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International Conference on Pairing-Based Cryptography

Pairing 2008: Pairing-Based Cryptography – Pairing 2008 pp 89–99Cite as

The Hidden Root Problem

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Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 5209))

Abstract

In this paper we study a novel computational problem called the Hidden Root Problem, which appears naturally when considering fault attacks on pairing based cryptosystems. Furthermore, a variant of this problem is one of the main obstacles for efficient pairing inversion. We present an algorithm to solve this problem over extension fields and investigate for which parameters the algorithm becomes practical.

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References

  1. Barreto, P.S.L.M., Galbraith, S.D., Ó’hÉigeartaigh, C., Scott, M.: Efficient pairing computation on supersingular abelian varieties. Designs, Codes and Cryptography 42(3), 239–271 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  2. Boneh, D., Venkatesan, R.: Hardness of computing the most significant bits of secret keys in Diffie-Hellman and related schemes. In: Koblitz, N. (ed.) CRYPTO 1996. LNCS, vol. 1109, pp. 129–142. Springer, Heidelberg (1996)

    Google Scholar 

  3. Boneh, D., Venkatesan, R.: Rounding in lattices and its cryptographic applications. In: Proc. 8th Annual ACM-SIAM Symp. on Discr. Algorithms, pp. 675–681. ACM, New York (1997)

    Google Scholar 

  4. Buchberger, B.: A theoretical basis for the reduction of polynomials to canonical forms. SIGSAM Bull 10(3), 19–29 (1976)

    Article  MathSciNet  Google Scholar 

  5. Courtois, N., Klimov, A., Patarin, J., Shamir, A.: Efficient algorithms for solving overdefined systems of multivariate polynomial equations. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 392–407. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  6. ElGamal, T.: A public key cryptosystem and a signature scheme based on discrete logarithms. IEEE Trans. Inform. Theory 31(4), 469–472 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  7. Galbraith, S.D., Hess, F., Vercauteren, F.: Aspects of pairing inversion (preprint 2008), http://eprint.iacr.org/2007/256

  8. Hess, F., Smart, N., Vercauteren, F.: The Eta-pairing revisited. IEEE Transactions on Information Theory 52(10), 4595–4602 (2006)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  10. Page, D., Vercauteren, F.: A Fault Attack on Pairing Based Cryptography. IEEE Transactions on Computers 55(9), 1075–1080 (2006)

    Article  Google Scholar 

  11. Hoffstein, J., Pipher, J., Silverman, J.H.: NTRU: a ring-based public key cryptosystem. In: Buhler, J.P. (ed.) ANTS 1998. LNCS, vol. 1423, pp. 267–288. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  12. Lenstra, A.K., Lenstra, H.W., Lovász, L.: Factoring polynomials with rational coefficients. Mathematische Annalen 261(4), 515–534 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  13. Rivest, R.L., Shamir, A., Adleman, L.: A method for obtaining digital signatures and public-key cryptosystems. Comm. ACM 21(2), 120–126 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  14. Shparlinski, I.E.: Playing ”Hide-and-Seek” in finite fields: Hidden number problem and its applications. In: Proc. 7th Spanish Meeting on Cryptology and Information Security, vol. 1, pp. 49–72. Univ. of Oviedo (2002)

    Google Scholar 

  15. Verheul, E.: Evidence that XTR Is More Secure than Supersingular Elliptic Curve Cryptosystems. In: Pfitzmann, B. (ed.) EUROCRYPT 2001. LNCS, vol. 2045, pp. 195–210. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  16. Verheul, E.: Evidence that XTR is more secure than supersingular elliptic curve cryptosystems. J. Crypt. 17(4), 277–296 (2004)

    MATH  MathSciNet  Google Scholar 

  17. Whelan, C., Scott, M.: The Importance of the Final Exponentiation in Pairings when Considering Fault Attacks. In: Takagi, T., Okamoto, T., Okamoto, E., Okamoto, T. (eds.) Pairing 2007. LNCS, vol. 4575, pp. 225–246. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  18. Whelan, C., Scott, M.: Side Channel Analysis of Practical Pairings: Which Path is more Secure? In: Nguyên, P.Q. (ed.) VIETCRYPT 2006. LNCS, vol. 4341, pp. 99–114. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

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Authors and Affiliations

  1. Department of Electrical Engineering, University of Leuven, Kasteelpark Arenberg 10, B-3001, Leuven-Heverlee, Belgium

    Frederik Vercauteren

Authors
  1. Frederik Vercauteren
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Steven D. Galbraith Kenneth G. Paterson

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© 2008 Springer-Verlag Berlin Heidelberg

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Vercauteren, F. (2008). The Hidden Root Problem. In: Galbraith, S.D., Paterson, K.G. (eds) Pairing-Based Cryptography – Pairing 2008. Pairing 2008. Lecture Notes in Computer Science, vol 5209. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85538-5_6

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  • DOI: https://doi.org/10.1007/978-3-540-85538-5_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-85503-3

  • Online ISBN: 978-3-540-85538-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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