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Pairings on Elliptic Curves over Finite Commutative Rings

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Cryptography and Coding (Cryptography and Coding 2005)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 3796))

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Abstract

The Weil and Tate pairings are defined for elliptic curves over fields, including finite fields. These definitions extend naturally to elliptic curves over ℤ/Nℤ, for any positive integer N, or more generally to elliptic curves over any finite commutative ring, and even the reduced Tate pairing makes sense in this more general setting.

This paper discusses a number of issues which arise if one tries to develop pairing-based cryptosystems on elliptic curves over such rings. We argue that, although it may be possible to develop some cryptosystems in this setting, there are obstacles in adapting many of the main ideas in pairing-based cryptography to elliptic curves over rings.

Our main results are: (i) an oracle that computes reduced Tate pairings over such rings (or even just over ℤ/Nℤ) can be used to factorise integers; and (ii) an oracle that determines whether or not the reduced Tate pairing of two points is trivial can be used to solve the quadratic residuosity problem.

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Authors and Affiliations

  1. Department of Mathematics, Royal Holloway, University of London, Egham, Surrey, TW20 0EX, UK

    Steven D. Galbraith & James F. McKee

Authors
  1. Steven D. Galbraith
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  2. James F. McKee
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Editors and Affiliations

  1. Dept. Computer Science, University of Bristol, Woodland Road, BS8 1UB, Bristol, United Kingdom

    Nigel P. Smart

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Galbraith, S.D., McKee, J.F. (2005). Pairings on Elliptic Curves over Finite Commutative Rings. In: Smart, N.P. (eds) Cryptography and Coding. Cryptography and Coding 2005. Lecture Notes in Computer Science, vol 3796. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11586821_26

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  • DOI: https://doi.org/10.1007/11586821_26

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-30276-6

  • Online ISBN: 978-3-540-32418-8

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