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

  • Javad DoliskaniEmail author
  • Éric Schost


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.


Finite field Algebraic closure Complexity Square root 

Mathematics Subject Classification

11Y16 12Y05 68W30 



The authors are supported by NSERC and the Canada Research Chairs program. We wish to thank the reviewers for their helpful remarks and suggestions.


  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). 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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