Computing the Cassels Pairing on Kolyvagin Classes in the Shafarevich-Tate Group

Authors:
Kirsten Eisenträger;Dimitar Jetchev;Kristin Lauter
Affiliations:
Department of Mathematics, The Pennsylvania State University 16802;Department of Mathematics, University of California, Berkeley 94720;Microsoft Research, Redmond 98052
Venue:
Pairing '08 Proceedings of the 2nd international conference on Pairing-Based Cryptography
Year:
2008

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Abstract

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.