### 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 language | English (US) |
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Title of host publication | Pairing-Based Cryptography - Pairing 2008 - Second International Conference, Proceedings |

Pages | 113-125 |

Number of pages | 13 |

DOIs | |

State | Published - Sep 25 2008 |

Event | 2nd International Conference on Pairing-Based Cryptography, Pairing 2008 - Egham, United Kingdom Duration: Sep 1 2008 → Sep 3 2008 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 5209 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 2nd International Conference on Pairing-Based Cryptography, Pairing 2008 |
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Country | United Kingdom |

City | Egham |

Period | 9/1/08 → 9/3/08 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*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

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*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

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 - 2008/9/25

Y1 - 2008/9/25

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.

UR - http://www.scopus.com/inward/record.url?scp=52149092782&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=52149092782&partnerID=8YFLogxK

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

ER -