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Square Roots Modulo p

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LATIN 2002: Theoretical Informatics (LATIN 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2286))

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Abstract

The algorithm of Tonelli and Shanks for computing square roots modulo a prime number is the most used, and probably the fastest among the known algorithms when averaged over all prime numbers. However, for some particular prime numbers, there are other algorithms which are considerably faster.

In this paper we compare the algorithm of Tonelli and Shanks with an algorithm based in quadratic field extensions due to Cipolla, and give an explicit condition on a prime number to decide which algorithm is faster. Finally, we show that there exists an infinite sequence of prime numbers for which the algorithm of Tonelli and Shanks is asymptotically worse.

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References

  1. Barban, M.B., Linnik, Y.V., Tshudakov, N.G.: On prime numbers in an arithmetic progression with a prime-power difference. Acta Arith. 9 (1964) 375–390

    Article  MathSciNet  Google Scholar 

  2. Bernstein, D.J.: Faster square roots in annoying finite fields, draft. Available from http://cr.yp.to/papers.html (2001)

  3. Cipolla, M.: Un metodo per la risoluzione della congruenza di secondo grado. Rend. Accad. Sci. Fis. Mat. Napoli 9 (1903) 154–163

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  4. Heath-Brown, D.R.: Zero-free regions for Dirichlet L-functions, and the least prime in an arithmetic progression. Proc. London Math. Soc. (3) 64 (1992) 265–338

    Article  MathSciNet  Google Scholar 

  5. Lindhurst, S.: An analysis of Shanks’s algorithm for computing square roots in finite fields. In: Number theory (Ottawa, ON, 1996). Amer. Math. Soc., Providence, RI (1999) 231–242

    Google Scholar 

  6. Linnik, U.V.: On the least prime in an arithmetic progression. I. The basic theorem. Rec. Math. [Mat. Sbornik] N.S. 15(57) (1944) 139–178

    MathSciNet  MATH  Google Scholar 

  7. Linnik, U.V.: On the least prime in an arithmetic progression. II. The Deuring-Heilbronn phenomenon. Rec. Math. [Mat. Sbornik] N.S. 15(57) (1944) 347–368

    MathSciNet  MATH  Google Scholar 

  8. Shanks, D.: Five number-theoretic algorithms. In: Proceedings of the Second Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1972). Utilitas Math., Winnipeg, Man. (1973) 51–70. Congressus Numerantium, No. VII

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  9. Tonelli, A.: Bemerkung über die Auflösung quadratischer Congruenzen. Göttinger Nachrichten (1891) 344–346

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Texas at Austin, 78712, Austin, Texas, USA

    Gonzalo Tornaría

Authors
  1. Gonzalo Tornaría
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Editor information

Editors and Affiliations

  1. Compaq Cambridge Research Laboratory, One Cambridge Center, 02142-1612, Cambridge, MA, USA

    Sergio Rajsbaum

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© 2002 Springer-Verlag Berlin Heidelberg

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Tornaría, G. (2002). Square Roots Modulo p . In: Rajsbaum, S. (eds) LATIN 2002: Theoretical Informatics. LATIN 2002. Lecture Notes in Computer Science, vol 2286. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45995-2_38

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  • DOI: https://doi.org/10.1007/3-540-45995-2_38

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-43400-9

  • Online ISBN: 978-3-540-45995-8

  • eBook Packages: Springer Book Archive

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