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Fast Tate Pairing Computation on Twisted Jacobi Intersections Curves

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Information Security and Cryptology (Inscrypt 2011)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 7537))

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Abstract

Recently there are lots of studies on the Tate pairing computation with different coordinate systems, such as twisted Edwards curves and Hessian curves coordinate systems. However, Jacobi intersections curves coordinate system, as another useful one, is overlooked in pairing-based cryptosystems.

This paper proposes the explicit formulae for the doubling and addition steps in Miller’s algorithm to compute the Tate pairing on twisted Jacobi intersections curves, as a larger class containing Jacobi intersections curves. Although these curves are not plane elliptic curves, our formulae are still very efficient and competitive with others. When the embedding degree is even, our doubling formulae are the fastest except for the formulae on Hessian/Selmer curves, and the parallel execution of our formulae are even more competitive with the Selmer curves case in the parallel manner. Besides, we give the detailed analysis of the fast variants of our formulae with other embedding degrees, such as the embedding degree 1, and the embedding degree dividing 4 and 6. At last, we analyze the relation between the Tate pairings on two isogenous elliptic curves, and show that the Tate pairing on twisted Jacobi intersections curves can be substituted for the Tate pairing on twisted Edwards curves completely.

This work was supported by the National 973 Program of China under Grant 2011CB302400, the National Natural Science Foundation of China under Grant 60970152, the Grand Project of Institute of Software under Grant YOCX285056.

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Authors and Affiliations

  1. SKLOIS, Institute of Software, Chinese Academy of Sciences, Beijing, China

    Xusheng Zhang, Shan Chen & Dongdai Lin

  2. Graduate University of Chinese Academy of Sciences, Beijing, China

    Xusheng Zhang & Shan Chen

Authors
  1. Xusheng Zhang
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  2. Shan Chen
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  3. Dongdai Lin
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Editor information

Editors and Affiliations

  1. Chinese Academy of Sciences, Institute of Information Engineering, 100190, Beijing, China

    Chuan-Kun Wu  & Dongdai Lin  & 

  2. Computer Science Department, Google Inc. and Columbia University, S.W. Mudd Building, NY 10027, New York, USA

    Moti Yung

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Zhang, X., Chen, S., Lin, D. (2012). Fast Tate Pairing Computation on Twisted Jacobi Intersections Curves. In: Wu, CK., Yung, M., Lin, D. (eds) Information Security and Cryptology. Inscrypt 2011. Lecture Notes in Computer Science, vol 7537. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34704-7_16

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  • DOI: https://doi.org/10.1007/978-3-642-34704-7_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-34703-0

  • Online ISBN: 978-3-642-34704-7

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