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Linear Recurrences with Polynomial Coefficients and Computation of the Cartier-Manin Operator on Hyperelliptic Curves

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Finite Fields and Applications (Fq 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2948))

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Abstract

We improve an algorithm originally due to Chudnovsky and Chudnovsky for computing one selected term in a linear recurrent sequence with polynomial coefficients. Using baby-steps / giant-steps techniques, the nth term in such a sequence can be computed in time proportional to \(\sqrt{n}\), instead of n for a naive approach.

As an intermediate result, we give a fast algorithm for computing the values taken by an univariate polynomial P on an arithmetic progression, taking as input the values of P on a translate on this progression.

We apply these results to the computation of the Cartier-Manin operator of a hyperelliptic curve. If the base field has characteristic p, this enables us to reduce the complexity of this computation by a factor of order \(\sqrt{p}\). We treat a practical example, where the base field is an extension of degree 3 of the prime field with p = 23232 – 5 elements.

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References

  1. Medicis, http://www.medicis.polytechnique.fr/

  2. Aho, A.V., Steiglitz, K., Ullman, J.D.: Evaluating polynomials at fixed sets of points. SIAM J. Comput. 4(4), 533–539 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bailey, D., Paar, C.: Optimal extension fields for fast arithmetic in public-key algorithms. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 472–485. Springer, Heidelberg (1998)

    Google Scholar 

  4. Borodin, A., Moenck, R.T.: Fast modular transforms. Comput. System Sci. 8(3), 366–386 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comp. 24(3-4), 235–265 (1997), See also http://www.maths.usyd.edu.au:8000/u/magma/

    Google Scholar 

  6. Bostan, A., Lecerf, G., Schost, É.: Tellegen’s principle into practice. In: Proceedings of ISSAC 2003, pp. 37–44. ACM Press, New York (2003)

    Chapter  Google Scholar 

  7. Bürgisser, P., Clausen, M., Shokrollahi, M.A.: Algebraic complexity theory. Grundlehren Math. Wiss., vol. 315. Springer, Heidelberg (1997)

    MATH  Google Scholar 

  8. Cantor, D.G., Kaltofen, E.: On fast multiplication of polynomials over arbitrary algebras. Acta Informatica 28(7), 693–701 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  9. Cartier, P.: Une nouvelle opération sur les formes différentielles. C. R. Acad. Sci. Paris 244, 426–428 (1957)

    MATH  MathSciNet  Google Scholar 

  10. Chudnovsky, D.V., Chudnovsky, G.V.: Approximations and complex multiplication according to Ramanujan. In: Ramanujan revisited, Urbana-Champaign, Ill, 1987, pp. 375–472. Academic Press, Boston (1988)

    Google Scholar 

  11. Flajolet, P., Salvy, B.: The SIGSAM challenges: Symbolic asymptotics in practice. SIGSAM Bull. 31(4), 36–47 (1997)

    Article  Google Scholar 

  12. Gaudry, P., Gürel, N.: Counting points in medium characteristic using Kedlaya’s algorithm. To appear in Experiment. Math.

    Google Scholar 

  13. Gaudry, P., Harley, R.: Counting points on hyperelliptic curves over finite fields. In: Bosma, W. (ed.) ANTS 2000. LNCS, vol. 1838, pp. 313–332. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  14. Gaudry, P., Schost, É.: Cardinality of a genus 2 hyperelliptic curve over GF(5 . 1024 + 41). e-mail to the NMBRTHRY mailing list (September 2002)

    Google Scholar 

  15. Hanrot, G., Quercia, M., Zimmermann, P.: The middle product algorithm, I. Speeding up the division and square root of power series (preprint)

    Google Scholar 

  16. Hasse, H., Witt, E.: Zyklische unverzweigte Erweiterungskörper vom primzahlgrade p über einem algebraischen Funktionenkörper der Charakteristik p. Monatsch. Math. Phys. 43, 477–492 (1936)

    MATH  MathSciNet  Google Scholar 

  17. Kaltofen, E., Corless, R.M., Jeffrey, D.J.: Challenges of symbolic computation: my favorite open problems. J. Symb. Comp. 29(6), 891–919 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kedlaya, K.: Countimg points on hyperelliptic curves using Monsky-Washnitzer. J. Ramanujan Math. Soc. 16, 323–338 (2001)

    MATH  MathSciNet  Google Scholar 

  19. Keller-Gehrig, W.: Fast algorithms for the characteristic polynomial. Theor. Comput. Sci. 36(2-3), 309–317 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  20. Manin, J.I.: The Hasse-Witt matrix of an algebraic curve. Trans. Amer. Math. Soc. 45, 245–264 (1965)

    Google Scholar 

  21. Matsuo, K., Chao, J., Tsujii, S.: An improved baby step giant step algorithm for point counting of hyperelliptic curves over finite fields. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 461–474. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  22. Moenck, R.T., Borodin, A.: Fast modular transforms via division. In: Thirteenth Annual IEEE Symposium on Switching and Automata Theory, Univ. Maryland, College Park, Md., pp. 90–96 (1972)

    Google Scholar 

  23. Pollard, J.M.: Theorems on factorization and primality testing. Proc. Cambridge Philos. Soc. 76, 521–528 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  24. Schönhage, A.: Schnelle Multiplikation von Polynomen über Körpern der Charakteristik 2. Acta Informatica 7, 395–398 (1977)

    Article  MATH  Google Scholar 

  25. Schönhage, A., Strassen, V.: Schnelle Multiplikation großer Zahlen. Computing 7, 281–292 (1971)

    Article  MATH  Google Scholar 

  26. Shoup, V.: NTL: A library for doing number theory, http://www.shoup.net

  27. Shoup, V.: A fast deterministic algorithm for factoring polynomials over finite fields of small characteristic. In: Proceedings of ISSAC 1991, pp. 14–21. ACM Press, New York (1991)

    Chapter  Google Scholar 

  28. Stöhr, K.-O., Voloch, J.: A formula for the Cartier operator on plane algebraic curves. J. Reine Angew. Math. 377, 49–64 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  29. Strassen, V.: Gaussian elimination is not optimal. Numer. Math. 13, 354–356 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  30. Strassen, V.: Einige Resultate über Berechnungskomplexität. Jber. Deutsch. Math.- Verein. 78(1), 1–8 (1976/1977)

    Google Scholar 

  31. von zur Gathen, J., Gerhard, J.: Modern computer algebra. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  32. Wan, D.: Computing zeta functions over finite fields. Contemp. Math. 225, 131–141 (1999)

    Google Scholar 

  33. Yui, N.: On the Jacobian varietes of hyperelliptic curves over fields of characteristic p > 2. J. Algebra 52, 378–410 (1978)

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Laboratoire STIX, École polytechnique, France

    Alin Bostan & Éric Schost

  2. IMAR, Romania

    Alin Bostan

  3. Laboratoire LIX, École polytechnique, France

    Pierrick Gaudry

Authors
  1. Alin Bostan
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  2. Pierrick Gaudry
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  3. Éric Schost
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Editor information

Editors and Affiliations

  1. Department of Mathematics, The Pennsylvania State University, University Park, 16802, PA, USA

    Gary L. Mullen

  2. University Paul Sabatier, AAECC/IRIT, 118 route de Narbonne, 31300, Toulouse cedex, France

    Alain Poli

  3. Sabancı University - FENS, Orhanli - Tuzla, 34956, Istanbul, Turkey

    Henning Stichtenoth

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© 2004 Springer-Verlag Berlin Heidelberg

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Bostan, A., Gaudry, P., Schost, É. (2004). Linear Recurrences with Polynomial Coefficients and Computation of the Cartier-Manin Operator on Hyperelliptic Curves. In: Mullen, G.L., Poli, A., Stichtenoth, H. (eds) Finite Fields and Applications. Fq 2003. Lecture Notes in Computer Science, vol 2948. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24633-6_4

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  • DOI: https://doi.org/10.1007/978-3-540-24633-6_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-21324-6

  • Online ISBN: 978-3-540-24633-6

  • eBook Packages: Springer Book Archive

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