); translation from Dokl. Akad. Nauk SSSR 156, 1308--1311 (1964; Zbl 0147.29602) and Sov. Math., Dokl. 29, 334--336 (1984); translation from Dokl. Akad. Nauk SSSR 275, 1041--1044 (1984; Zbl 0594.12014)].
reviewed by: Tanja Lange
Mathematics Subject Classification ID: 11T06
Mathematics Subject Classification ID: 12E05
Mathematics Subject Classification ID: 11Y16
zbMATH DE Number: 1763661
zbMATH Keywords: irreducible polynomials
zbMATH Keywords: composite finite fields
zbMATH Keywords: extension fields
zbMATH Keywords: Varshamov operator
zbMATH Keywords: self-dual polynomials
author: Melsik K. Kuyregyan
MaRDI profile type: Publication
full work available at URL: ffta.2001.0323
OpenAlex ID: W2033273497
cites work: Q3236675
cites work: Q3754705
cites work: Q3772321
cites work: Q3288136
cites work: Q3676235
cites work: Q5525475
cites work: Linear Recurring Sequences
cites work: Q3216750
cites work: The explicit construction of irreducible polynomials over finite fields
cites work: Q3136478
DOI: 10.1006/FFTA.2001.0323
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description: scientific article; zbMATH DE number 1763661