Transformations on Irreducible Binary Polynomials

  • Jean-Francis Michon
  • Philippe Ravache
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6338)


Using the natural action of \(GL_2(\mathbb F_2)\simeq \frak S_3\) over \(\mathbb F_2[X]\), one can define different classes of polynomials strongly analogous to self-reciprocal irreducible polynomials. We give transformations to construct polynomials of each kind of invariance and we deal with the question of explicit infinite sequences of invariant irreducible polynomials in \(\mathbb F_2[X]\). We generalize results obtained by Varshamov, Wiedemann, Meyn and Cohen and we give sequences of invariant irreducible polynomials. Moreover we explain what happens when the given constructions fail. We also give a result on the order of the polynomials of one of the classes: the alternate irreducible polynomials.


irreducible polynomials finite fields sequences of irreducible invariant polynomials 


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  1. 1.
    Cohen, S.D.: The Explicit Construction of Irreducible Polynomials over Finite Fields. Designs, Codes and Cryptography 2, 169–174 (1992)zbMATHCrossRefGoogle Scholar
  2. 2.
    Kyuregyan, M.: Recurrent Methods for Constructing Irreducible Polynomials over GF(2s). Finite Fields and Their Applications 8(3), 52–68 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Kyuregyan, M.: Iterated Constructions of Irreducible Polynomials over Finite Fields with Linearly Independant Roots. Finite Fields and Their Applications 10(1), 323–341 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Meyn, H.: On the Construction of Irreducible Self-reciprocal Polynomials over Finite Fields. Appl. Algebra Eng. Comm. Comp. 1, 43–53 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Meyn, H., Götz, W.: Self-reciprocal Polynomials over Finite Fields. Publ. IRMA Strasbourg 413/S-21, 82–90 (1990)Google Scholar
  6. 6.
    Michon, J.-F., Ravache, P.: On Different Families of Invariant Irreducible Polynomials over GF(2). Finite Fields and Their Applications 16(3), 163–174 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
  8. 8.
    Varshamov, R.R.: A General Method of Synthesis for Irreducible Polynomials over Galois Fields. Soviet Math. Dokl. 29, 334–336 (1984)zbMATHGoogle Scholar
  9. 9.
    Wiedemann, D.: An Iterated Quadratic Extension of GF(2). Fibonacci Quart 26, 290–295 (1988)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Jean-Francis Michon
    • 1
  • Philippe Ravache
    • 1
  1. 1.LITIS EA 4108Université de RouenSaint-Étienne du Rouvray cedexFrance

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