Advertisement

A Certifying Compiler for Zero-Knowledge Proofs of Knowledge Based on Σ-Protocols

  • José Bacelar Almeida
  • Endre Bangerter
  • Manuel Barbosa
  • Stephan Krenn
  • Ahmad-Reza Sadeghi
  • Thomas Schneider
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6345)

Abstract

Zero-knowledge proofs of knowledge (ZK-PoK) are important building blocks for numerous cryptographic applications. Although ZK-PoK have a high potential impact, their real world deployment is typically hindered by their significant complexity compared to other (non-interactive) crypto primitives. Moreover, their design and implementation are time-consuming and error-prone.

We contribute to overcoming these challenges as follows: We present a comprehensive specification language and a compiler for ZK-PoK protocols based on Σ-protocols. The compiler allows the fully automatic translation of an abstract description of a proof goal into an executable implementation. Moreover, the compiler overcomes various restrictions of previous approaches, e.g., it supports the important class of exponentiation homomorphisms with hidden-order co-domain, needed for privacypreserving applications such as DAA. Finally, our compiler is certifying, in the sense that it automatically produces a formal proof of the soundness of the compiled protocol for a large class of protocols using the Isabelle/HOL theorem prover.

Keywords

Zero-Knowledge Protocol Compiler Formal Verification 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Almeida, J., Bangerter, E., Barbosa, M., Krenn, S., Sadeghi, A.R., Schneider, T.: A certifying compiler for zero-knowledge proofs of knowledge based on Σ-protocols. Cryptology ePrint Archive, Report 2010/339 (2010)Google Scholar
  2. 2.
    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)Google Scholar
  3. 3.
    Han, W., Chen, K., Zheng, D.: Receipt-freeness for Groth e-voting schemes. Journal of Information Science and Engineering 25, 517–530 (2009)MathSciNetGoogle Scholar
  4. 4.
    Kikuchi, H., Nagai, K., Ogata, W., Nishigaki, M.: Privacy-preserving similarity evaluation and application to remote biometrics authentication. Soft Computing 14, 529–536 (2010)CrossRefGoogle Scholar
  5. 5.
    Camenisch, J.: Group Signature Schemes and Payment Systems Based on the Discrete Logarithm Problem. PhD thesis, ETH Zurich, Konstanz (1998)Google Scholar
  6. 6.
    Camenisch, J., Michels, M.: Proving in zero-knowledge that a number is the product of two safe primes. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 107–122. Springer, Heidelberg (1999)Google Scholar
  7. 7.
    Brands, S.: Untraceable off-line cash in wallet with observers. In: Stinson, D.R. (ed.) CRYPTO 1993. LNCS, vol. 773, pp. 302–318. Springer, Heidelberg (1994)Google Scholar
  8. 8.
    Lindell, Y., Pinkas, B., Smart, N.P.: Implementing two-party computation efficiently with security against malicious adversaries. In: Ostrovsky, R., De Prisco, R., Visconti, I. (eds.) SCN 2008. LNCS, vol. 5229, pp. 2–20. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  9. 9.
    Brickell, E., Camenisch, J., Chen, L.: Direct anonymous attestation. In: ACM CCS 2004, pp. 132–145. ACM Press, New York (2004)CrossRefGoogle Scholar
  10. 10.
    Camenisch, J., Herreweghen, E.V.: Design and implementation of the idemix anonymous credential system. In: ACM CCS 2002, pp. 21–30. ACM Press, New York (2002)CrossRefGoogle Scholar
  11. 11.
    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
  12. 12.
    Bangerter, E., Camenisch, J., Maurer, U.: 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)CrossRefGoogle Scholar
  13. 13.
    Schnorr, C.: Efficient signature generation by smart cards. Journal of Cryptology 4, 161–174 (1991)zbMATHCrossRefGoogle Scholar
  14. 14.
    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
  15. 15.
    Camenisch, J., Lysyanskaya, A.: An efficient system for non-transferable anonymous credentials with optional anonymity revocation. In: Pfitzmann, B. (ed.) EUROCRYPT 2001. LNCS, vol. 2045, pp. 93–118. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  16. 16.
    Lipmaa, H.: On diophantine complexity and statistical zeroknowledge arguments. In: Laih, C.-S. (ed.) ASIACRYPT 2003. LNCS, vol. 2894, pp. 398–415. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  17. 17.
    Paulson, L.: Isabelle: a Generic Theorem Prover. Volume 828 of LNCS. Springer (1994)zbMATHGoogle Scholar
  18. 18.
    MacKenzie, P., Oprea, A., Reiter, M.K.: Automatic generation of two-party computations. In: ACM CCS 2003, pp. 210–219. ACM, New York (2003)CrossRefGoogle Scholar
  19. 19.
    Malkhi, D., Nisan, N., Pinkas, B., Sella, Y.: Fairplay — a secure two-party computation system. In: USENIX Security 2004 (2004)Google Scholar
  20. 20.
    Damgård, I., Geisler, M., Krøigaard, M., Nielsen, J.B.: Asynchronous multiparty computation: Theory and implementation. In: Jarecki, S., Tsudik, G. (eds.) Public Key Cryptography – PKC 2009. LNCS, vol. 5443, pp. 160–179. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  21. 21.
    Briner, T.: Compiler for zero-knowledge proof-of-knowledge protocols. Master’s thesis, ETH Zurich (2004)Google Scholar
  22. 22.
    Camenisch, J., Rohe, M., Sadeghi, A.R.: Sokrates - a compiler framework for zero-knowledge protocols. In: WEWoRC 2005 (2005)Google Scholar
  23. 23.
    Bangerter, E., Camenisch, J., Krenn, S., Sadeghi, A.R., Schneider, T.: Automatic generation of sound zero-knowledge protocols. Cryptology ePrint Archive, Report 2008/471, Poster Session of EUROCRYPT 2009 (2008)Google Scholar
  24. 24.
    Bangerter, E., Briner, T., Heneka, W., Krenn, S., Sadeghi, A.R., Schneider, T.: Automatic generation of Σ-protocols. In: EuroPKI 2009 (to appear, 2009)Google Scholar
  25. 25.
    Bangerter, E., Krenn, S., Sadeghi, A.R., Schneider, T., Tsay, J.K.: On the design and implementation of efficient zero-knowledge proofs of knowledge. In: Software Performance Enhancements for Encryption and Decryption and Cryptographic Compilers – SPEED-CC 2009, October 12-13 (2009)Google Scholar
  26. 26.
    Meiklejohn, S., Erway, C., Küpçü, A., Hinkle, T., Lysyanskaya, A.: ZKPDL: A language-based system for efficient zero-knowledge proofs and electronic cash. In: USENIX 10 (to appear, 2010)Google Scholar
  27. 27.
    Rivest, R., Shamir, A., Adleman, L.: A method for obtaining digital signatures and public-key cryptosystems. Communications of the ACM 21, 120–126 (1978)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Backes, M., Maffei, M., Unruh, D.: Zero-knowledge in the applied pi-calculus and automated verification of the direct anonymous attestation protocol. In: IEEE Symposium on Security and Privacy – SP 2008, pp. 202–215. IEEE, Los Alamitos (2008)CrossRefGoogle Scholar
  29. 29.
    Baskar, A., Ramanujam, R., Suresh, S.P.: A dolev-yao model for zero knowledge. In: Datta, A. (ed.) ASIAN 2009. LNCS, vol. 5913, pp. 137–146. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  30. 30.
    Blanchet, B.: ProVerif: Cryptographic protocol verifier in the formal model (2010)Google Scholar
  31. 31.
    Backes, M., Hritcu, C., Maffei, M.: Type-checking zero-knowledge. In: ACM CCS 2008, pp. 357–370. ACM, New York (2008)CrossRefGoogle Scholar
  32. 32.
    Backes, M., Unruh, D.: Computational soundness of symbolic zero-knowledge proofs against active attackers. In: IEEE Computer Security Foundations Symposium - CSF 2008, 255–269 Preprint on IACR ePrint 2008/152 (2008)Google Scholar
  33. 33.
    Barthe, G., Hedin, D., Zanella Béguelin, S., Grégoire, B., Heraud, S.: A machine-checked formalization of Σ-protocols. In: 23rd IEEE Computer Security Foundations Symposium, CSF 2010, IEEE, Los Alamitos (2010)Google Scholar
  34. 34.
    Barthe, G., Grégoire, B., Béguelin, S.: Formal certification of code-based cryptographic proofs. In: ACM SIGPLAN-SIGACT POPL 2009, pp. 90–101 (2009)Google Scholar
  35. 35.
    Goubault-Larrecq, J., Parrennes, F.: Cryptographic protocol analysis on real C code. In: Cousot, R. (ed.) VMCAI 2005. LNCS, vol. 3385, pp. 363–379. Springer, Heidelberg (2005)Google Scholar
  36. 36.
    Bhargavan, K., Fournet, C., Gordon, A., Tse, S.: Verified interoperable implementations of security protocols. ACM Trans. Program. Lang. Syst. 31(1), 1–61 (2008)CrossRefGoogle Scholar
  37. 37.
    Bhargavan, K., Fournet, C., Corin, R., Zalinescu, E.: Cryptographically verified implementations for TLS. In: ACM CCS 2008, pp. 459–468. ACM, New York (2008)CrossRefGoogle Scholar
  38. 38.
    Blanchet, B.: An efficient cryptographic protocol verifier based on prolog rules. In: Workshop on Computer Security Foundations – CSFW 2001, p. 82. IEEE, Los Alamitos (2001)Google Scholar
  39. 39.
    Blanchet, B.: A computationally sound mechanized prover for security protocols. In: IEEE Symposium on Security and Privacy – SP 2006, pp. 140–154. IEEE, Los Alamitos (2006)CrossRefGoogle Scholar
  40. 40.
    Camenisch, J., Stadler, M.: Efficient group signature schemes for large groups (extended abstract). In: Kaliski Jr., B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 410–424. Springer, Heidelberg (1997)Google Scholar
  41. 41.
    Cramer, R.: Modular Design of Secure yet Practical Cryptographic Protocols. PhD thesis, CWI and University of Amsterdam (1997)Google Scholar
  42. 42.
    Damgård, I.: On Σ-protocols, Lecture on Cryptologic Protocol Theory, Faculty of Science, University of Aarhus (2004)Google Scholar
  43. 43.
    Guillou, L., Quisquater, J.J.: A “paradoxical” identity-based signature scheme resulting from zero-knowledge. In: Goldwasser, S. (ed.) CRYPTO 1988. LNCS, vol. 403, pp. 216–231. Springer, Heidelberg (1990)Google Scholar
  44. 44.
    Cramer, R., Shoup, V.: A practical public key cryptosystem provably secure against adaptive chosen ciphertext attack. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 13–25. Springer, Heidelberg (1998)Google Scholar
  45. 45.
    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)Google Scholar
  46. 46.
    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. 77–85. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  47. 47.
    Bangerter, E.: Efficient Zero-Knowledge Proofs of Knowledge for Homomorphisms. PhD thesis, Ruhr-University Bochum (2005)Google Scholar
  48. 48.
    Smart, N.P. (ed.): Final Report on Unified Theoretical Framework of Efficient Zero-Knowledge Proofs of Knowledge. CACE project deliverable (2009)Google Scholar
  49. 49.
    Cramer, R., Damgård, I., Schoenmakers, B.: Proofs 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
  50. 50.
    Shamir, A.: How to share a secret. Communications of the ACM 22, 612–613 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  51. 51.
    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
  52. 52.
    Brands, S.: Rapid demonstration of linear relations connected by boolean operators. In: Fumy, W. (ed.) EUROCRYPT 1997. LNCS, vol. 1233, pp. 318–333. Springer, Heidelberg (1997)Google Scholar
  53. 53.
    Bresson, E., Stern, J.: Proofs of knowledge for non-monotone discrete-log formulae and applications. In: Chan, A.H., Gligor, V.D. (eds.) ISC 2002. LNCS, vol. 2433, pp. 272–288. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  54. 54.
    Granlund, T.: The GNU MP Bignum Library (2010), http://gmplib.org/
  55. 55.
    Nipkow, T., Paulson, L.: Isabelle (2010), http://isabelle.in.tun.de
  56. 56.
    Nipkow, T., Paulson, L., Wenzel, M.: Isabelle/HOL: a proof assistant for higher-order logic. LNCS, vol. 2283. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  57. 57.
    Ballarin, C., Kammüller, F., Paulson, L.: The Isabelle/HOL Algebra Library (2008), http://isabelle.in.tum.de/library/HOL/HOL-Algebra/document.pdf

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • José Bacelar Almeida
    • 1
  • Endre Bangerter
    • 2
  • Manuel Barbosa
    • 1
  • Stephan Krenn
    • 3
  • Ahmad-Reza Sadeghi
    • 4
  • Thomas Schneider
    • 4
  1. 1.Universidade do MinhoPortugal
  2. 2.Bern University of Applied SciencesBiel-BienneSwitzerland
  3. 3.Bern University of Applied Sciences, Biel-Bienne, Switzerland, and, University of FribourgSwitzerland
  4. 4.Horst Görtz Institute for IT-SecurityRuhr-University BochumGermany

Personalised recommendations