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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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References

  1. Boneh, D., Franklin, M.: Identity-based encryption from the Weil pairing. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 213–229. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  2. Boneh, D., Lynn, B., Shacham, H.: Short signatures from the Weil pairing. J. Crypt 17(4), 297–319 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cassels, J.W.S.: Lectures on Elliptic Curves. LMS Student Texts, Cambridge, vol. 24 (1991)

    Google Scholar 

  4. Demytko, N.: A new elliptic curve based analogue of RSA. In: Helleseth, T. (ed.) EUROCRYPT 1993. LNCS, vol. 765, pp. 40–49. Springer, Heidelberg (1994)

    Google Scholar 

  5. Frey, G., Rück, H.-G.: A remark concerning m-divisibility and the discrete logarithm problem in the divisor class group of curves. Math. Comp. 52, 865–874 (1994)

    Article  Google Scholar 

  6. Frey, G., Müller, M., Rück, H.-G.: The Tate pairing and the discrete logarithm applied to elliptic curve cryptosystems. IEEE Trans. Inf. Th. 45, 1717–1719 (1999)

    Article  MATH  Google Scholar 

  7. Galbraith, S.D.: Elliptic curve Paillier schemes. J. Crypt. 15(2), 129–138 (2002)

    MATH  MathSciNet  Google Scholar 

  8. Galbraith, S.D., Harrison, K., Soldera, D.: Implementing the Tate pairing. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 324–337. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  9. Girault, M.: An Identity-Based Identification Scheme Based on Discrete Logarithms Modulo a Composite Number. In: Damgård, I.B. (ed.) EUROCRYPT 1990. LNCS, vol. 473, pp. 481–486. Springer, Heidelberg (1991)

    Google Scholar 

  10. Joux, A.: A One Round Protocol for Tripartite Diffie-Hellman. In: Bosma, W. (ed.) ANTS 2000. LNCS, vol. 1838, pp. 385–394. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  11. Kunihiro, N., Koyama, K.: Equivalence of counting the number of points on elliptic curve over the ring Z_n and factoring n. In: Nyberg, K. (ed.) EUROCRYPT 1998. LNCS, vol. 1403, pp. 47–58. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  12. Koyama, K., Maurer, U.M., Okamoto, T., Vanstone, S.A.: New public-key schemes based on elliptic curves over the ring Z n . In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 252–266. Springer, Heidelberg (1992)

    Google Scholar 

  13. Lenstra Jr., H.W.: Factoring integers with elliptic curves. Annals of Mathematics 126, 649–673 (1987)

    Article  MathSciNet  Google Scholar 

  14. Lenstra Jr., H.W.: Elliptic curves and number theoretic algorithms. In: Proc. International Congr. Math., pp. 99–120. AMS, Berkeley (1986/1988)

    Google Scholar 

  15. Lim, C.H., Lee, P.J.: Security and performance of server-aided RSA computation protocols. In: Coppersmith, D. (ed.) CRYPTO 1995. LNCS, vol. 963, pp. 70–83. Springer, Heidelberg (1995)

    Google Scholar 

  16. Mao, W.: Verifiable partial sharing of integer factors. In: Tavares, S., Meijer, H. (eds.) SAC 1998. LNCS, vol. 1556, pp. 94–105. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  17. Martin, S., Morillo, P., Villar, J.L.: Computing the order of points on an elliptic curve modulo N is as difficult as factoring N. Applied Math. Letters 14, 341–346 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  18. McKee, J.F.: Subtleties in the distribution of the numbers of points on elliptic curves over a finite prime field. J. London Math. Soc. (2) 59, 448–460 (1999)

    Google Scholar 

  19. McKee, J.F., Pinch, R.G.E.: Old and new deterministic factoring algorithms. In: Cohen, H. (ed.) ANTS 1996. LNCS, vol. 1122, pp. 217–224. Springer, Heidelberg (1996)

    Google Scholar 

  20. McKee, J.F., Pinch, R.G.E.: Further attacks on server-aided RSA cryptosystems (1998) (unpublished manuscript)

    Google Scholar 

  21. Meyer, B., Mueller, V.: A public key cryptosystem based on elliptic curves over Z /n Z equivalent to factoring. In: Maurer, U.M. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 49–59. Springer, Heidelberg (1996)

    Google Scholar 

  22. Miller, V.S.: Short programs for functions on curves (1986) (unpublished manuscript)

    Google Scholar 

  23. Miller, V.S.: The Weil pairing, and its efficient calculation. J. Crypt. 17(4), 235–261 (2004)

    Article  MATH  Google Scholar 

  24. Okamoto, T., Uchiyama, S.: Security of an identity-based cryptosystem and the related reductions. In: Nyberg, K. (ed.) EUROCRYPT 1998. LNCS, vol. 1403, pp. 546–560. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  25. Pollard, J.M.: A Monte Carlo method for factorisation. BIT 15, 331–334 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  26. Pollard, J.M.: Monte Carlo methods for index computations (mod p). Math. Comp. 32, 918–924 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  27. Rabin, M.O.: Digitalized signatures and public-key functions as intractable as factorization, Technical report TR-212. MIT Laboratory for Computer Science (1979)

    Google Scholar 

  28. Scott, M., Barreto, P.S.L.M.: Compressed pairings. In: Franklin, M. (ed.) CRYPTO 2004. LNCS, vol. 3152, pp. 140–156. Springer, Heidelberg (2004)

    Google Scholar 

  29. Shanks, D.: Class number, a theory of factorisation and genera. In: Lewis, D.J. (ed.) Number theory institute 1969, Proceedings of symposia in pure mathematics, Providence RI, vol. 20, pp. 415–440. AMS (1971)

    Google Scholar 

  30. Turk, J.W.M.: Fast arithmetic operations on numbers and polynomials. In: Lenstra Jr., H.W., Tijdeman, R. (eds.) Computational methods in number theory, Part 1, Mathematical Centre Tracts, vol. 154, Amsterdam (1984)

    Google Scholar 

  31. Vanstone, S.A., Zuccherato, R.J.: Elliptic curve cryptosystems using curves of smooth order over the ring Z n . IEEE Trans. Inform. Theory 43(4), 1231–1237 (1997)

    Article  MATH  MathSciNet  Google Scholar 

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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

  • eBook Packages: Computer ScienceComputer Science (R0)

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