### Abstract

The hardness of the elliptic curve discrete logarithm problem (ECDLP) on a finite field is essential for the security of all elliptic curve cryptographic schemes. The ECDLP on a field K is as follows: given an elliptic curve E over K, a point S ∈ E(K), and a point T ∈ E(K) with T ∈ (S), find the integer d such that T = dS. A number of ways of approaching the solution to the ECDLP on a finite field is known, for example, the MOV attack [5], and the anomalous attack [7,10]. In this paper, we propose an algorithm to solve the ECDLP on the p-adic field Q_{p}. Our method is to use the theory of formal groups associated to elliptic curves, which is used for the anomalous attack proposed by Smart [10], and Satoh and Araki [7].

Original language | English |
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Title of host publication | Information Security Practice and Experience - 6th International Conference, ISPEC 2010, Proceedings |

Pages | 110-122 |

Number of pages | 13 |

DOIs | |

Publication status | Published - Dec 23 2010 |

Externally published | Yes |

Event | 6th International Conference on Information Security Practice and Experience, ISPEC 2010 - Seoul, Korea, Republic of Duration: May 12 2010 → May 13 2010 |

### 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 | 6047 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 6th International Conference on Information Security Practice and Experience, ISPEC 2010 |
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Country | Korea, Republic of |

City | Seoul |

Period | 5/12/10 → 5/13/10 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Information Security Practice and Experience - 6th International Conference, ISPEC 2010, Proceedings*(pp. 110-122). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6047 LNCS). https://doi.org/10.1007/978-3-642-12827-1_9

**The elliptic curve discrete logarithm problems over the p-adic field and formal groups.** / Yasuda, Masaya.

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

*Information Security Practice and Experience - 6th International Conference, ISPEC 2010, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 6047 LNCS, pp. 110-122, 6th International Conference on Information Security Practice and Experience, ISPEC 2010, Seoul, Korea, Republic of, 5/12/10. https://doi.org/10.1007/978-3-642-12827-1_9

}

TY - GEN

T1 - The elliptic curve discrete logarithm problems over the p-adic field and formal groups

AU - Yasuda, Masaya

PY - 2010/12/23

Y1 - 2010/12/23

N2 - The hardness of the elliptic curve discrete logarithm problem (ECDLP) on a finite field is essential for the security of all elliptic curve cryptographic schemes. The ECDLP on a field K is as follows: given an elliptic curve E over K, a point S ∈ E(K), and a point T ∈ E(K) with T ∈ (S), find the integer d such that T = dS. A number of ways of approaching the solution to the ECDLP on a finite field is known, for example, the MOV attack [5], and the anomalous attack [7,10]. In this paper, we propose an algorithm to solve the ECDLP on the p-adic field Qp. Our method is to use the theory of formal groups associated to elliptic curves, which is used for the anomalous attack proposed by Smart [10], and Satoh and Araki [7].

AB - The hardness of the elliptic curve discrete logarithm problem (ECDLP) on a finite field is essential for the security of all elliptic curve cryptographic schemes. The ECDLP on a field K is as follows: given an elliptic curve E over K, a point S ∈ E(K), and a point T ∈ E(K) with T ∈ (S), find the integer d such that T = dS. A number of ways of approaching the solution to the ECDLP on a finite field is known, for example, the MOV attack [5], and the anomalous attack [7,10]. In this paper, we propose an algorithm to solve the ECDLP on the p-adic field Qp. Our method is to use the theory of formal groups associated to elliptic curves, which is used for the anomalous attack proposed by Smart [10], and Satoh and Araki [7].

UR - http://www.scopus.com/inward/record.url?scp=78650279088&partnerID=8YFLogxK

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U2 - 10.1007/978-3-642-12827-1_9

DO - 10.1007/978-3-642-12827-1_9

M3 - Conference contribution

AN - SCOPUS:78650279088

SN - 3642128262

SN - 9783642128264

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

SP - 110

EP - 122

BT - Information Security Practice and Experience - 6th International Conference, ISPEC 2010, Proceedings

ER -