Advertisement

Online/Offline OR Composition of Sigma Protocols

  • Michele CiampiEmail author
  • Giuseppe Persiano
  • Alessandra Scafuro
  • Luisa Siniscalchi
  • Ivan Visconti
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9666)

Abstract

Proofs of partial knowledge allow a prover to prove knowledge of witnesses for k out of n instances of NP languages. Cramer, Schoenmakers and Damgård [10] provided an efficient construction of a 3-round public-coin witness-indistinguishable (kn)-proof of partial knowledge for any NP language, by cleverly combining n executions of \(\varSigma \)-protocols for that language. This transform assumes that all n instances are fully specified before the proof starts, and thus directly rules out the possibility of choosing some of the instances after the first round.

Very recently, Ciampi et al. [6] provided an improved transform where one of the instances can be specified in the last round. They focus on (1, 2)-proofs of partial knowledge with the additional feature that one instance is defined in the last round, and could be adaptively chosen by the verifier. They left as an open question the existence of an efficient (1, 2)-proof of partial knowledge where no instance is known in the first round. More in general, they left open the question of constructing an efficient (kn)-proof of partial knowledge where knowledge of all n instances can be postponed. Indeed, this property is achieved only by inefficient constructions requiring NP reductions [19].

In this paper we focus on the question of achieving adaptive-input proofs of partial knowledge. We provide through a transform the first efficient construction of a 3-round public-coin witness-indistinguishable (kn)-proof of partial knowledge where all instances can be decided in the third round. Our construction enjoys adaptive-input witness indistinguishability. Additionally, the proof of knowledge property remains also if the adversarial prover selects instances adaptively at last round as long as our transform is applied to a proof of knowledge belonging to the widely used class of proofs of knowledge described in [9, 21]. Since knowledge of instances and witnesses is not needed before the last round, we have that the first round can be precomputed and in the online/offline setting our performance is similar to the one of [10].

Our new transform relies on the DDH assumption (in contrast to the transforms of [6, 10] that are unconditional).

Keywords

\(\varSigma \)-protocols WI PoKs Delayed and adaptive input 

Notes

Acknowledgments

We thank the anonymous reviewers of Eurocrypt 2016 for many insightful comments and suggestions. This work has been supported in part by “GNCS - INdAM” and in part by the EU COST Action IC1306.

References

  1. 1.
    Bernhard, D., Pereira, O., Warinschi, B.: How not to prove yourself: pitfalls of the fiat-shamir heuristic and applications to helios. In: Proceedings of the Advances in Cryptology - ASIACRYPT 2012–18th International Conference on the Theory and Application of Cryptology and Information Security, Beijing, China, 2–6 December 2012, pp. 626–643 (2012)Google Scholar
  2. 2.
    Blundo, C., Persiano, G., Sadeghi, A.-R., Visconti, I.: Improved security notions and protocols for non-transferable identification. In: Jajodia, S., Lopez, J. (eds.) ESORICS 2008. LNCS, vol. 5283, pp. 364–378. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  3. 3.
    Catalano, D., Dodis, Y., Visconti, I.: Mercurial commitments: minimal assumptions and efficient constructions. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 120–144. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. 4.
    Catalano, D., Visconti, I.: Hybrid trapdoor commitments and their applications. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 298–310. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  5. 5.
    Catalano, D., Visconti, I.: Hybrid commitments and their applications to zero-knowledge proof systems. Theor. Comput. Sci. 374(1–3), 229–260 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ciampi, M., Persiano, G., Scafuro, A., Siniscalchi, L., Visconti, I.: Improved OR-composition of sigma-protocols. In: Kushilevitz, E., Malkin, T. (eds.) TCC 2016-A. LNCS, vol. 9563, pp. 112–141. Springer, Heidelberg (2016)CrossRefGoogle Scholar
  7. 7.
    Ciampi, M., Persiano, G., Scafuro, A., Siniscalchi, L., Visconti, I.: Online/offline OR composition of sigma protocols. Cryptology ePrint Archive, Report 2016/175 (2016). http://eprint.iacr.org/
  8. 8.
    Ciampi, M., Persiano, G., Siniscalchi, L., Visconti, I.: A transform for NIZK almost as efficient and general as the fiat-shamir transform without programmable random oracles. In: Kushilevitz, E., Malkin, T. (eds.) TCC 2016-A. LNCS, vol. 9563, pp. 83–111. Springer, Heidelberg (2016)CrossRefGoogle Scholar
  9. 9.
    Cramer, R., Damgård, I.B.: Zero-knowledge proofs for finite field arithmetic or: can zero-knowledge be for free? In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 424–441. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  10. 10.
    Cramer, R., Damgård, I.B., Schoenmakers, B.: Proof of partial knowledge and simplified design of witness hiding protocols. In: Desmedt, Y.G. (ed.) CRYPTO 1994. LNCS, vol. 839, pp. 174–187. Springer, Heidelberg (1994)Google Scholar
  11. 11.
    Damgård, I.: On \(\Sigma \)-protocol (2010). http://www.cs.au.dk/~ivan/Sigma.pdf
  12. 12.
    Damgård, I., Groth, J.: Non-interactive and reusable non-malleable commitment schemes. In: Proceedings of the 35th Annual ACM Symposium on Theory of Computing, June 9–11, 2003, San Diego, CA, USA, pp. 426–437 (2003)Google Scholar
  13. 13.
    Damgård, I.B., Nielsen, J.B.: Perfect hiding and perfect binding universally composable commitment schemes with constant expansion factor. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 581–596. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  14. 14.
    Fiat, A., Shamir, A.: How to prove yourself: practical solutions to identification and signature problems. In: Odlyzko, A.M. (ed.) CRYPTO 1986. LNCS, vol. 263, pp. 186–194. Springer, Heidelberg (1987)Google Scholar
  15. 15.
    Garay, J.A., MacKenzie, P., Yang, K.: Strengthening zero-knowledge protocols using signatures. J. Cryptology 19(2), 169–209 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Guillou, L.C., Quisquater, J.-J.: A practical zero-knowledge protocol fitted to security microprocessor minimizing both transmission and memory. In: Günther, C.G. (ed.) EUROCRYPT 1988. LNCS, vol. 330, pp. 123–128. Springer, Heidelberg (1988)Google Scholar
  17. 17.
    Hazay, C., Lindell, Y.: Efficient Secure Two-Party Protocols - Techniques and Constructions. Information Security and Cryptography. Springer, Heidelberg (2010)CrossRefzbMATHGoogle Scholar
  18. 18.
    Katz, J., Ostrovsky, R.: Round-optimal secure two-party computation. In: Franklin, M. (ed.) CRYPTO 2004. LNCS, vol. 3152, pp. 335–354. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  19. 19.
    Lapidot, D., Shamir, A.: Publicly verifiable non-interactive zero-knowledge proofs. In: Menezes, A., Vanstone, S.A. (eds.) CRYPTO 1990. LNCS, vol. 537, pp. 353–365. Springer, Heidelberg (1991)Google Scholar
  20. 20.
    Lindell, Y.: An efficient transform from sigma protocols to NIZK with a CRS and non-programmable random oracle. In: Dodis, Y., Nielsen, J.B. (eds.) TCC 2015, Part I. LNCS, vol. 9014, pp. 93–109. Springer, Heidelberg (2015)Google Scholar
  21. 21.
    Maurer, U.: Zero-knowledge proofs of knowledge for group homomorphisms. Des. Codes Crypt. 77(2), 1–14 (2015)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Micciancio, D., Petrank, E.: Simulatable commitments and efficient concurrent zero-knowledge. In: Proceedings of the Advances in Cryptology - EUROCRYPT 2003, International Conference on the Theory and Applications of Cryptographic Techniques, Warsaw, Poland, 4–8 May 2003, pp. 140–159 (2003)Google Scholar
  23. 23.
    Ostrovsky, R., Pandey, O., Visconti, I.: Efficiency preserving transformations for concurrent non-malleable zero knowledge. In: Micciancio, D. (ed.) TCC 2010. LNCS, vol. 5978, pp. 535–552. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  24. 24.
    Ostrovsky, R., Richelson, S., Scafuro, A.: Round-optimal black-box two-party computation. In: Gennaro, R., Robshaw, M. (eds.) CRYPTO 2015. LNCS, vol. 9216, pp. 339–358. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  25. 25.
    Pass, R.: Simulation in quasi-polynomial time, and its application to protocol composition. In: Biham, E. (ed.) EUROCRYPT 2003. LNCS, vol. 2656, pp. 160–176. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  26. 26.
    Santis, A.D., Crescenzo, G.D., Persiano, G., Yung, M.: On monotone formula closure of SZK. In: 35th Annual Symposium on Foundations of Computer Science, Santa Fe, New Mexico, USA, 20–22 November 1994, pp. 454–465 (1994)Google Scholar
  27. 27.
    Schnorr, C.-P.: Efficient identification and signatures for smart cards. In: Brassard, G. (ed.) CRYPTO 1989. LNCS, vol. 435, pp. 239–252. Springer, Heidelberg (1990)Google 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.
    Yung, M., Zhao, Y.: Generic and practical resettable zero-knowledge in the bare public-key model. In: Proceedings of the Advances in Cryptology - EUROCRYPT 2007, 26th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Barcelona, Spain, 20–24 May 2007, pp. 129–147 (2007)Google Scholar

Copyright information

© International Association for Cryptologic Research 2016

Authors and Affiliations

  • Michele Ciampi
    • 1
    Email author
  • Giuseppe Persiano
    • 2
  • Alessandra Scafuro
    • 3
  • Luisa Siniscalchi
    • 1
  • Ivan Visconti
    • 1
  1. 1.DIEMUniversity of SalernoSalernoItaly
  2. 2.DISA-MISUniversity of SalernoSalernoItaly
  3. 3.Boston University and Northeastern UniversityBostonUSA

Personalised recommendations