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Efficient Binary Conversion for Paillier Encrypted Values

  • Berry Schoenmakers
  • Pim Tuyls
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4004)

Abstract

We consider the framework of secure n-party computation based on threshold homomorphic cryptosystems as put forth by Cramer, Damgård, and Nielsen at Eurocrypt 2001. When used with Paillier’s cryptosystem, this framework allows for efficient secure evaluation of any arithmetic circuit defined over ℤ N , where N is the RSA modulus of the underlying Paillier cryptosystem.

In this paper, we extend the scope of the framework by considering the problem of converting a given Paillier encryption of a value x ∈ ℤ N into Paillier encryptions of the bits of x. We present solutions for the general case in which x can be any integer in {0,1,...,N – 1}, and for the restricted case in which x < N/(n2 κ ) for a security parameter κ. In the latter case, we show how to extract the ℓ least significant bits of x (in encrypted form) in time proportional to ℓ, typically saving a factor of log2 N /ℓ compared to the general case.

Thus, intermediate computations that rely in an essential way on the binary representations of their input values can be handled without enforcing that the entire computation is done bitwise. Typical examples involve the relational operators such as < and =. As a specific scenario we will consider the setting for (approximate) matching of biometric templates, given as bit strings.

Keywords

Security Parameter Security Proof Homomorphic Encryption Random Oracle Model Biometric Authentication 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. [ACS02]
    Algesheimer, J., Camenisch, J., Shoup, V.: Efficient computation modulo a shared secret with application to the generation of shared safe-prime products. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 417–432. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  2. [Bou00]
    Boudot, F.: Efficient proofs that a committed number lies in an interval. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 431–444. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  3. [CDN01]
    Cramer, R., Damgård, I., Nielsen, J.B.: Multiparty computation from threshold homomorphic encryption. In: Pfitzmann, B. (ed.) EUROCRYPT 2001. LNCS, vol. 2045, pp. 280–300. Springer, Heidelberg (2001), Full version, eprint.iacr.org/2000/055 CrossRefGoogle Scholar
  4. [CFSY96]
    Cramer, R., Franklin, M., Schoenmakers, B., Yung, M.: Multi-authority secret ballot elections with linear work. In: Maurer, U.M. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 72–83. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  5. [CS02]
    Cramer, R., Shoup, V.: Universal hash proofs and a paradigm for adaptive chosen ciphertext secure public-key encryption. In: Knudsen, L.R. (ed.) EUROCRYPT 2002. LNCS, vol. 2332, pp. 45–64. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  6. [DF02]
    Damgård, I., Fujisaki, E.: A statistically-hiding integer commitment scheme based on groups with hidden order. In: Zheng, Y. (ed.) ASIACRYPT 2002. LNCS, vol. 2501, pp. 125–142. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  7. [DFK+06]
    Damgård, I., Fitzi, M., Kiltz, E., Nielsen, J.B., Toft, T.: Unconditionally secure constant-rounds multi-party computation for equality, comparison, bits and exponentiation. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 285–304. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  8. [DJ01]
    Damgård, I., Jurik, M.: A generalisation, a simplification and some applications of Paillier’s probabilistic public-key system. In: Kim, K.-c. (ed.) PKC 2001. LNCS, vol. 1992, pp. 119–136. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  9. [DJ02]
    Damgård, I., Jurik, M.: Client/server tradeoffs for online elections. In: Naccache, D., Paillier, P. (eds.) PKC 2002. LNCS, vol. 2274, pp. 125–140. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  10. [DJ03]
    Damgård, I., Jurik, M.: A length-flexible threshold cryptosystem with applications. In: Safavi-Naini, R., Seberry, J. (eds.) ACISP 2003. LNCS, vol. 2727, pp. 350–364. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  11. [DRS04]
    Dodis, Y., Reyzin, M., Smith, A.: Fuzzy extractors: How to generate strong keys from biometrics and other noisy data. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 523–540. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  12. [FO97]
    Fujisaki, E., Okamoto, T.: Statistical zero knowledge protocols to prove modular polynomial relations. In: Kaliski Jr., B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 16–30. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  13. [FPS00]
    Fouque, P.-A., Poupard, G., Stern, J.: Sharing decryption in the context of voting or lotteries. In: Frankel, Y. (ed.) FC 2000. LNCS, vol. 1962, pp. 90–104. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  14. [GMPY06]
    Garay, J., MacKenzie, P., Prabhakaran, M., Yang, K.: Resource fairness and composability of cryptographic protocols. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 404–428. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  15. [KAMR04]
    Kerschbaum, F., Atallah, M.J., M’Raïhi, D., Rice, J.R.: Private fingerprint verification without local storage. In: Zhang, D., Jain, A.K. (eds.) ICBA 2004. LNCS, vol. 3072, pp. 387–394. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  16. [Lip03]
    Lipmaa, H.: On diophantine complexity and statistical zero-knowledge arguments. In: Laih, C.-S. (ed.) ASIACRYPT 2003. LNCS, vol. 2894, pp. 398–415. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  17. [Pai99]
    Paillier, P.: Public-key cryptosystems based on composite degree residuosity classes. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 223–238. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  18. [ST04]
    Schoenmakers, B., Tuyls, P.: Practical two-party computation based on the conditional gate. In: Lee, P.J. (ed.) ASIACRYPT 2004. LNCS, vol. 3329, pp. 119–136. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  19. [TG04]
    Tuyls, P., Goseling, J.: Capacity and examples of template protecting biometric authentication systems. In: Maltoni, D., Jain, A.K. (eds.) BioAW 2004. LNCS, vol. 3087, pp. 158–170. Springer, Heidelberg (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Berry Schoenmakers
    • 1
  • Pim Tuyls
    • 2
  1. 1.Dept. of Mathematics and Computing ScienceTU EindhovenEindhovenThe Netherlands
  2. 2.Philips Research Labs, Prof.EindhovenThe Netherlands

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