Succinct Non-Interactive Zero Knowledge Arguments from Span Programs and Linear Error-Correcting Codes

  • Helger Lipmaa
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8269)


Gennaro, Gentry, Parno and Raykova proposed an efficient NIZK argument for Circuit-SAT, based on non-standard tools like conscientious and quadratic span programs. We propose a new linear PCP for the Circuit-SAT, based on a combination of standard span programs (that verify the correctness of every individual gate) and high-distance linear error-correcting codes (that check the consistency of wire assignments). This allows us to simplify all steps of the argument, which results in significantly improved efficiency. We then construct an NIZK Circuit-SAT argument based on existing techniques.


Circuit-SAT linear error-correcting codes linear PCP non-interactive zero knowledge polynomial algebra quadratic span program span program verifiable computation 


  1. 1.
    Bitansky, N., Canetti, R., Chiesa, A., Tromer, E.: From Extractable Collision Resistance to Succinct Non-Interactive Arguments of Knowledge, and Back Again. In: Goldwasser, S. (ed.) ITCS 2012, pp. 326–349. ACM Press (2012)Google Scholar
  2. 2.
    Bitansky, N., Chiesa, A., Ishai, Y., Paneth, O., Ostrovsky, R.: Succinct Non-interactive Arguments via Linear Interactive Proofs. In: Sahai, A. (ed.) TCC 2013. LNCS, vol. 7785, pp. 315–333. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  3. 3.
    Blum, M., Feldman, P., Micali, S.: Non-Interactive Zero-Knowledge and Its Applications. In: STOC 1988, pp. 103–112. ACM Press (1988)Google Scholar
  4. 4.
    Brassard, G., Chaum, D., Crépeau, C.: Minimum Disclosure Proofs of Knowledge. Journal of Computer and System Sciences 37(2), 156–189 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chaabouni, R., Lipmaa, H., Zhang, B.: A Non-interactive Range Proof with Constant Communication. In: Keromytis, A.D. (ed.) FC 2012. LNCS, vol. 7397, pp. 179–199. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  6. 6.
    Di Crescenzo, G., Lipmaa, H.: Succinct NP Proofs from an Extractability Assumption. In: Beckmann, A., Dimitracopoulos, C., Löwe, B. (eds.) CiE 2008. LNCS, vol. 5028, pp. 175–185. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  7. 7.
    Dodunekov, S., Landgev, I.: On Near-MDS Codes. Journal of Geometry 54(1-2), 30–43 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dwork, C., Naor, M.: Zaps and Their Applications. In: FOCS 2000, pp. 283–293. IEEE Computer Society Press (2000)Google Scholar
  9. 9.
    Elkin, M.: An Improved Construction of Progression-Free Sets. Israel J. of Math. 184, 93–128 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fauzi, P., Lipmaa, H., Zhang, B.: Efficient Modular NIZK Arguments from Shift and Product. In: Abdalla, M. (ed.) CANS 2013. LNCS, vol. 8257, pp. 92–121. Springer, Heidelberg (2013)Google Scholar
  11. 11.
    Gál, A.: A Characterization of Span Program Size and Improved Lower Bounds for Monotone Span Programs. Computational Complexity 10(4), 277–296 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gathen, J., Gerhard, J.: Modern Computer Algebra, 2nd edn. Cambridge University Press (2003)Google Scholar
  13. 13.
    Gennaro, R., Gentry, C., Parno, B.: Non-interactive Verifiable Computing: Outsourcing Computation to Untrusted Workers. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 465–482. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  14. 14.
    Gennaro, R., Gentry, C., Parno, B., Raykova, M.: Quadratic Span Programs and Succinct NIZKs without PCPs. Tech. Rep. 2012/215, IACR (April 19, 2012), (last retrieved version from June 18, 2012)
  15. 15.
    Gennaro, R., Gentry, C., Parno, B., Raykova, M.: Quadratic Span Programs and Succinct NIZKs without PCPs. In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 626–645. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  16. 16.
    Groth, J.: Short Pairing-Based Non-interactive Zero-Knowledge Arguments. In: Abe, M. (ed.) ASIACRYPT 2010. LNCS, vol. 6477, pp. 321–340. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  17. 17.
    Hoover, H.J., Klawe, M.M., Pippenger, N.: Bounding Fan-out in Logical Networks. Journal of the ACM 31(1), 13–18 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Karchmer, M., Wigderson, A.: On Span Programs. In: Structure in Complexity Theory Conference 1993, pp. 102–111. IEEE Computer Society Press (1993)Google Scholar
  19. 19.
    Lipmaa, H.: Progression-Free Sets and Sublinear Pairing-Based Non-Interactive Zero-Knowledge Arguments. In: Cramer, R. (ed.) TCC 2012. LNCS, vol. 7194, pp. 169–189. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  20. 20.
    Lipmaa, H.: Succinct Non-Interactive Zero Knowledge Arguments from Span Programs and Linear Error-Correcting Codes. Tech. Rep. 2013/121, IACR (February 28, 2013),
  21. 21.
    Lipmaa, H., Zhang, B.: A More Efficient Computationally Sound Non-Interactive Zero-Knowledge Shuffle Argument. In: Visconti, I., De Prisco, R. (eds.) SCN 2012. LNCS, vol. 7485, pp. 477–502. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  22. 22.
    Micali, S.: CS Proofs. In: Goldwasser, S. (ed.) FOCS 1994, pp. 436–453. IEEE, IEEE Computer Society Press (1994)Google Scholar
  23. 23.
    Parno, B., Gentry, C., Howell, J., Raykova, M.: Pinocchio: Nearly Practical Verifiable Computation. In: IEEE Symposium on Security and Privacy, pp. 238–252. IEEE Computer SocietyGoogle Scholar
  24. 24.
    Reichardt, B.: Reflections for Quantum Query Algorithms. In: Randall, D. (ed.) SODA 2011, pp. 560–569. SIAM (2011)Google Scholar
  25. 25.
    Valiant, L.G.: Universal Circuits (Preliminary Report). In: STOC 1976, pp. 196–203. ACM (1976)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Helger Lipmaa
    • 1
  1. 1.Institute of Computer ScienceUniversity of TartuEstonia

Personalised recommendations