IEICE TRANSACTIONS on Fundamentals
Construction of Pairing-Friendly Hyperelliptic Curves Based on the Closed Formulae of the Order of the Jacobian Group
Summary :
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 y2=x5+c. By using the formula, we develop an analogue of the Cocks-Pinch method for curves of type y2=x5+c. For the case of genus four, we also develop an analogue of the Cocks-Pinch method for curves of type y2=x9+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 y2=x5+c. By using the formula, we develop an analogue of the Cocks-Pinch method for curves of type y2=x5+c. For the case of genus four, we also develop an analogue of the Cocks-Pinch method for curves of type y2=x9+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 y2=x5+c. By using the formula, we develop an analogue of the Cocks-Pinch method for curves of type y2=x5+c. For the case of genus four, we also develop an analogue of the Cocks-Pinch method for curves of type y2=x9+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},}
Copy
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 y2=x5+c. By using the formula, we develop an analogue of the Cocks-Pinch method for curves of type y2=x5+c. For the case of genus four, we also develop an analogue of the Cocks-Pinch method for curves of type y2=x9+cx. In particular, we construct the first examples of pairing-friendly hyperelliptic curves of genus four with ordinary Jacobians.
ER -
English
