Tate Pairing Implementation for Hyperelliptic y
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, Tate Pairing Implementation
for Hyperelliptic Curves y 2 = xp − x + d
Iwan Duursma1 and Hyang-Sook Lee2,
1
Department of Mathematics, University of Illinois at Urbana-Champaign
Urbana IL 61801, USA
2
Department of Mathematics, Ewha Womans University
Seoul, 120-750, Korea
Abstract. The Weil and Tate pairings have been used recently to build
new schemes in cryptography. It is known that the Weil pairing takes
longer than twice the running time of the Tate pairing. Hence it is neces-
sary to develop more efficient implementations of the Tate pairing for the
practical application of pairing based cryptosystems. In 2002, Barreto et
al. and Galbraith et al. provided new algorithms for the fast computation
of the Tate pairing in characteristic three. In this paper, we give a closed
formula for the Tate pairing on the hyperelliptic curve y 2 = xp − x + d in
characteristic p. This result improves the implementations in [BKLS02],
[GHS02] for the special case p = 3.
1 Introduction
Pairings were first used in cryptography as a cryptanalytic tool for reducing
the discrete log problem on some elliptic curves to the discrete log problem in
a finite field. There are two reduction types. One uses the Weil pairing and
is called the MOV reduction [MOV93], the other uses the Tate pairing and
is called the FR reduction [FR94]. Positive cryptographic applications based
on pairings arose from the work of Joux [J00], who gave a simple one round
tripartite Diffie-Hellman protocol on supersingular curves. Curve based pairings,
such as the Weil pairing and Tate pairing, provide a good setting for the so-
called bilinear Diffie-Hellman problem. Many cryptographic schemes based on
the pairings have been developed recently, such as identity based encryption
[BF01], identity based signature schemes [SOK00], [CC03], [H02a], [P02], and
identity based authenticated key agreement [S02]. For the practical application
of those systems it is important to have efficient implementations of the pairings.
According to [G01], the Tate pairing can be computed more efficiently than the
Weil pairing. The recent papers [BKLS02], [GHS02] provide fast computations
of the Tate pairing in characteristic three.
Our main result in this paper is a closed expression for the Tate pairing on
the hyperelliptic curve defined by the equation C d /k : y 2 = xp − x + d, for a
Supported by Korea Research Foundation Grant (KRF-2002-070-C00010)
C.S. Laih (Ed.): ASIACRYPT 2003, LNCS 2894, pp. 111–123, 2003.
c International Association for Cryptologic Research 2003
2 = x p – x + d 1st edition by Iwan Duursma,
Hyang Sook Lee ISBN 3540205920 9783540205920 pdf
download
https://ebookball.com/product/tate-pairing-implementation-for-
hyperelliptic-y-2-x-p-aeur-x-d-1st-edition-by-iwan-duursma-hyang-
sook-lee-isbn-3540205920-9783540205920-8946/
Explore and download more ebooks or textbooks
at ebookball.com
, Get Your Digital Files Instantly: PDF, ePub, MOBI and More
Quick Digital Downloads: PDF, ePub, MOBI and Other Formats
The AGM-X 0(N) Heegner Point Lifting Algorithm and Elliptic Curve
Point Counting 1st edition by David Kohel ISBN 3540205920
9783540205920
https://ebookball.com/product/the-agm-x-0-n-heegner-point-
lifting-algorithm-and-elliptic-curve-point-counting-1st-edition-
by-david-kohel-isbn-3540205920-9783540205920-11308/
U X L encyclopedia of water science 1st Edition by Lee Lerner, Brenda
Wilmoth Lerner 0787676756 978-0787676759
https://ebookball.com/product/u-x-l-encyclopedia-of-water-
science-1st-edition-by-lee-lerner-brenda-wilmoth-
lerner-0787676756-978-0787676759-19350/
Electricity AC DC Motors Controls and Maintenace 10th Edition by
Jeffrey Keljik 1111646759 9781111646752
https://ebookball.com/product/electricity-ac-dc-motors-controls-
and-maintenace-10th-edition-by-jeffrey-
keljik-1111646759-9781111646752-17638/
3D 2D Image Registration The Impact of X Ray Views and Their Number
1st Edition by Dejan Tomazevic, Bostjan Likar, Franjo Pernus ISBN
9783540757573
https://ebookball.com/product/3d-2d-image-registration-the-
impact-of-x-ray-views-and-their-number-1st-edition-by-dejan-
tomazevic-bostjan-likar-franjo-pernus-isbn-9783540757573-11208/
,Mac OS X Server User Managment 1st edition by Apple Inc
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(Ebook PDF) High Power Converters and AC Drives 2nd edition by Bin Wu,
Mehdi Narimani 1119156068 9781119156062 full chapters
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and-ac-drives-2nd-edition-by-bin-wu-mehdi-
narimani-1119156068-9781119156062-full-chapters-14628/
An Efficient Public Key Trace and Revoke Scheme Secure against
Adaptive Chosen Ciphertext Attack 1st edition by Chong Hee Kim, Yong
Ho Hwang, Pil Joong Lee ISBN 3540205920 9783540205920
https://ebookball.com/product/an-efficient-public-key-trace-and-
revoke-scheme-secure-against-adaptive-chosen-ciphertext-
attack-1st-edition-by-chong-hee-kim-yong-ho-hwang-pil-joong-lee-
isbn-3540205920-9783540205920-11948/
On Diophantine Complexity and Statistical Zero Knowledge Arguments 1st
edition by ISBN Helger Lipmaa 3540205920 9783540205920
https://ebookball.com/product/on-diophantine-complexity-and-
statistical-zero-knowledge-arguments-1st-edition-by-isbn-helger-
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Verifiable Homomorphic Oblivious Transfer and Private Equality Test
1st edition by Helger Lipmaa ISBN 3540205920 9783540205920
https://ebookball.com/product/verifiable-homomorphic-oblivious-
transfer-and-private-equality-test-1st-edition-by-helger-lipmaa-
isbn-3540205920-9783540205920-9424/
, Tate Pairing Implementation
for Hyperelliptic Curves y 2 = xp − x + d
Iwan Duursma1 and Hyang-Sook Lee2,
1
Department of Mathematics, University of Illinois at Urbana-Champaign
Urbana IL 61801, USA
2
Department of Mathematics, Ewha Womans University
Seoul, 120-750, Korea
Abstract. The Weil and Tate pairings have been used recently to build
new schemes in cryptography. It is known that the Weil pairing takes
longer than twice the running time of the Tate pairing. Hence it is neces-
sary to develop more efficient implementations of the Tate pairing for the
practical application of pairing based cryptosystems. In 2002, Barreto et
al. and Galbraith et al. provided new algorithms for the fast computation
of the Tate pairing in characteristic three. In this paper, we give a closed
formula for the Tate pairing on the hyperelliptic curve y 2 = xp − x + d in
characteristic p. This result improves the implementations in [BKLS02],
[GHS02] for the special case p = 3.
1 Introduction
Pairings were first used in cryptography as a cryptanalytic tool for reducing
the discrete log problem on some elliptic curves to the discrete log problem in
a finite field. There are two reduction types. One uses the Weil pairing and
is called the MOV reduction [MOV93], the other uses the Tate pairing and
is called the FR reduction [FR94]. Positive cryptographic applications based
on pairings arose from the work of Joux [J00], who gave a simple one round
tripartite Diffie-Hellman protocol on supersingular curves. Curve based pairings,
such as the Weil pairing and Tate pairing, provide a good setting for the so-
called bilinear Diffie-Hellman problem. Many cryptographic schemes based on
the pairings have been developed recently, such as identity based encryption
[BF01], identity based signature schemes [SOK00], [CC03], [H02a], [P02], and
identity based authenticated key agreement [S02]. For the practical application
of those systems it is important to have efficient implementations of the pairings.
According to [G01], the Tate pairing can be computed more efficiently than the
Weil pairing. The recent papers [BKLS02], [GHS02] provide fast computations
of the Tate pairing in characteristic three.
Our main result in this paper is a closed expression for the Tate pairing on
the hyperelliptic curve defined by the equation C d /k : y 2 = xp − x + d, for a
Supported by Korea Research Foundation Grant (KRF-2002-070-C00010)
C.S. Laih (Ed.): ASIACRYPT 2003, LNCS 2894, pp. 111–123, 2003.
c International Association for Cryptologic Research 2003
