### Abstract

In 2018, Amadori et al. proposed a new variant of index calculus to solve the elliptic curve discrete logarithm problem (ECDLP), using Semaev’s summation polynomials. The variant drastically decreases the number of required Gröbner basis computations, and it outperforms other index calculus algorithms for the ECDLP over prime fields. In this paper, we provide several improvements to accelerate to solve systems of multivariate equations arising in the variant. A main improvement is to apply the hybrid method, which mixes exhaustive search and Gröbner bases techniques to solve multivariate systems over finite fields. We also make use of symmetries of summation polynomials. We show experimental results of our improvements, and give their complexity analysis to discuss a limitation of our acceleration in both theory and practice.

Original language | English |
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Title of host publication | Cryptology and Network Security - 17th International Conference, CANS 2018, Proceedings |

Editors | Panos Papadimitratos, Jan Camenisch |

Publisher | Springer Verlag |

Pages | 377-393 |

Number of pages | 17 |

ISBN (Print) | 9783030004330 |

DOIs | |

Publication status | Published - Jan 1 2018 |

Event | 17th International Conference on Cryptology and Network Security, CANS 2018 - Naples, Italy Duration: Sep 30 2018 → Oct 3 2018 |

### 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 | 11124 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 17th International Conference on Cryptology and Network Security, CANS 2018 |
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Country | Italy |

City | Naples |

Period | 9/30/18 → 10/3/18 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Cryptology and Network Security - 17th International Conference, CANS 2018, Proceedings*(pp. 377-393). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11124 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-030-00434-7_19

**Acceleration of index calculus for solving ECDLP over prime fields and its limitation.** / Kudo, Momonari; Yokota, Yuki; Takahashi, Yasushi; Yasuda, Masaya.

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

*Cryptology and Network Security - 17th International Conference, CANS 2018, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 11124 LNCS, Springer Verlag, pp. 377-393, 17th International Conference on Cryptology and Network Security, CANS 2018, Naples, Italy, 9/30/18. https://doi.org/10.1007/978-3-030-00434-7_19

}

TY - GEN

T1 - Acceleration of index calculus for solving ECDLP over prime fields and its limitation

AU - Kudo, Momonari

AU - Yokota, Yuki

AU - Takahashi, Yasushi

AU - Yasuda, Masaya

PY - 2018/1/1

Y1 - 2018/1/1

N2 - In 2018, Amadori et al. proposed a new variant of index calculus to solve the elliptic curve discrete logarithm problem (ECDLP), using Semaev’s summation polynomials. The variant drastically decreases the number of required Gröbner basis computations, and it outperforms other index calculus algorithms for the ECDLP over prime fields. In this paper, we provide several improvements to accelerate to solve systems of multivariate equations arising in the variant. A main improvement is to apply the hybrid method, which mixes exhaustive search and Gröbner bases techniques to solve multivariate systems over finite fields. We also make use of symmetries of summation polynomials. We show experimental results of our improvements, and give their complexity analysis to discuss a limitation of our acceleration in both theory and practice.

AB - In 2018, Amadori et al. proposed a new variant of index calculus to solve the elliptic curve discrete logarithm problem (ECDLP), using Semaev’s summation polynomials. The variant drastically decreases the number of required Gröbner basis computations, and it outperforms other index calculus algorithms for the ECDLP over prime fields. In this paper, we provide several improvements to accelerate to solve systems of multivariate equations arising in the variant. A main improvement is to apply the hybrid method, which mixes exhaustive search and Gröbner bases techniques to solve multivariate systems over finite fields. We also make use of symmetries of summation polynomials. We show experimental results of our improvements, and give their complexity analysis to discuss a limitation of our acceleration in both theory and practice.

UR - http://www.scopus.com/inward/record.url?scp=85057335292&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85057335292&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-00434-7_19

DO - 10.1007/978-3-030-00434-7_19

M3 - Conference contribution

AN - SCOPUS:85057335292

SN - 9783030004330

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 377

EP - 393

BT - Cryptology and Network Security - 17th International Conference, CANS 2018, Proceedings

A2 - Papadimitratos, Panos

A2 - Camenisch, Jan

PB - Springer Verlag

ER -