Aude Le Gluher | Searcharxiv

arXiv2021

Refined Analysis of the Asymptotic Complexity of the Number Field Sieve

The classical heuristic complexity of the Number Field Sieve (NFS) is the solution of an optimization problem that involves an unknown function, usually noted $o(1)$ and called $ξ(N)$ throughout this paper, which tends to zero as the entry $N$ grows. The aim of this paper is to find optimal asymptotic choices of the parameters of NFS as $N$ grows, in order to minimize its heuristic asymptotic computational cost. This amounts to minimizing a function of the parameters of NFS bound together by a non-linear constraint. We provide precise asymptotic estimates of the minimizers of this optimization problem, which yield refined formulas for the asymptotic complexity of NFS. One of the main outcomes of this analysis is that $ξ(N)$ has a very slow rate of convergence: We prove that it is equivalent to $4{\log}{\log}{\log}\,N/(3{\log}{\log}\,N)$. Moreover, $ξ(N)$ has an unpredictable behavior for practical estimates of the complexity. Indeed, we provide an asymptotic series expansion of $ξ$ and numerical experiments indicate that this series starts converging only for $N>\exp(\exp(25))$, far beyond the practical range of NFS. This raises doubts on the relevance of NFS running time estimates that are based on setting $ξ=0$ in the asymptotic formula.

cs.CR↗

arXiv2020

A Fast Randomized Geometric Algorithm for Computing Riemann-Roch Spaces

We propose a probabilistic 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 $O(\max(\mathrm{deg}(\mathcal C)^{2ω}, \mathrm{deg}(D_+)^ω))$ arithmetic operations in $k$, where $ω$ 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 $O(g^ω)$ operations in $k$, which equals the best known complexity for this problem.

cs.SC↗