Designs, Codes and Cryptography

, Volume 74, Issue 3, pp 559–569 | Cite as

Computing in degree \(2^k\)-extensions of finite fields of odd characteristic

Article

Abstract

We show how to perform basic operations (arithmetic, square roots, computing isomorphisms) over finite fields of the form \(\mathbb F _{q^{2^k}}\) in essentially linear time.

Keywords

Finite field Algebraic closure Complexity Square root 

Mathematics Subject Classification

11Y16 12Y05 68W30 

References

  1. 1.
    Bostan A., Chowdhury M.F.I., van der Hoeven J., Schost É.: Homotopy methods for multiplication modulo triangular sets. J. Symb. Comput. 46(12), 1378–1402 (2011).Google Scholar
  2. 2.
    Brent R.P., Kung H.T.: Fast algorithms for manipulating formal power series. J. Assoc. Comput. Mach. 25(4), 581–595 (1978).Google Scholar
  3. 3.
    Cantor D.G., Kaltofen E.: On fast multiplication of polynomials over arbitrary algebras. Acta Inform. 28(7), 693–701 (1991).Google Scholar
  4. 4.
    Cipolla, M.: Un metodo per la risoluzione della congruenza di secondo grado. Napoli Rend. 9, 153–163 (1903)Google Scholar
  5. 5.
    De Feo L., Schost É.: Fast arithmetics in Artin–Schreier towers over finite fields. J. Symb. Comput. 47(7), 771–792 (2012).Google Scholar
  6. 6.
    Doliskani J., Schost É.: Taking roots over high extensions of finite fields. Math. Comput. (to appear) (2012).Google Scholar
  7. 7.
    Feng W., Nogami Y., Morikawa Y.: A fast square root computation using the Frobenius mapping. In: Information and Communications Security. Lecture Notes in Computer Science, vol. 2836, pp. 1–10. Springer, Heidelberg (2003).Google Scholar
  8. 8.
    von zur Gathen J., Gerhard J.: Modern Computer Algebra, 2nd edn. Cambridge University Press, Cambridge (2003).Google Scholar
  9. 9.
    von zur Gathen J., Shoup V.: Computing Frobenius maps and factoring polynomials. Comput. Complex. 2(3):187–224, (1992).Google Scholar
  10. 10.
    Gaudry P., Schost É.: Genus 2 point counting over prime fields. J. Symb. Comput. 47(4), 368–400 (2012).Google Scholar
  11. 11.
    Kaltofen E., Shoup V.: Fast polynomial factorization over high algebraic extensions of finite fields. In: ISSAC’97, pp. 184–188. ACM, New York (1997).Google Scholar
  12. 12.
    Kedlaya K.S., Umans C.: Fast polynomial factorization and modular composition. SIAM J. Comput. 40(6), 1767–1802 (2011).Google Scholar
  13. 13.
    Lang S.: Algebra, Graduate Texts in Mathematics vol. 211, 3rd edn. Springer, New York (2002).Google Scholar
  14. 14.
    Schoof R.: Elliptic curves over finite fields and the computation of square roots mod \(p\). Math. Comput. 44, 483–494 (1985).Google Scholar
  15. 15.
    Shanks D.: Five number-theoretic algorithms. In: Proceedings of the Second Manitoba Conference on Numerical Mathematics, pp. 51–70 (1972).Google Scholar
  16. 16.
    Shoup, V.: A library for doing number theory (NTL). http://www.shoup.net/ntl/. Accessed July 2013.
  17. 17.
    Shoup V.: Fast construction of irreducible polynomials over finite fields. J. Symb. Comput. 17(5), 371–391 (1994).Google Scholar
  18. 18.
    Tonelli, A. : Bemerkung über die Auflösung quadratischer Congruenzen. Göttinger Nachrichten, pp. 344–346 (1891).Google Scholar
  19. 19.
    Wang F., Nogami Y., Morikawa Y.: An efficient square root computation in finite fields \({GF}(p^{2^d})\). IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E88-A(10), 2792–2799 (2005).Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Western UniversityLondonUSA

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