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.
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机译：在先前与F. Abu Salem的联合文章中，我们为C_ {3,4}曲线上的“典型”除数类的Jacobian群算术提供了有效的算法，其他作者的结果也有所改进。那时，我们只能说一般的除数是典型的，因此如果在很大的有限域上实现这些算法，就不太可能遇到。本文确定了这些典型除数的明确特征，因为g> = 1的任意平滑投影曲线至少具有加法点。我们给出了具有这些典型除数的Jacobian群算术的通用算法，不仅证明了各种除数是典型的算法是正确的，而且我们算法的成功提供了保证结果输出正确以及输出和/或输出的保证。除数也是典型的。这些结果尤其适用于我们针对C_ {3,4}曲线的较早算法。作为副产品，我们在以前的C_ {3,4}曲线算法中获得了大约15％的进一步加速。
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