On Jacobian group arithmetic for typical divisors on curves

In a previous joint article with F. Abu Salem, we gave efficient algorithms for Jacobian group arithmetic of “typical” divisor classes on C3,4 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 that if our algorithms can be carried out, then this provides a guarantee that the resulting output is correct and that the resulting divisor is also typical. These results apply in particular to our earlier algorithms for C3,4 curves. As a byproduct, we obtain a further speedup of approximately 15% on our previous algorithms for C3,4 curves.