An O(M(n) logn) Algorithm for the Jacobi Symbol

  • Richard P. Brent
  • Paul Zimmermann
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6197)

Abstract

The best known algorithm to compute the Jacobi symbol of two n-bit integers runs in time O(M(n)logn), using Schönhage’s fast continued fraction algorithm combined with an identity due to Gauss. We give a different O(M(n)logn) algorithm based on the binary recursive gcd algorithm of Stehlé and Zimmermann. Our implementation — which to our knowledge is the first to run in time O(M(n)logn) — is faster than GMP’s quadratic implementation for inputs larger than about 10000 decimal digits.

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References

  1. 1.
    Bach, E.: A note on square roots in finite fields. IEEE Trans. on Information Theory 36(6), 1494–1498 (1990)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bach, E., Shallit, J.O.: Algorithmic Number Theory: Efficient Algorithms, vol. 1. MIT Press, Cambridge (1996) (Solution to problem 5.52)MATHGoogle Scholar
  3. 3.
    Bachmann, P.: Niedere Zahlentheorie, Teubner, Leipzig, vol. 1 (1902); Reprinted by Chelsea, New York (1968)Google Scholar
  4. 4.
    Brent, R.P.: Twenty years’ analysis of the binary Euclidean algorithm. In: Davies, J., Roscoe, A.W., Woodcock, J. (eds.) Millennial Perspectives in Computer Science: Proceedings of the 1999 Oxford - Microsoft Symposium in honour of Professor Sir Antony Hoare, Palgrave, New York, pp. 41–53 (2000), http://wwwmaths.anu.edu.au/~brent/pub/pub183.html
  5. 5.
    Daireaux, B., Maume-Deschamps, V., Vallée, B.: The Lyapunov tortoise and the dyadic hare. In: Proceedings of the 2005 International Conference on Analysis of Algorithms, DMTCS Proc. AD, pp. 71–94 (2005), http://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/issue/view/81
  6. 6.
    Gauss, C.F.: Theorematis fundamentalis in doctrina de residuis quadraticis, demonstrationes et ampliatones novæ. Comm. Soc. Reg. Sci. Gottingensis Rec. 4 (presented February 10, 1817) (1818); Reprinted in Carl Friedrich Gauss Werke, Bd. 2: Höhere Arithmetik, Göttingen, pp. 47–64 (1876) Google Scholar
  7. 7.
    Knuth, D.E.: The Art of Computer Programming. In: Seminumerical Algorithms, 3rd edn., vol. 2, Addison-Wesley, Reading (1997)Google Scholar
  8. 8.
    Möller, N.: On Schönhage’s algorithm and subquadratic integer GCD computation. Mathematics of Computation 77(261), 589–607 (2008)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Schönhage, A.: Schnelle Berechnung von Kettenbruchentwicklungen. Acta Informatica 1, 139–144 (1971)MATHCrossRefGoogle Scholar
  10. 10.
    Schönhage, A.: Personal communication by email (December 2009)Google Scholar
  11. 11.
    Schönhage, A., Grotefeld, A.F.W., Vetter, E.: Fast Algorithms: A Multitape Turing Machine Implementation. BI-Wissenschaftsverlag, Mannheim (1994)MATHGoogle Scholar
  12. 12.
    Shallit, J., Sorenson, J.: A binary algorithm for the Jacobi symbol. ACM SIGSAM Bulletin 27(1), 4–11 (1993), http://euclid.butler.edu/~sorenson/papers/binjac.ps CrossRefGoogle Scholar
  13. 13.
    Stehlé, D., Zimmermann, P.: A binary recursive gcd algorithm. In: Buell, D.A. (ed.) ANTS 2004. LNCS, vol. 3076, pp. 411–425. Springer, Heidelberg (2004)Google Scholar
  14. 14.
    Vallée, B.: A unifying framework for the analysis of a class of Euclidean algorithms. In: Gonnet, G.H., Viola, A. (eds.) LATIN 2000. LNCS, vol. 1776, pp. 343–354. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  15. 15.
    Weilert, A.: Fast Computation of the Biquadratic Residue Symbol. Journal of Number Theory 96, 133–151 (2002)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Richard P. Brent
    • 1
  • Paul Zimmermann
    • 2
  1. 1.Australian National UniversityCanberraAustralia
  2. 2.INRIA Nancy - Grand EstVillers-lès-NancyFrance

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