TY - GEN
T1 - Computing the cassels pairing on Kolyvagin classes in the shafarevich-tate group
AU - Eisenträger, Kirsten
AU - Jetchev, Dimitar
AU - Lauter, Kristin
PY - 2008
Y1 - 2008
N2 - 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.
AB - 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.
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U2 - 10.1007/978-3-540-85538-5_8
DO - 10.1007/978-3-540-85538-5_8
M3 - Conference contribution
AN - SCOPUS:52149092782
SN - 3540855033
SN - 9783540855033
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 113
EP - 125
BT - Pairing-Based Cryptography - Pairing 2008 - Second International Conference, Proceedings
T2 - 2nd International Conference on Pairing-Based Cryptography, Pairing 2008
Y2 - 1 September 2008 through 3 September 2008
ER -