Asymptotically fast group operations on Jacobians of general curves

Abstract

Abstract. Let C be a curve of genus g over a field k. We describe probabilistic algorithms for addition and inversion of the classes of rational divisors in the Jacobian of C. After a precomputation, which is done only once for the curve C, the algorithms use only linear algebra in vector spaces of dimension at most O(g log g), and so take O(g 3+ ) field operations in k, using Gaussian elimination. Using fast algorithms for the linear algebra, one can improve this time to O(g 2.376 ). This represents a significant improvement over the previous record of O(g 4 ) field operations (also after a precomputation) for general curves of genus g.

On Jacobian group arithmetic for typical divisors on curves

“…It is convenient to show first that L is semi-typical, even though this is implied by the full result. Indeed, as L varies in Pic d C, the line bundle , and our next goal is to show that the above inclusion is an equality for most D. This is similar to the proof of Lemma 4.10 in [7], and is in essence the base point free pencil trick. We have as usual dim …”

“…When D is good, it is well known (and basically tautological) that the choice of such a D is unique when D is reduced; in Proposition 3.5, we recall the definition of a reduced divisor, and later show in Corollary 3.7 that typical divisors are always reduced. Hence typical elements of Pic 0 (C) have a unique representation by a good divisor D of degree g, and we do not need to go through the more elaborate tests for equality used in the general algorithms of [6,7]. We still need divisors of degrees d ≥ g to represent various intermediate results in our algorithms, so we carry out the discussion below for general d.…”

“…We have dim V = 3, and we construct in that article a matrix M whose columns represent the images in V of the basis {t, xt, yt, x 2 t, xyt} for tW 7 . Thus an element of ker M describes a linear combination of c 0 t + c 1 xt + · · · , that lies in tW 7 ∩ (sW 8 ⊕ W 8 ), hence simultaneously describes a combination = c 0 + c 1 x + · · · ∈ W .…”

“…This yields (ii) and (iii). The divisor of G 1 ∈ W 7 A must have the form A + E, and {F, G 1 } are an IGS for A, so E is disjoint from D. Finally, the divisor of H 1 follows from the fact that…”

“…The above articles, however, do not include a test to verify whether the input divisors or output data are in fact typical, so that in principle the algorithms might return wrong results without this being detected during the computation. It is desirable to know when such algorithms have failed, so that one can redo the computation using slightly slower algorithms that work for all divisors (for example, those in [6,7]). …”