The number of irreducible polynomials over finite fields with vanishing trace and reciprocal trace

Creative Commons License

Çakıroğlu Y., Yayla O., Yilmaz E. S.

DESIGNS, CODES, AND CRYPTOGRAPHY, vol.90, no.10, pp.2407-2417, 2022 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 90 Issue: 10
  • Publication Date: 2022
  • Doi Number: 10.1007/s10623-022-01088-2
  • Journal Name: DESIGNS, CODES, AND CRYPTOGRAPHY
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, PASCAL, Applied Science & Technology Source, Compendex, Computer & Applied Sciences, INSPEC, MathSciNet, zbMATH
  • Page Numbers: pp.2407-2417
  • Keywords: Irreducible polynomials, Finite fields, Trace function, Algebraic curves, CURVES, ENUMERATION, GF(2)
  • Middle East Technical University Affiliated: Yes

Abstract

We present the formula for the number of monic irreducible polynomials of degree n over the finite field F-q where the coefficients of x(n)(-1) and x vanish for n >= 3. In particular, we give a relation between rational points of algebraic curves over finite fields and the number of elements a is an element of F-qn for which Trace(a) = 0 and Trace(a(-1)) = 0.