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Fast addition on non-hyperelliptic genus 3 curves

Stéphane Flon , Roger Oyono and Christophe Ritzenthaler

Abstract: We present a fast addition algorithm in the Jacobian of a genus 3 non-hyperelliptic curve over a field of any characteristic. When the curve has a rational flex and char(k) > 5, the computational cost for addition is 148M + 15SQ + 2I and 165M + 20SQ + 2I for doubling. An appendix focuses on the computation of flexes in all characteristics. For large odd q, we also show that the set of rational points of a non-hyperelliptic curve of genus 3 can not be an arc.

This paper apears in :

  • S. Flon, R. Oyono, C. Ritzenthaler: Fast addition on non-hyperelliptic genus 3 curves, Algebraic geometry and its applications, Proceedings of the first SAGA conference, Ser. Number theory and its applications, World Sci. Publ., Hackensack, NJ, pp. 1 -- 28, 2008.


    The following files contain the implementation (in MAGMA) of the arithmetic on the Jacobian of non-hyperelliptic curves of genus 3 via explicit formulae. For the moment we only implemented it when the characteristic of base field is at least 7.

    All the programs and documentation are in the following zip archive, quartic.zip (???k).

    We have separated it into four files:

    "input" The file where the algorithm gets its general setting (the base field and the curve)
    "addition" The core of the algorithm
    "fieldarithmetic" The file for the optimisation of polynomial arithmetic in char(k) > 5
    "scalarproduct" The file to perform multiplication of a divsor with a scalar

    Now we get to the VFAQ (Virtually Frequently Asked Questions):


    To run the programm: Run magma in the directory where the files are, and then type: load "input";

    To change the input (that is the base field and the curve), edit the file "input" and enter almost what you like.

    Enjoy! If you have any suggestion, the authors invite you to write them to:

    Stéphane Flon
    Roger Oyono
    Christophe Ritzenthaler

    e-mail : Roger Oyono

    Last update November 17, 2011