Efficiency Preserving Transformations for Concurrent Non-malleable Zero Knowledge

  • Rafail Ostrovsky
  • Omkant Pandey
  • Ivan Visconti
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5978)

Abstract

Ever since the invention of Zero-Knowledge by Goldwasser, Micali, and Rackoff [1], Zero-Knowledge has become a central building block in cryptography - with numerous applications, ranging from electronic cash to digital signatures. The properties of Zero-Knowledge range from the most simple (and not particularly useful in practice) requirements, such as honest-verifier zero-knowledge to the most demanding (and most useful in applications) such as non-malleable and concurrent zero-knowledge. In this paper, we study the complexity of efficient zero-knowledge reductions, from the first type to the second type. More precisely, under a standard complexity assumption (ddh), on input a public-coin honest-verifier statistical zero knowledge argument of knowledge π′ for a language L we show a compiler that produces an argument system π for L that is concurrent non-malleable zero-knowledge (under non-adaptive inputs – which is the best one can hope to achieve [2,3]). If κ is the security parameter, the overhead of our compiler is as follows:

  • The round complexity of π is \(r+\tilde{O}(\log\kappa)\) rounds, where r is the round complexity of π′.

  • The new prover \(\mathcal{P}\) (resp., the new verifier \(\mathcal{V}\)) incurs an additional overhead of (at most) \(r+{\kappa\cdot\tilde{O}(\log^2\kappa)}\) modular exponentiations. If tags of length \(\tilde{O}(\log\kappa)\) are provided, the overhead is only \(r+{\tilde{O}(\log^2\kappa)}\) modular exponentiations.

The only previous concurrent non-malleable zero-knowledge (under non-adaptive inputs) was achieved by Barak, Prabhakaran and Sahai [4]. Their construction, however, mainly focuses on a feasibility result rather than efficiency, and requires expensive \({\mathcal{NP}}\)-reductions.

References

  1. 1.
    Goldwasser, S., Micali, S., Rackoff, C.: The knowledge complexity of interactive proof-systems. In: Proc. 17th STOC, pp. 291–304 (1985)Google Scholar
  2. 2.
    Lindell, Y.: General composition and universal composability in secure multi-party computation. In: Proc. 44th FOCS, pp. 394–403 (2003)Google Scholar
  3. 3.
    Lindell, Y.: Lower bounds for concurrent self composition. In: Naor, M. (ed.) TCC 2004. LNCS, vol. 2951, pp. 203–222. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  4. 4.
    Barak, B., Prabhakaran, M., Sahai, A.: Concurrent non-malleable zero knowledge. In: FOCS 2006 (2006); Full version on Cryptology ePrint Archive report, http://eprint.iacr.org/
  5. 5.
    Dolev, D., Dwork, C., Naor, M.: Nonmalleable cryptography. SIAM Journal on Computing 30(2), 391–437 (2000); (electronic) Preliminary version in STOC 1991 (1991)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Garay, J.A., MacKenzie, P.D., Yang, K.: Strengthening zero-knowledge protocols using signatures. In: Biham, E. (ed.) EUROCRYPT 2003. LNCS, vol. 2656, pp. 177–194. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  7. 7.
    MacKenzie, P., Yang, K.: On Simulation-Sound Trapdoor Commitments. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 382–400. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  8. 8.
    Gennaro, R.: Multi-trapdoor Commitments and Their Applications to Proof s of Knowledge Secure Under Concurrent Man-in-the-Middle Attacks. In: Franklin, M. (ed.) CRYPTO 2004. LNCS, vol. 3152, pp. 220–236. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  9. 9.
    Damgård, I., Nielsen, J.B., Orlandi, C.: On the necessary and sufficient assumptions for uc computation. In: Micciancio, D. (ed.) TCC 2010. LNCS, vol. 5978. Springer, Heidelberg (2010)Google Scholar
  10. 10.
    Pedersen, T.P.: Non-interactive and information-theoretic secure verifiable secret sharing. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 129–140. Springer, Heidelberg (1992)Google Scholar
  11. 11.
    Canetti, R., Goldreich, O., Goldwasser, S., Micali, S.: Resettable zero-knowledge. In: Proc. 32th STOC, pp. 235–244 (2000)Google Scholar
  12. 12.
    Micciancio, D., Petrank, E.: Simulatable commitments and efficient concurrent zero-knowledge. In: Biham, E. (ed.) EUROCRYPT 2003. LNCS, vol. 2656, pp. 140–159. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  13. 13.
    Lin, H., Pass, R., Venkitasubramaniam, M.: Concurrent non-malleable commitments from any one-way function. In: Canetti, R. (ed.) TCC 2008. LNCS, vol. 4948, pp. 571–588. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  14. 14.
    Pass, R., Rosen, A.: New and improved constructions of non-malleable cryptographic protocols. In: Proc. 37th STOC (2005)Google Scholar
  15. 15.
    De Santis, A., Di Crescenzo, G., Ostrovsky, R., Persiano, G., Sahai, A.: Robust non-interactive zero knowledge. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 566–598. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  16. 16.
    Mohassel, P., Franklin, M.K.: Efficiency tradeoffs for malicious two-party computation. In: Yung, M., Dodis, Y., Kiayias, A., Malkin, T.G. (eds.) PKC 2006. LNCS, vol. 3958, pp. 458–473. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  17. 17.
    Woodruff, D.P.: Revisiting the efficiency of malicious two-party computation. In: Naor, M. (ed.) EUROCRYPT 2007. LNCS, vol. 4515, pp. 79–96. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  18. 18.
    Lindell, Y., Pinkas, B.: An efficient protocol for secure two-party computation in the presence of malicious adversaries. In: Naor, M. (ed.) EUROCRYPT 2007. LNCS, vol. 4515, pp. 52–78. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  19. 19.
    Goyal, V., Mohassel, P., Smith, A.: Efficient two party and multi party computation against covert adversaries. In: Smart, N.P. (ed.) EUROCRYPT 2008. LNCS, vol. 4965, pp. 289–306. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  20. 20.
    Chase, M., Lysyanskaya, A.: Simulatable vrfs with applications to multi-theorem nizk. In: Menezes, A. (ed.) CRYPTO 2007. LNCS, vol. 4622, pp. 303–322. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  21. 21.
    Groth, J., Ostrovsky, R., Sahai, A.: Perfect non-interactive zero knowledge for NP. In: Vaudenay, S. (ed.) EUROCRYPT 2006. LNCS, vol. 4004, pp. 339–358. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  22. 22.
    Goldreich, O.: Foundations of Cryptography: Basic Tools. Cambridge University Press, Cambridge (2001)CrossRefMATHGoogle Scholar
  23. 23.
    Schnorr, C.P.: Efficient identification and signatures for smart cards (abstract). In: Quisquater, J.-J., Vandewalle, J. (eds.) EUROCRYPT 1989. LNCS, vol. 434, pp. 688–689. Springer, Heidelberg (1990)CrossRefGoogle Scholar
  24. 24.
    Richardson, R., Kilian, J.: On the concurrent composition of zero-knowledge proofs. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 415–432. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  25. 25.
    Prabhakaran, M., Rosen, A., Sahai, A.: Concurrent zero knowledge with logarithmic round-complexity. In: FOCS, pp. 366–375 (2002)Google Scholar
  26. 26.
    Di Crescenzo, G., Persiano, G., Visconti, I.: Constant-round resettable zero knowledge with concurrent soundness in the bare public-key model. In: Franklin, M. (ed.) CRYPTO 2004. LNCS, vol. 3152, pp. 237–253. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  27. 27.
    Di Crescenzo, G., Visconti, I.: Concurrent zero knowledge in the public-key model. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 816–827. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  28. 28.
    Visconti, I.: Efficient zero knowledge on the internet. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4052, pp. 22–33. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  29. 29.
    Ostrovsky, R., Persiano, G., Visconti, I.: Constant-round concurrent nmwi and its relation to nmzk. Technical Report ECCC Report TR06-095, ECCC (2006)Google Scholar
  30. 30.
    Ostrovsky, R., Persiano, G., Visconti, I.: Constant-round concurrent non-malleable zero knowledge in the bare public-key model. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part II. LNCS, vol. 5126, pp. 548–559. Springer, Heidelberg (2008)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Rafail Ostrovsky
    • 1
  • Omkant Pandey
    • 1
  • Ivan Visconti
    • 2
  1. 1.University of CaliforniaLos AngelesUSA
  2. 2.University of SalernoItaly

Personalised recommendations