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Published in:
2008 | OriginalPaper | Chapter
Computing the Cassels Pairing on Kolyvagin Classes in the Shafarevich-Tate Group
Authors : Kirsten Eisenträger, Dimitar Jetchev, Kristin Lauter
Published in: Pairing-Based Cryptography – Pairing 2008
Publisher: Springer Berlin Heidelberg
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Kolyvagin has shown how to study the Shafarevich-Tate group of elliptic curves over imaginary quadratic fields via Kolyvagin classes constructed from Heegner points. One way to produce explicit non-trivial elements of the Shafarevich-Tate group is by proving that a locally trivial Kolyvagin class is globally non-trivial, which is difficult in practice. We provide a method for testing whether an explicit element of the Shafarevich-Tate group represented by a Kolyvagin class is globally non-trivial by determining whether the Cassels pairing between the class and another locally trivial Kolyvagin class is non-zero. Our algorithm explicitly computes Heegner points over ring class fields to produce the Kolyvagin classes and uses the efficiently computable cryptographic Tate pairing.