A Fast Randomized Geometric Algorithm for Computing Riemann-Roch Spaces

Aude Le Gluher 1 Pierre-Jean Spaenlehauer 1
1 CARAMBA - Cryptology, arithmetic : algebraic methods for better algorithms
LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry, Inria Nancy - Grand Est
Abstract : We propose a probabilistic Las Vegas variant of Brill-Noether's algorithm for computing a basis of the Riemann-Roch space $L(D)$ associated to a divisor $D$ on a projective nodal plane curve $\mathcal C$ over a sufficiently large perfect field $k$. Our main result shows that this algorithm requires at most $O(\max(\mathrm{deg}(\mathcal C)^{2\omega}, \mathrm{deg}(D_+)^\omega))$ arithmetic operations in $k$, where $\omega$ is a feasible exponent for matrix multiplication and $D_+$ is the smallest effective divisor such that $D_+\geq D$. This improves the best known upper bounds on the complexity of computing Riemann-Roch spaces. Our algorithm may fail, but we show that provided that a few mild assumptions are satisfied, the failure probability is bounded by $O(\max(\mathrm{deg}(\mathcal C)^4, \mathrm{deg}(D_+)^2)/\lvert \mathcal E\rvert)$, where $\mathcal E$ is a finite subset of $k$ in which we pick elements uniformly at random. We provide a freely available C++/NTL implementation of the proposed algorithm and we present experimental data. In particular, our implementation enjoys a speedup larger than 6 on many examples (and larger than 200 on some instances over large finite fields) compared to the reference implementation in the Magma computer algebra system. As a by-product, our algorithm also yields a method for computing the group law on the Jacobian of a smooth plane curve of genus $g$ within $O(g^\omega)$ operations in $k$, which slightly improves in this context the best known complexity $O(g^{\omega+\varepsilon})$ of Khuri-Makdisi's algorithm.
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https://hal.inria.fr/hal-01930573
Contributor : Pierre-Jean Spaenlehauer <>
Submitted on : Thursday, November 22, 2018 - 10:07:41 AM
Last modification on : Thursday, May 16, 2019 - 11:02:07 AM

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  • HAL Id : hal-01930573, version 1
  • ARXIV : 1811.08237

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Aude Le Gluher, Pierre-Jean Spaenlehauer. A Fast Randomized Geometric Algorithm for Computing Riemann-Roch Spaces. 2018. ⟨hal-01930573⟩

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