Efficient NIZK Arguments via Parallel Verification of Benes Networks

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8642)


We work within the recent paradigm, started by Groth (ASIACRYPT 2010), of constructing short non-interactive zero knowledge arguments from a small number basic arguments in a modular fashion. The main technical result of this paper is a new permutation argument, by using product and shift arguments of Lipmaa (2014) and a parallelizable variant of the Beneš network. We use it to design a short non-interactive zero knowledge argument for the NP-complete language CircuitSAT with Θ(n log2 n) prover’s computational complexity, where n is the size of the circuit. The permutation argument can be naturally used to design direct NIZK arguments for many other NP-complete languages.


Beneš networks modular NIZK arguments perfect zero knowledge product argument shift argument shuffle 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abe, M.: Mix-Networks on Permutation Networks. In: Lam, K.-Y., Okamoto, E., Xing, C. (eds.) ASIACRYPT 1999. LNCS, vol. 1716, pp. 258–273. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  2. 2.
    Abe, M., Hoshino, F.: Remarks on Mix-Network Based on Permutation Networks. In: Kim, K.-C. (ed.) PKC 2001. LNCS, vol. 1992, pp. 317–324. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  3. 3.
    Ben-Sasson, E., Chiesa, A., Genkin, D., Tromer, E., Virza, M.: SNARKs for C: Verifying Program Executions Succinctly and in Zero Knowledge. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013, Part II. LNCS, vol. 8043, pp. 90–108. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  4. 4.
    Beneš, V.E.: Mathematical Theory of Connecting Networks and Telephone Traffic. Academic Press (August 28, 1965)Google Scholar
  5. 5.
    Bitansky, N., Chiesa, A., Ishai, Y., Ostrovsky, R., Paneth, O.: 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
  6. 6.
    Blelloch, G.: Vector Models for Data-Parallel Computing. MIT Press (1990)Google Scholar
  7. 7.
    Blum, M., Feldman, P., Micali, S.: Non-Interactive Zero-Knowledge and Its Applications. In: STOC 1988, pp. 103–112. ACM Press (1988)Google Scholar
  8. 8.
    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
  9. 9.
    Clos, C.: A Study of Non-Blocking Switching Networks. Bell System Technical Journal 32(2), 406–424 (1953)CrossRefGoogle Scholar
  10. 10.
    Damgård, I.: Towards Practical Public Key Systems Secure against Chosen Ciphertext Attacks. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 445–456. Springer, Heidelberg (1992)Google Scholar
  11. 11.
    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
  12. 12.
    Fauzi, P., Lipmaa, H., Zhang, B.: Efficient Modular NIZK Arguments from Shift and Product. In: Abdalla, M., Nita-Rotaru, C., Dahab, R. (eds.) CANS 2013. LNCS, vol. 8257, pp. 92–121. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  13. 13.
    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
  14. 14.
    Goldwasser, S., Micali, S., Rackoff, C.: The Knowledge Complexity of Interactive Proof-Systems. In: Sedgewick, R. (ed.) STOC 1985, pp. 291–304. ACM Press (1985)Google Scholar
  15. 15.
    Golle, P., Jarecki, S., Mironov, I.: Cryptographic Primitives Enforcing Communication and Storage Complexity. In: Blaze, M. (ed.) FC 2002. LNCS, vol. 2357, pp. 120–135. Springer, Heidelberg (2003)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.
    Groth, J., Lu, S.: A Non-interactive Shuffle with Pairing Based Verifiability. In: Kurosawa, K. (ed.) ASIACRYPT 2007. LNCS, vol. 4833, pp. 51–67. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  18. 18.
    Hwang, F.K.M.: The Mathematical Theory of Nonblocking Switching Networks, 2nd edn. Series on Applied Mathematics, vol. 15. World Scientific Publishing Co Pte Ltd. (October 1, 2004)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. In: Sako, K., Sarkar, P. (eds.) ASIACRYPT 2013, Part I. LNCS, vol. 8269, pp. 41–60. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  21. 21.
    Lipmaa, H.: Almost Optimal Short Adaptive Non-Interactive Zero Knowledge. Tech. Rep. 2014/396, International Association for Cryptologic Research (2014),
  22. 22.
    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
  23. 23.
    Nassimi, D., Sahni, S.: Parallel Algorithms to Set Up the Benes Permutation Network. IEEE Trans. Computers 31(2), 148–154 (1982)CrossRefzbMATHGoogle Scholar
  24. 24.
    Opferman, D.C., Tsao-Wu, N.T.: On a Class of Rearrangeable Switching Networks. Part I: Control Algorithm. Bell System Technical Journal 50(5), 1579–1600 (1971)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Pippenger, N.: On the Evaluation of Powers and Monomials. SIAM J. Comput. 9(2), 230–250 (1980)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Pratt, V.R., Stockmeyer, L.J.: A Characterization of the Power of Vector Machines. Journal of Computer and System Sciences 12(2), 198–221 (1976)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Straus, E.G.: Addition Chains of Vectors. American Mathematical Monthly 70, 806–808 (1964)MathSciNetGoogle Scholar
  28. 28.
    Waksman, A.: A Permutation Network. Journal of the ACM 15(1), 159–163 (1968)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Institute of Computer ScienceUniversity of TartuEstonia

Personalised recommendations