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

  • Alin Bostan
  • Pierrick Gaudry
  • Éric Schost
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2948)

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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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Alin Bostan
    • 1
    • 2
  • Pierrick Gaudry
    • 3
  • Éric Schost
    • 1
  1. 1.Laboratoire STIXÉcole polytechniqueFrance
  2. 2.IMARRomania
  3. 3.Laboratoire LIXÉcole polytechniqueFrance

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