A New Type of Fast Endomorphisms on Jacobians of Hyperelliptic Curves and Their Cryptographic Application

Katsuyuki TAKASHIMA

doi:10.1093/ietfec/e89-a.1.124

A New Type of Fast Endomorphisms on Jacobians of Hyperelliptic Curves and Their Cryptographic Application

Katsuyuki TAKASHIMA

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The Gallant-Lambert-Vanstone method [14](GLV method for short) is a scalar multiplication method for elliptic curve cryptography (ECC). In WAP WTLS [49], SEC 2 [44], ANSI X9.62 [1] and X9.63 [2], several domain parameters for applications of the GLV method are described. Curves with those parameters have efficiently-computable endomorphisms. Recently the GLV method for Jacobians of hyperelliptic curve (HEC) has also been studied. In this paper, we discuss applications of the GLV method to curves with real multiplication (RM). It is the first time to use RM for efficient scalar multiplication as far as we know. We describe the general algorithm for using such RM, and we show that some genus 2 curves with RM have enough effciency to be used in the GLV method as in the previous CM case. Moreover, we will see that such RM curves can be obtained abundantly unlike the previously proposed CM curves of genus 2.

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E89-A No.1 pp.124-133

Publication Date: 2006/01/01

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Online ISSN: 1745-1337

DOI: 10.1093/ietfec/e89-a.1.124

Type of Manuscript: Special Section PAPER (Special Section on Cryptography and Information Security)

Category: Elliptic Curve Cryptography

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Katsuyuki TAKASHIMA, "A New Type of Fast Endomorphisms on Jacobians of Hyperelliptic Curves and Their Cryptographic Application" in IEICE TRANSACTIONS on Fundamentals, vol. E89-A, no. 1, pp. 124-133, January 2006, doi: 10.1093/ietfec/e89-a.1.124.
Abstract: The Gallant-Lambert-Vanstone method [14](GLV method for short) is a scalar multiplication method for elliptic curve cryptography (ECC). In WAP WTLS [49], SEC 2 [44], ANSI X9.62 [1] and X9.63 [2], several domain parameters for applications of the GLV method are described. Curves with those parameters have efficiently-computable endomorphisms. Recently the GLV method for Jacobians of hyperelliptic curve (HEC) has also been studied. In this paper, we discuss applications of the GLV method to curves with real multiplication (RM). It is the first time to use RM for efficient scalar multiplication as far as we know. We describe the general algorithm for using such RM, and we show that some genus 2 curves with RM have enough effciency to be used in the GLV method as in the previous CM case. Moreover, we will see that such RM curves can be obtained abundantly unlike the previously proposed CM curves of genus 2.
URL: https://globals.ieice.org/en_transactions/fundamentals/10.1093/ietfec/e89-a.1.124/_p

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@ARTICLE{e89-a_1_124,
author={Katsuyuki TAKASHIMA, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={A New Type of Fast Endomorphisms on Jacobians of Hyperelliptic Curves and Their Cryptographic Application},
year={2006},
volume={E89-A},
number={1},
pages={124-133},
abstract={The Gallant-Lambert-Vanstone method [14](GLV method for short) is a scalar multiplication method for elliptic curve cryptography (ECC). In WAP WTLS [49], SEC 2 [44], ANSI X9.62 [1] and X9.63 [2], several domain parameters for applications of the GLV method are described. Curves with those parameters have efficiently-computable endomorphisms. Recently the GLV method for Jacobians of hyperelliptic curve (HEC) has also been studied. In this paper, we discuss applications of the GLV method to curves with real multiplication (RM). It is the first time to use RM for efficient scalar multiplication as far as we know. We describe the general algorithm for using such RM, and we show that some genus 2 curves with RM have enough effciency to be used in the GLV method as in the previous CM case. Moreover, we will see that such RM curves can be obtained abundantly unlike the previously proposed CM curves of genus 2.},
keywords={},
doi={10.1093/ietfec/e89-a.1.124},
ISSN={1745-1337},
month={January},}

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TY - JOUR
TI - A New Type of Fast Endomorphisms on Jacobians of Hyperelliptic Curves and Their Cryptographic Application
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 124
EP - 133
AU - Katsuyuki TAKASHIMA
PY - 2006
DO - 10.1093/ietfec/e89-a.1.124
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E89-A
IS - 1
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - January 2006
AB - The Gallant-Lambert-Vanstone method [14](GLV method for short) is a scalar multiplication method for elliptic curve cryptography (ECC). In WAP WTLS [49], SEC 2 [44], ANSI X9.62 [1] and X9.63 [2], several domain parameters for applications of the GLV method are described. Curves with those parameters have efficiently-computable endomorphisms. Recently the GLV method for Jacobians of hyperelliptic curve (HEC) has also been studied. In this paper, we discuss applications of the GLV method to curves with real multiplication (RM). It is the first time to use RM for efficient scalar multiplication as far as we know. We describe the general algorithm for using such RM, and we show that some genus 2 curves with RM have enough effciency to be used in the GLV method as in the previous CM case. Moreover, we will see that such RM curves can be obtained abundantly unlike the previously proposed CM curves of genus 2.
ER -