Advertisement

A Swan-like note for a family of binary pentanomials

  • Giorgos Kapetanakis
Original Paper
  • 7 Downloads

Abstract

In this note, we employ the techniques of Swan (Pac J Math 12(3):1099–1106, 1962) with the purpose of studying the parity of the number of the irreducible factors of the penatomial \(X^n+X^{3s}+X^{2s}+X^{s}+1\in \mathbb {F}_2[X]\), where s is even and \(n>3s\). Our results imply that if \(n \not \equiv \pm 1 \pmod {8}\), then the polynomial in question is reducible.

Keywords

Swan-like Binary field Pentanomial 

Mathematics Subject Classification

11T06 11C08 

Notes

Acknowledgements

This work was initiated during the author’s visit to the Federal University of Santa Catarina. The author is grateful to the anonymous reviewers for their valuable comments.

References

  1. 1.
    Ahmandi, O.: Self-reciprocal irreducible pentanomials over \({\mathbb{F}}_{2}\). Des. Codes Cryptogr. 38(3), 395–397 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Banegas, G., Custódio, R., Panario, D.: A new class of irreducible pentanomials for polynomial based-multipliers in binary fields. J. Cryptogr. Eng. (2018).  https://doi.org/10.1007/s13389-018-0197-6
  3. 3.
    Bluher, A.W.: A Swan-like theorem. Finite Fields Appl. 12(1), 128–138 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Fredricksen, H., Hales, A., Sweet, M.: A generalization of Swan’s theorem. Math. Comput. 46(173), 321–331 (1986)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Hales, A., Newhart, D.: Swan’s theorem for binary tetranomials. Finite Fields Appl. 12(2), 301–311 (2006)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hanson, B., Panario, D., Thomson, D.: Swan-like results for binomials and trinomials over finite fields of odd characteristic. Des. Codes Cryptogr. 61(3), 273–283 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Institute of Electrical and Electronics Engineering (IEEE): Standard specifications for public key cryptography. In: Standard P1363-2000. Draft D13 (2000)Google Scholar
  8. 8.
    Koepf, W., Kim, R.: The parity of the number of irreducible factors for some pentanomials. Finite Fields Appl. 15(5), 585–603 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Lidl, R., Niederreiter, H.: Finite Fields, volume 20 of Enyclopedia of Mathematics and Its Applications (2nd edn). Cambridge University Press, Cambridge (1997)Google Scholar
  10. 10.
    Mullen, G.L., Panario, D.: Handbook of Finite Fields. CRC Press, Boca Raton (2013)CrossRefGoogle Scholar
  11. 11.
    Reyhani-Masoleh, A., Hasan, M.A.: Low complexity bit parallel architectures for polynomial basis multiplication over \({\mathit{GF}}(2^m)\). IEEE Trans. Comput. 53(8), 945–959 (2004)CrossRefGoogle Scholar
  12. 12.
    Rodríguez-Henríquez, F., Kaya Koç, Çetin: Parallel multipliers based on special irreducible pentanomials. IEEE Trans. Comput. 52(12), 1535–1542 (2003)CrossRefGoogle Scholar
  13. 13.
    Swan, R.G.: Factorization of polynomials over finite fields. Pac. J. Math. 12(3), 1099–1106 (1962)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Wu, H., Hasan, M.A.: Low complexity bit-parallel multipliers for a class of finite fields. IEEE Trans. Comput. 47(8), 883–887 (1998)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Zhang, T., Parhi, K.K.: Systematic design of original and modified Mastrovito multipliers for general irreducible polynomials. IEEE Trans. Comput. 50(7), 734–749 (2001)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Engineering and Natural SciencesSabancı ÜniversitesiTuzlaTurkey

Personalised recommendations