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In a previous joint article with F. Abu Salem, we gave efficient algorithmsfor Jacobian group arithmetic of "typical" divisor classes on C_{3,4} curves,improving on similar results by other authors. At that time, we could onlystate that a generic divisor was typical, and hence unlikely to be encounteredif one implemented these algorithms over a very large finite field. Thisarticle pins down an explicit characterization of these typical divisors, foran arbitrary smooth projective curve of genus g >= 1 having at least onerational point. We give general algorithms for Jacobian group arithmetic withthese typical divisors, and prove not only that the algorithms are correct ifvarious divisors are typical, but also that the success of our algorithmsprovides a guarantee that the resulting output is correct and that theresulting input and/or output divisors are also typical. These results apply inparticular to our earlier algorithms for C_{3,4} curves. As a byproduct, weobtain a further speedup of approximately 15% on our previous algorithms forC_{3,4} curves.

On Jacobian group arithmetic for typical divisors on curves