Document Open Access Logo

Robustness for Space-Bounded Statistical Zero Knowledge

Authors Eric Allender , Jacob Gray, Saachi Mutreja, Harsha Tirumala , Pengxiang Wang

Thumbnail PDF


  • Filesize: 0.91 MB
  • 21 pages

Document Identifiers

Author Details

Eric Allender
  • Rutgers University, Piscataway, NJ, USA
Jacob Gray
  • University of Massachusetts, Amherst, MA, USA
Saachi Mutreja
  • University of California, Berkeley, CA, USA
Harsha Tirumala
  • Rutgers University, Piscataway, NJ, USA
Pengxiang Wang
  • University of Michigan, Ann Arbor, MI, USA


This work was done in part while EA and HT were visiting the Simons Institute for the Theory of Computing. This work was carried out while JG, SM, and PW were participants in the 2022 DIMACS REU program at Rutgers University. We thank Yuval Ishai for helpful conversations about SREN, and we thank Markus Lohrey, Sam Buss, and Dave Barrington for useful discussions about Lemma 34. We also thank the anonymous referees for helpful comments.

Cite AsGet BibTex

Eric Allender, Jacob Gray, Saachi Mutreja, Harsha Tirumala, and Pengxiang Wang. Robustness for Space-Bounded Statistical Zero Knowledge. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 56:1-56:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


We show that the space-bounded Statistical Zero Knowledge classes SZK_L and NISZK_L are surprisingly robust, in that the power of the verifier and simulator can be strengthened or weakened without affecting the resulting class. Coupled with other recent characterizations of these classes [Eric Allender et al., 2023], this can be viewed as lending support to the conjecture that these classes may coincide with the non-space-bounded classes SZK and NISZK, respectively.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
  • Interactive Proofs


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Eric Allender, John Gouwar, Shuichi Hirahara, and Caleb Robelle. Cryptographic hardness under projections for time-bounded Kolmogorov complexity. Theoretical Computer Science, 940:206-224, 2023. URL:
  2. Eric Allender, Jacob Gray, Saachi Mutreja, Harsha Tirumala, and Pengxiang Wang. Robustness for space-bounded statistical zero knowledge. Electron. Colloquium Comput. Complex., TR22-138, 2022. URL:
  3. Eric Allender and Shuichi Hirahara. New insights on the (non-) hardness of circuit minimization and related problems. ACM Transactions on Computation Theory (TOCT), 11(4):1-27, 2019. Google Scholar
  4. Eric Allender, Shuichi Hirahara, and Harsha Tirumala. Kolmogorov complexity characterizes statistical zero knowledge. In 14th Innovations in Theoretical Computer Science Conference (ITCS), volume 251 of LIPIcs, pages 3:1-3:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL:
  5. Eric Allender and Ian Mertz. Complexity of regular functions. Journal of Computer and System Sciences, 104:5-16, 2019. Language and Automata Theory and Applications - LATA 2015. URL:
  6. Eric Allender, Klaus Reinhardt, and Shiyu Zhou. Isolation, matching, and counting uniform and nonuniform upper bounds. Journal of Computer and System Sciences, 59(2):164-181, 1999. URL:
  7. Benny Applebaum, Yuval Ishai, and Eyal Kushilevitz. Cryptography in NC⁰. SIAM Journal on Computing, 36(4):845-888, 2006. URL:
  8. V. Arvind and T. C. Vijayaraghavan. Classifying problems on linear congruences and abelian permutation groups using logspace counting classes. computational complexity, 19(1):57-98, November 2009. URL:
  9. Samuel R. Buss. The Boolean formula value problem is in ALOGTIME. In Proceedings of the 19th Annual ACM Symposium on Theory of Computing (STOC), pages 123-131. ACM, 1987. URL:
  10. Samuel R Buss. Algorithms for Boolean formula evaluation and for tree contraction. Arithmetic, Proof Theory, and Computational Complexity, 23:96-115, 1993. Google Scholar
  11. Ronald Cramer, Serge Fehr, Yuval Ishai, and Eyal Kushilevitz. Efficient multi-party computation over rings. In Proc. International Conference on the Theory and Applications of Cryptographic Techniques; Advances in Cryptology (EUROCRYPT), volume 2656 of Lecture Notes in Computer Science, pages 596-613. Springer, 2003. URL:
  12. Zeev Dvir, Dan Gutfreund, Guy N Rothblum, and Salil P Vadhan. On approximating the entropy of polynomial mappings. In Second Symposium on Innovations in Computer Science, pages 460-475. Tsinghua University Press, 2011. Google Scholar
  13. Moses Ganardi and Markus Lohrey. A universal tree balancing theorem. ACM Transactions on Computation Theory, 11(1):1:1-1:25, 2019. URL:
  14. Oded Goldreich, Amit Sahai, and Salil Vadhan. Can statistical zero knowledge be made non-interactive? or On the relationship of SZK and NISZK. In Annual International Cryptology Conference, pages 467-484. Springer, 1999. URL:
  15. Oded Goldreich, Amit Sahai, and Salil P. Vadhan. Honest-verifier statistical zero-knowledge equals general statistical zero-knowledge. In Proceedings of the 30th Annual ACM Symposium on the Theory of Computing (STOC), pages 399-408. ACM, 1998. URL:
  16. Ulrich Hertrampf, Steffen Reith, and Heribert Vollmer. A note on closure properties of logspace MOD classes. Information Processing Letters, 75(3):91-93, 2000. URL:
  17. Yuval Ishai and Eyal Kushilevitz. Perfect constant-round secure computation via perfect randomizing polynomials. In Proc. International Conference on Automata, Languages, and Programming (ICALP), volume 2380 of Lecture Notes in Computer Science, pages 244-256. Springer, 2002. URL:
  18. Richard M. Karp, Eli Upfal, and Avi Wigderson. Constructing a perfect matching is in random NC. Combinatorica, 6(1):35-48, 1986. URL:
  19. Nathan Linial, Yishay Mansour, and Noam Nisan. Constant depth circuits, Fourier transform, and learnability. J. ACM, 40(3):607-620, 1993. URL:
  20. Pierre McKenzie and Stephen A. Cook. The parallel complexity of Abelian permutation group problems. SIAM Journal on Computing, 16(5):880-909, 1987. URL:
  21. Ketan Mulmuley, Umesh V. Vazirani, and Vijay V. Vazirani. Matching is as easy as matrix inversion. In Proceedings of the 19th Annual ACM Symposium on Theory of Computing (STOC), pages 345-354. ACM, 1987. URL:
  22. Tatsuaki Okamoto. On relationships between statistical zero-knowledge proofs. Journal of Computer and System Sciences, 60(1):47-108, 2000. URL:
  23. Chris Peikert and Vinod Vaikuntanathan. Noninteractive statistical zero-knowledge proofs for lattice problems. In Proc. Advances in Cryptology: 28th Annual International Cryptology Conference (CRYPTO), volume 5157 of Lecture Notes in Computer Science, pages 536-553. Springer, 2008. URL:
  24. Vishal Ramesh, Sasha Sami, and Noah Singer. Simple reductions to circuit minimization: DIMACS REU report. Technical report, DIMACS, Rutgers University, 2021. Internal document. Google Scholar
  25. Amit Sahai and Salil P. Vadhan. A complete problem for statistical zero knowledge. J. ACM, 50(2):196-249, 2003. URL:
  26. Jacobo Torán. On the hardness of graph isomorphism. SIAM Journal on Computing, 33(5):1093-1108, 2004. URL:
  27. Heribert Vollmer. Introduction to circuit complexity: a uniform approach. Springer Science & Business Media, 1999. URL:
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail