On Jacobian group arithmetic for typical divisors on curves
Abstract
In a previous joint article with F. Abu Salem, we gave efficient algorithms for Jacobian group arithmetic of "typical" divisor classes on C_{3,4} curves, improving on similar results by other authors. At that time, we could only state that a generic divisor was typical, and hence unlikely to be encountered if one implemented these algorithms over a very large finite field. This article pins down an explicit characterization of these typical divisors, for an arbitrary smooth projective curve of genus g >= 1 having at least one rational point. We give general algorithms for Jacobian group arithmetic with these typical divisors, and prove not only that the algorithms are correct if various divisors are typical, but also that the success of our algorithms provides a guarantee that the resulting output is correct and that the resulting input and/or output divisors are also typical. These results apply in particular to our earlier algorithms for C_{3,4} curves. As a byproduct, we obtain a further speedup of approximately 15% on our previous algorithms for C_{3,4} curves.
Cite
@article{arxiv.1310.6324,
title = {On Jacobian group arithmetic for typical divisors on curves},
author = {Kamal Khuri-Makdisi},
journal= {arXiv preprint arXiv:1310.6324},
year = {2019}
}
Comments
29 pages, further editing to take into account referee reports. Section 3 reworked to better separate the general results from those for C_{3,4} curves, which are now marked as examples. Section 4 now sets off the results more clearly instead of including them in narrative form in the text