Computing the cassels pairing on Kolyvagin classes in the shafarevich-tate group

Kirsten Eisenträger, Dimitar Jetchev, Kristin Lauter

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Citation (Scopus)

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.

Original languageEnglish (US)
Title of host publicationPairing-Based Cryptography - Pairing 2008 - Second International Conference, Proceedings
Pages113-125
Number of pages13
DOIs
StatePublished - Sep 25 2008
Event2nd International Conference on Pairing-Based Cryptography, Pairing 2008 - Egham, United Kingdom
Duration: Sep 1 2008Sep 3 2008

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume5209 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other2nd International Conference on Pairing-Based Cryptography, Pairing 2008
CountryUnited Kingdom
CityEgham
Period9/1/089/3/08

Fingerprint

Pairing
Computing
Testing
Trivial
Tate Pairing
Imaginary Quadratic Field
Class
Elliptic Curves
Ring

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Eisenträger, K., Jetchev, D., & Lauter, K. (2008). Computing the cassels pairing on Kolyvagin classes in the shafarevich-tate group. In Pairing-Based Cryptography - Pairing 2008 - Second International Conference, Proceedings (pp. 113-125). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5209 LNCS). https://doi.org/10.1007/978-3-540-85538-5_8
Eisenträger, Kirsten ; Jetchev, Dimitar ; Lauter, Kristin. / Computing the cassels pairing on Kolyvagin classes in the shafarevich-tate group. Pairing-Based Cryptography - Pairing 2008 - Second International Conference, Proceedings. 2008. pp. 113-125 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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Eisenträger, K, Jetchev, D & Lauter, K 2008, Computing the cassels pairing on Kolyvagin classes in the shafarevich-tate group. in Pairing-Based Cryptography - Pairing 2008 - Second International Conference, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 5209 LNCS, pp. 113-125, 2nd International Conference on Pairing-Based Cryptography, Pairing 2008, Egham, United Kingdom, 9/1/08. https://doi.org/10.1007/978-3-540-85538-5_8

Computing the cassels pairing on Kolyvagin classes in the shafarevich-tate group. / Eisenträger, Kirsten; Jetchev, Dimitar; Lauter, Kristin.

Pairing-Based Cryptography - Pairing 2008 - Second International Conference, Proceedings. 2008. p. 113-125 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5209 LNCS).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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Eisenträger K, Jetchev D, Lauter K. Computing the cassels pairing on Kolyvagin classes in the shafarevich-tate group. In Pairing-Based Cryptography - Pairing 2008 - Second International Conference, Proceedings. 2008. p. 113-125. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-540-85538-5_8