### Abstract

Recent innovative public key cryptographic applications such as ID-based cryptography are based on pairing cryptography. They efficiently use some torsion group structures constructed on certain elliptic curves defined over finite fields. For this purpose, this paper shows that a relation between group order of elliptic curve and extension degree of definition field especially from the viewpoint of torsion structure for pairing- based cryptographic use. In detail, it is shown that the order of elliptic curve over r ^{i}-th extension field denoted by #E(F _{q}ri ) is divisible by r ^{2i} and it has the torsion structure denoted by Z _{r}i Z _{r}i when the base order of elliptic curve denoted by #E(Fq) is divisible by r^i and the order of the multiplicative group of the definition field is also divisible by r^i, where r denotes the order of one cyclic group in the torsion structure.

Original language | English |
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Title of host publication | 2012 World Telecommunications Congress, WTC 2012 |

Publication status | Published - 2012 |

Event | 2012 World Telecommunications Congress, WTC 2012 - Miyazaki, Japan Duration: Mar 5 2012 → Mar 6 2012 |

### Other

Other | 2012 World Telecommunications Congress, WTC 2012 |
---|---|

Country | Japan |

City | Miyazaki |

Period | 3/5/12 → 3/6/12 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Networks and Communications

### Cite this

*2012 World Telecommunications Congress, WTC 2012*[6170444]

**A relation between group order of elliptic curve and extension degree of definition field.** / Sumo, Taichi; Mori, Yuki; Nogami, Yasuyuki; Matsushima, Tomoko; Uehara, Satoshi.

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

*2012 World Telecommunications Congress, WTC 2012.*, 6170444, 2012 World Telecommunications Congress, WTC 2012, Miyazaki, Japan, 3/5/12.

}

TY - GEN

T1 - A relation between group order of elliptic curve and extension degree of definition field

AU - Sumo, Taichi

AU - Mori, Yuki

AU - Nogami, Yasuyuki

AU - Matsushima, Tomoko

AU - Uehara, Satoshi

PY - 2012

Y1 - 2012

N2 - Recent innovative public key cryptographic applications such as ID-based cryptography are based on pairing cryptography. They efficiently use some torsion group structures constructed on certain elliptic curves defined over finite fields. For this purpose, this paper shows that a relation between group order of elliptic curve and extension degree of definition field especially from the viewpoint of torsion structure for pairing- based cryptographic use. In detail, it is shown that the order of elliptic curve over r i-th extension field denoted by #E(F qri ) is divisible by r 2i and it has the torsion structure denoted by Z ri Z ri when the base order of elliptic curve denoted by #E(Fq) is divisible by r^i and the order of the multiplicative group of the definition field is also divisible by r^i, where r denotes the order of one cyclic group in the torsion structure.

AB - Recent innovative public key cryptographic applications such as ID-based cryptography are based on pairing cryptography. They efficiently use some torsion group structures constructed on certain elliptic curves defined over finite fields. For this purpose, this paper shows that a relation between group order of elliptic curve and extension degree of definition field especially from the viewpoint of torsion structure for pairing- based cryptographic use. In detail, it is shown that the order of elliptic curve over r i-th extension field denoted by #E(F qri ) is divisible by r 2i and it has the torsion structure denoted by Z ri Z ri when the base order of elliptic curve denoted by #E(Fq) is divisible by r^i and the order of the multiplicative group of the definition field is also divisible by r^i, where r denotes the order of one cyclic group in the torsion structure.

UR - http://www.scopus.com/inward/record.url?scp=84860365569&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84860365569&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84860365569

SN - 9784885522574

BT - 2012 World Telecommunications Congress, WTC 2012

ER -