On Jacobian group arithmetic for typical divisors on curves

On Jacobian group arithmetic for typical divisors on curves In a previous joint article with Abu Salem, we gave efficient algorithms for Jacobian group arithmetic of “typical” divisor classes on $$C_{3,4}$$ C 3 , 4 curves, improving on similar results by other authors. At that time, we could only state that a general 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 $$g \ge 1$$ 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 $$C_{3,4}$$ C 3 , 4 curves. As a byproduct, we obtain a further speedup of approximately 15% on our previous algorithms for $$C_{3,4}$$ C 3 , 4 curves. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Research in Number Theory Springer Journals

On Jacobian group arithmetic for typical divisors on curves

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Publisher
Springer International Publishing
Copyright
Copyright © 2018 by SpringerNature
Subject
Mathematics; Number Theory
eISSN
2363-9555
D.O.I.
10.1007/s40993-018-0101-6
Publisher site
See Article on Publisher Site

Abstract

In a previous joint article with Abu Salem, we gave efficient algorithms for Jacobian group arithmetic of “typical” divisor classes on $$C_{3,4}$$ C 3 , 4 curves, improving on similar results by other authors. At that time, we could only state that a general 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 $$g \ge 1$$ 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 $$C_{3,4}$$ C 3 , 4 curves. As a byproduct, we obtain a further speedup of approximately 15% on our previous algorithms for $$C_{3,4}$$ C 3 , 4 curves.

Journal

Research in Number TheorySpringer Journals

Published: Jan 31, 2018

References

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