On the Portability of Generalized Schnorr Proofs

  • Jan Camenisch
  • Aggelos Kiayias
  • Moti Yung
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5479)

Abstract

The notion of Zero Knowledge Proofs (of knowledge) [ZKP] is central to cryptography; it provides a set of security properties that proved indispensable in concrete protocol design. These properties are defined for any given input and also for any auxiliary verifier private state, as they are aimed at any use of the protocol as a subroutine in a bigger application. Many times, however, moving the theoretical notion to practical designs has been quite problematic. This is due to the fact that the most efficient protocols fail to provide the above ZKP properties for all possible inputs and verifier states. This situation has created various problems to protocol designers who have often either introduced imperfect protocols with mistakes or with lack of security arguments, or they have been forced to use much less efficient protocols in order to achieve the required properties. In this work we address this issue by introducing the notion of “protocol portability,” a property that identifies input and verifier state distributions under which a protocol becomes a ZKP when called as a subroutine in a sequential execution of a larger application. We then concentrate on the very efficient and heavily employed “Generalized Schnorr Proofs” (GSP) and identify the portability of such protocols. We also point to previous protocol weaknesses and errors that have been made in numerous applications throughout the years, due to employment of GSP instances while lacking the notion of portability (primarily in the case of unknown order groups). This demonstrates that cryptographic application designers who care about efficiency need to consider our notion carefully. We provide a compact specification language for GSP protocols that protocol designers can employ. Our specification language is consistent with the ad-hoc notation that is currently widely used and it offers automatic derivation of the proof protocol while dictating its portability (i.e., the proper initial state and inputs) and its security guarantees. Finally, as a second alternative to designers wishing to use GSPs, we present a modification of GSP protocols that is unconditionally portable (i.e., ZKP) and is still quite efficient. Our constructions are the first such protocols proven secure in the standard model (as opposed to the random oracle model).

References

  1. 1.
    Ateniese, G., Camenisch, J.L., Joye, M., Tsudik, G.: A Practical and Provably Secure Coalition-Resistant Group Signature Scheme. In: Bellare, M. (ed.) CRYPTO 2000. LNCS, vol. 1880, p. 255. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  2. 2.
    Ateniese, G., Camenisch, J., Joye, M., Tsudik, G.: Remarks on ”analysis of one popular group signature scheme” in asiacrypt 2006. Cryptology ePrint Archive, Report 2006/464 (2006), http://eprint.iacr.org/
  3. 3.
    Ateniese, G., Song, D.X., Tsudik, G.: Quasi-efficient revocation in group signatures. In: Blaze, M. (ed.) FC 2002. LNCS, vol. 2357, pp. 183–197. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  4. 4.
    Bangerter, E.: On Efficient Zero-Knowledge Proofs of Knowledge. PhD thesis, Ruhr U. Bochum (2005)Google Scholar
  5. 5.
    Bangerter, E., Camenisch, J.L., Maurer, U.M.: Efficient proofs of knowledge of discrete logarithms and representations in groups with hidden order. In: Vaudenay, S. (ed.) PKC 2005. LNCS, vol. 3386, pp. 154–171. Springer, Heidelberg (2005), http://www.zurich.ibm.com/~/jca/papers/bacama05.pdf CrossRefGoogle Scholar
  6. 6.
    Bellare, M., Goldreich, O.: On Defining Proofs of Knowledge. In: Brickell, E.F. (ed.) CRYPTO 1992. LNCS, vol. 740, pp. 390–420. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  7. 7.
    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
  8. 8.
    Bresson, E., Stern, J.: Efficient revocation in group signatures. In: Kim, K.-c. (ed.) PKC 2001. LNCS, vol. 1992, pp. 190–206. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  9. 9.
    Brickell, E., Camenisch, J., Chen, L.: Direct anonymous attestation. In: Proc. 11th ACM Conference on Computer and Communications Security, pp. 225–234. ACM Press, New York (2004)Google Scholar
  10. 10.
    Bussard, L., Molva, R., Roudier, Y.: History-Based Signature or How to Trust Anonymous Documents. In: Jensen, C., Poslad, S., Dimitrakos, T. (eds.) iTrust 2004. LNCS, vol. 2995, pp. 78–92. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  11. 11.
    Bussard, L., Roudier, Y., Molva, R.: Untraceable secret credentials: Trust establishment with privacy. In: PerCom Workshops, pp. 122–126. IEEE Computer Society, Los Alamitos (2004)Google Scholar
  12. 12.
    Camenisch, J., Kiayias, A., Yung, M.: On the portability of generalized schnorr proofs. Technical report, Cryptology ePrint Archive (2009)Google Scholar
  13. 13.
    Camenisch, J.L., Shoup, V.: Practical Verifiable Encryption and Decryption of Discrete Logarithms. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 126–144. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  14. 14.
    Camenisch, J.L., Stadler, M.A.: Efficient Group Signature Schemes for Large Groups. In: Kaliski Jr., B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 410–424. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  15. 15.
    Camenisch, J.L.: Group Signature Schemes and Payment Systems Based on the Discrete Logarithm Problem. PhD thesis, ETH Zürich, Diss. ETH No. 12520. Hartung Gorre Verlag, Konstanz (1998)Google Scholar
  16. 16.
    Cao, Z.: Analysis of One Popular Group Signature Scheme. In: Lai, X., Chen, K. (eds.) ASIACRYPT 2006. LNCS, vol. 4284, pp. 460–466. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  17. 17.
    Chan, A.H., Frankel, Y., Tsiounis, Y.: Easy Come - Easy Go Divisible Cash. In: Nyberg, K. (ed.) EUROCRYPT 1998. LNCS, vol. 1403, pp. 561–575. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  18. 18.
    Chan, A.H., Frankel, Y., Tsiounis, Y.: Easy come - easy go divisible cash. GTE Technical Report (1998), http://www.ccs.neu.edu/home/yiannis/pubs.html
  19. 19.
    Cramer, R., Shoup, V.: Signature schemes based on the strong rsa assumption. ACM Trans. Inf. Syst. Secur. 3(3), 161–185 (2000)CrossRefGoogle Scholar
  20. 20.
    Damgård, I.B.: Efficient concurrent zero-knowledge in the auxiliary string model. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 418–430. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  21. 21.
    Damgård, I.B., 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
  22. 22.
    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)CrossRefGoogle Scholar
  23. 23.
    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
  24. 24.
    Furukawa, J., Yonezawa, S.: Group Signatures with Separate and Distributed Authorities. In: Blundo, C., Cimato, S. (eds.) SCN 2004. LNCS, vol. 3352, pp. 77–90. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  25. 25.
    Gaud, M., Traoré, J.: On the Anonymity of Fair Offline E-cash Systems. In: Wright, R.N. (ed.) FC 2003. LNCS, vol. 2742, pp. 34–50. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  26. 26.
    Goldreich, O., Micali, S., Wigderson, A.: How to play any mental game. In: STOC 1987: Proceedings of the nineteenth annual ACM conference on Theory of computing, pp. 218–229. ACM Press, New York (1987)CrossRefGoogle Scholar
  27. 27.
    Goldreich, O.: The Foundations of Cryptography, vol. 2. Cambridge University Press, Cambridge (1999)MATHGoogle Scholar
  28. 28.
    Goldwasser, S., Micali, S., Rackoff, C.: The knowledge complexity of interactive proof systems. SIAM Journal on Computing 18(1), 186–208 (1989)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Kunz-Jacques, S., Martinet, G., Poupard, G., Stern, J.: Cryptanalysis of an Efficient Proof of Knowledge of Discrete Logarithm. In: Yung, M., Dodis, Y., Kiayias, A., Malkin, T.G. (eds.) PKC 2006. LNCS, vol. 3958, pp. 27–43. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  30. 30.
    Van Le, T., Nguyen, K.Q., Varadharajan, V.: How to Prove That a Committed Number Is Prime. In: Lam, K.-Y., Okamoto, E., Xing, C. (eds.) ASIACRYPT 1999. LNCS, vol. 1716, pp. 208–218. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  31. 31.
    Lysyanskaya, A., Ramzan, Z.: Group Blind Digital Signatures: A Scalable Solution to Electronic Cash. In: Hirschfeld, R. (ed.) FC 1998. LNCS, vol. 1465, pp. 184–197. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  32. 32.
    MacKenzie, P.D., Reiter, M.K.: Two-party generation of DSA signatures. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 137–154. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  33. 33.
    Mykletun, E., Narasimha, M., Tsudik, G.: Signature Bouquets: Immutability for Aggregated/Condensed Signatures. In: Samarati, P., Ryan, P.Y.A., Gollmann, D., Molva, R. (eds.) ESORICS 2004. LNCS, vol. 3193, pp. 160–176. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  34. 34.
    Nakanishi, T., Shiota, M., Sugiyama, Y.: An Efficient Online Electronic Cash with Unlinkable Exact Payments. In: Zhang, K., Zheng, Y. (eds.) ISC 2004. LNCS, vol. 3225, pp. 367–378. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  35. 35.
    Pointcheval, D., Stern, J.: Security arguments for digital signatures and blind signatures. Journal of Cryptology 13(3), 361–396 (2000)CrossRefMATHGoogle Scholar
  36. 36.
    De Santis, A., Micali, S., Persiano, G.: Noninteractive Zero-Knowledge Proof Systems. In: Pomerance, C. (ed.) CRYPTO 1987. LNCS, vol. 293, pp. 52–72. Springer, Heidelberg (1988)Google Scholar
  37. 37.
    Schnorr, C.P.: Efficient signature generation by smart cards. Journal of Cryptology 4(3), 161–174 (1991)CrossRefMATHGoogle Scholar
  38. 38.
    Song, D.X.: Practical forward secure group signature schemes. In: Proc. 8th ACM Conference on Computer and Communications Security, pp. 225–234. ACM press, New York (2001)Google Scholar
  39. 39.
    Susilo, W., Mu, Y.: On the Security of Nominative Signatures. In: Boyd, C., González Nieto, J.M. (eds.) ACISP 2005. LNCS, vol. 3574, pp. 329–335. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  40. 40.
    Tang, C., Liu, Z., Wang, M.: A verifiable secret sharing scheme with statistical zero-knowledge. Cryptology ePrint Archive, Report 2003/222 (2003), http://eprint.iacr.org/
  41. 41.
    Tsang, P.P., Wei, V.K.: Short Linkable Ring Signatures for E-Voting, E-Cash and Attestation. In: Deng, R.H., Bao, F., Pang, H., Zhou, J. (eds.) ISPEC 2005. LNCS, vol. 3439, pp. 48–60. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  42. 42.
    Tsang, P.P., Wei, V.K., Chan, T.K., Au, M.H., Liu, J.K., Wong, D.S.: Separable Linkable Threshold Ring Signatures. In: Canteaut, A., Viswanathan, K. (eds.) INDOCRYPT 2004. LNCS, vol. 3348, pp. 384–398. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  43. 43.
    Wei, V.K.: Tracing-by-linking group signatures. In: Zhou, J., López, J., Deng, R.H., Bao, F. (eds.) ISC 2005. LNCS, vol. 3650, pp. 149–163. Springer, Heidelberg (2005)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Jan Camenisch
    • 1
  • Aggelos Kiayias
    • 2
  • Moti Yung
    • 3
  1. 1.IBM ResearchZurichSwitzerland
  2. 2.Computer Science and EngineeringUniversity of ConnecticutStorrsUSA
  3. 3.Google Inc. and Computer ScienceColumbia UniversityNew YorkUSA

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