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International Conference on Information Security and Cryptology

Inscrypt 2013: Information Security and Cryptology pp 185–200Cite as

Pairing Computation on Edwards Curves with High-Degree Twists

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Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 8567))

Abstract

Elliptic curve can be seen as the intersection of two quadratic surfaces in space. In this paper, we used the geometry approach to explain the group law for general elliptic curves given by intersection of two quadratic surfaces, then we construct the Miller function over the intersection of quadratic surfaces. As an example, we obtain the Miller function of Tate pairing computation on twisted Edwards curves. Then we present the explicit formulae for pairing computation on Edwards curves. Our formulae for the doubling step are a littler faster than that proposed by Arène et al.. Moreover, when \(j=1728\) and \(j=0\) we consider quartic and sextic twists to improve the efficiency respectively. Finally, we present the formulae of refinements technique on Edwards curves to obtain gain up when the embedding degree is odd.

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Acknowledgment

This work was supported by National Natural Science Foundation of China (No. 11101002, No. 11271129 and No. 61370187) and Beijing Natural Science Foundation (No. 1132009).

Author information

Authors and Affiliations

  1. LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, China

    Liangze Li & Fan Zhang

  2. College of Sciences, North China University of Technology, Beijing, 100144, China

    Hongfeng Wu

  3. Beijing International Center for Mathematical Research, Beijing, 100871, China

    Liangze Li

Authors
  1. Liangze Li
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  2. Hongfeng Wu
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  3. Fan Zhang
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Corresponding author

Correspondence to Hongfeng Wu .

Editor information

Editors and Affiliations

  1. Chinese Academy of Sciences, Beijing, China

    Dongdai Lin

  2. University of Texas, San Antonio, Texas, USA

    Shouhuai Xu

  3. Columbia University, New York, New York, USA

    Moti Yung

A Examples of Pairing-Friendly Edwards Curves

A Examples of Pairing-Friendly Edwards Curves

We list some pairing friendly Edwards curves with various k=6,12,24. We use construction 6.6 in [10] to present it. \(h=\#S_{1,d}(\mathbb {F}_p)/r\), \(\rho =\log _2(p)/\log _2(r)\).

$$\begin{aligned} k=&6, \rho =1.99, \lceil \log _2(p)\rceil =511, \lceil \log _2(r)\rceil =257, \lceil \log _2(p^k)\rceil =3063,\\ p=&4469269309980865699858008332735282459011729442283504212242920046\\&5254107669101255894363776709837049695943172869161549919107677836\\&20776600027887471085196217,\\ r=&1157920892373161954235709850086879132491274769309617791781887340\\&53461721558841,\\ h=&2^6\cdot 3\cdot 11^4\cdot 31^2\cdot 15659837533^2\cdot 241375889423392081986527^2,\\ d=&3664251552441012307564539365366691396566209647164298880039621750\\&3855065157940074941206695810480629869345608774421066373731513792\\&25747580224215243612885716.\\ \end{aligned}$$
$$\begin{aligned} k=&12, \rho =1.48, \lceil \log _2(p)\rceil =239, \lceil \log _2(r)\rceil =161, \lceil \log _2(p^k)\rceil =2861,\\ p=&5889490310694441330739011548712381814951849552463124431529211730\\&78632117,\\ r=&1461501653010476419563824324075703470606892615001,\\ h=&2^4\cdot 3\cdot 13^2\cdot 19^2\cdot 331^2\cdot 1120711^2,\\ d=&3039686049194322977578848038674418249362581181730689600918590539\\&56432956.\\ \end{aligned}$$
$$\begin{aligned} k=&12, \rho =1.49, \lceil \log _2(p)\rceil =383, \lceil \log _2(r)\rceil =257, \lceil \log _2(p^k)\rceil =4589,\\ p=&1313400206546489077704631059395345592330370814691407061669418717\\&8169845236078372714249135715340284274851981554471437,\\ r=&1157920892373165737821551871767212460418194942614239462794724036\\&61265709211401,\\ h=&2^4\cdot 3^5\cdot 3245503^2\cdot 52627646891^2,\\ d=&2086750387520096896070418610187776681469852959441702575044395173\\&987802972703740715028995508138402551966362217924268.\\ \end{aligned}$$
$$\begin{aligned} k=&24, \rho =1.24, \lceil \log _2(p)\rceil =319, \lceil \log _2(r)\rceil =257, \lceil \log _2(p^k)\rceil =7642,\\ p=&7120003282946788688767832825047892963122039770343506948090350241\\&49143440464464180057177127640101,\\ r=&1157926942199022831048968574721142864333630419694136944823750216\\&16015000100401,\\ h=&2^4\cdot 3^3\cdot 5^4\cdot 17^2\cdot 280717^2,\\ d=&6563654562067688285838956119740898916476600058476145431602456870\\&2651596101614445130173618550273.\\ \end{aligned}$$

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Li, L., Wu, H., Zhang, F. (2014). Pairing Computation on Edwards Curves with High-Degree Twists. In: Lin, D., Xu, S., Yung, M. (eds) Information Security and Cryptology. Inscrypt 2013. Lecture Notes in Computer Science(), vol 8567. Springer, Cham. https://doi.org/10.1007/978-3-319-12087-4_12

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  • DOI: https://doi.org/10.1007/978-3-319-12087-4_12

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  • Publisher Name: Springer, Cham

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  • Online ISBN: 978-3-319-12087-4

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