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An explicit construction of pairing-friendly hyperelliptic curves with ordinary Jacobians was firstly given by D. Freeman for the genus two case. In this paper, we give an explicit construction of pairing-friendly hyperelliptic curves of genus two and four with ordinary Jacobians based on the closed formulae for the order of the Jacobian of special hyperelliptic curves. For the case of genus two, we prove the closed formula for curves of type *y*^{2}=*x*^{5}+*c*. By using the formula, we develop an analogue of the Cocks-Pinch method for curves of type *y*^{2}=*x*^{5}+*c*. For the case of genus four, we also develop an analogue of the Cocks-Pinch method for curves of type *y*^{2}=*x*^{9}+*cx*. In particular, we construct the first examples of pairing-friendly hyperelliptic curves of genus four with ordinary Jacobians.

- Publication
- IEICE TRANSACTIONS on Fundamentals Vol.E93-A No.6 pp.1132-1139

- Publication Date
- 2010/06/01

- Publicized

- Online ISSN
- 1745-1337

- DOI
- 10.1587/transfun.E93.A.1132

- Type of Manuscript
- Special Section PAPER (Special Section on Discrete Mathematics and Its Applications)

- Category
- Cryptography and Information Security

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Aya COMUTA, Mitsuru KAWAZOE, Tetsuya TAKAHASHI, Isamu YOSHIZAWA, "Construction of Pairing-Friendly Hyperelliptic Curves Based on the Closed Formulae of the Order of the Jacobian Group" in IEICE TRANSACTIONS on Fundamentals,
vol. E93-A, no. 6, pp. 1132-1139, June 2010, doi: 10.1587/transfun.E93.A.1132.

Abstract: An explicit construction of pairing-friendly hyperelliptic curves with ordinary Jacobians was firstly given by D. Freeman for the genus two case. In this paper, we give an explicit construction of pairing-friendly hyperelliptic curves of genus two and four with ordinary Jacobians based on the closed formulae for the order of the Jacobian of special hyperelliptic curves. For the case of genus two, we prove the closed formula for curves of type *y*^{2}=*x*^{5}+*c*. By using the formula, we develop an analogue of the Cocks-Pinch method for curves of type *y*^{2}=*x*^{5}+*c*. For the case of genus four, we also develop an analogue of the Cocks-Pinch method for curves of type *y*^{2}=*x*^{9}+*cx*. In particular, we construct the first examples of pairing-friendly hyperelliptic curves of genus four with ordinary Jacobians.

URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.E93.A.1132/_p

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@ARTICLE{e93-a_6_1132,

author={Aya COMUTA, Mitsuru KAWAZOE, Tetsuya TAKAHASHI, Isamu YOSHIZAWA, },

journal={IEICE TRANSACTIONS on Fundamentals},

title={Construction of Pairing-Friendly Hyperelliptic Curves Based on the Closed Formulae of the Order of the Jacobian Group},

year={2010},

volume={E93-A},

number={6},

pages={1132-1139},

abstract={An explicit construction of pairing-friendly hyperelliptic curves with ordinary Jacobians was firstly given by D. Freeman for the genus two case. In this paper, we give an explicit construction of pairing-friendly hyperelliptic curves of genus two and four with ordinary Jacobians based on the closed formulae for the order of the Jacobian of special hyperelliptic curves. For the case of genus two, we prove the closed formula for curves of type *y*^{2}=*x*^{5}+*c*. By using the formula, we develop an analogue of the Cocks-Pinch method for curves of type *y*^{2}=*x*^{5}+*c*. For the case of genus four, we also develop an analogue of the Cocks-Pinch method for curves of type *y*^{2}=*x*^{9}+*cx*. In particular, we construct the first examples of pairing-friendly hyperelliptic curves of genus four with ordinary Jacobians.},

keywords={},

doi={10.1587/transfun.E93.A.1132},

ISSN={1745-1337},

month={June},}

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TY - JOUR

TI - Construction of Pairing-Friendly Hyperelliptic Curves Based on the Closed Formulae of the Order of the Jacobian Group

T2 - IEICE TRANSACTIONS on Fundamentals

SP - 1132

EP - 1139

AU - Aya COMUTA

AU - Mitsuru KAWAZOE

AU - Tetsuya TAKAHASHI

AU - Isamu YOSHIZAWA

PY - 2010

DO - 10.1587/transfun.E93.A.1132

JO - IEICE TRANSACTIONS on Fundamentals

SN - 1745-1337

VL - E93-A

IS - 6

JA - IEICE TRANSACTIONS on Fundamentals

Y1 - June 2010

AB - An explicit construction of pairing-friendly hyperelliptic curves with ordinary Jacobians was firstly given by D. Freeman for the genus two case. In this paper, we give an explicit construction of pairing-friendly hyperelliptic curves of genus two and four with ordinary Jacobians based on the closed formulae for the order of the Jacobian of special hyperelliptic curves. For the case of genus two, we prove the closed formula for curves of type *y*^{2}=*x*^{5}+*c*. By using the formula, we develop an analogue of the Cocks-Pinch method for curves of type *y*^{2}=*x*^{5}+*c*. For the case of genus four, we also develop an analogue of the Cocks-Pinch method for curves of type *y*^{2}=*x*^{9}+*cx*. In particular, we construct the first examples of pairing-friendly hyperelliptic curves of genus four with ordinary Jacobians.

ER -