Document Open Access Logo

Efficient Batch Verification for UP

Authors Omer Reingold, Guy N. Rothblum, Ron D. Rothblum

PDF
Thumbnail PDF

File

LIPIcs.CCC.2018.22.pdf
  • Filesize: 0.67 MB
  • 23 pages

Document Identifiers

Author Details

Omer Reingold
  • Stanford University, Palo Alto CA, USA
Guy N. Rothblum
  • Weizmann Institute, Rehovot, Israel
Ron D. Rothblum
  • MIT and Northeastern University, Cambridge and Boston MA, USA

Funding

  • Reingold, Omer: Supported by NSF grant CCF-1749750.
  • Rothblum, Ron D.: Research supported in part by NSF Grants CNS-1350619 and CNS-1414119, Alfred P. Sloan Research Fellowship, Microsoft Faculty Fellowship and in part by the Defense Advanced Research Projects Agency (DARPA), the U.S. Army Research Office under contracts W911NF-15-C-0226 and W911NF-15-C-0236 and by the Cybersecurity and Privacy Institute at Northeastern University.

Cite As Get BibTex

Omer Reingold, Guy N. Rothblum, and Ron D. Rothblum. Efficient Batch Verification for UP. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 22:1-22:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.CCC.2018.22

Abstract

Consider a setting in which a prover wants to convince a verifier of the correctness of k NP statements. For example, the prover wants to convince the verifier that k given integers N_1,...,N_k are all RSA moduli (i.e., products of equal length primes). Clearly this problem can be solved by simply having the prover send the k NP witnesses, but this involves a lot of communication. Can interaction help? In particular, is it possible to construct interactive proofs for this task whose communication grows sub-linearly with k?
Our main result is such an interactive proof for verifying the correctness of any k UP statements (i.e., NP statements that have a unique witness). The proof-system uses only a constant number of rounds and the communication complexity is k^delta * poly(m), where delta>0 is an arbitrarily small constant, m is the length of a single witness, and the poly term refers to a fixed polynomial that only depends on the language and not on delta. The (honest) prover strategy can be implemented in polynomial-time given access to the k (unique) witnesses.
Our proof leverages "interactive witness verification" (IWV), a new type of proof-system that may be of independent interest. An IWV is a proof-system in which the verifier needs to verify the correctness of an NP statement using: (i) a sublinear number of queries to an alleged NP witness, and (ii) a short interaction with a powerful but untrusted prover. In contrast to the setting of PCPs and Interactive PCPs, here the verifier only has access to the raw NP witness, rather than some encoding thereof.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Interactive Proof
  • Batch Verification
  • Unique Solution

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and the hardness of approximation problems. J. ACM, 45(3):501-555, 1998. URL: http://dx.doi.org/10.1145/278298.278306.
  2. Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and the hardness of approximation problems. J. ACM, 45(3):501-555, 1998. URL: http://dx.doi.org/10.1145/278298.278306.
  3. Sanjeev Arora and Shmuel Safra. Probabilistic checking of proofs; A new characterization of NP. In 33rd Annual Symposium on Foundations of Computer Science, Pittsburgh, Pennsylvania, USA, 24-27 October 1992, pages 2-13. IEEE Computer Society, 1992. URL: http://dx.doi.org/10.1109/SFCS.1992.267824.
  4. László Babai, Lance Fortnow, Leonid A. Levin, and Mario Szegedy. Checking computations in polylogarithmic time. In Cris Koutsougeras and Jeffrey Scott Vitter, editors, Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, May 5-8, 1991, New Orleans, Louisiana, USA, pages 21-31. ACM, 1991. URL: http://dx.doi.org/10.1145/103418.103428.
  5. László Babai, Lance Fortnow, and Carsten Lund. Non-deterministic exponential time has two-prover interactive protocols. Computational Complexity, 1:3-40, 1991. URL: http://dx.doi.org/10.1007/BF01200056.
  6. László Babai and Shlomo Moran. Arthur-merlin games: A randomized proof system, and a hierarchy of complexity classes. J. Comput. Syst. Sci., 36(2):254-276, 1988. URL: http://dx.doi.org/10.1016/0022-0000(88)90028-1.
  7. Michael Ben-Or, Shafi Goldwasser, Joe Kilian, and Avi Wigderson. Multi-prover interactive proofs: How to remove intractability assumptions. In Janos Simon, editor, Proceedings of the 20th Annual ACM Symposium on Theory of Computing, May 2-4, 1988, Chicago, Illinois, USA, pages 113-131. ACM, 1988. URL: http://dx.doi.org/10.1145/62212.62223.
  8. Eli Ben-Sasson, Alessandro Chiesa, and Nicholas Spooner. Interactive oracle proofs. Cryptology ePrint Archive, Report 2016/116, 2016. URL: http://eprint.iacr.org/.
  9. Itay Berman, Ron D. Rothblum, and Vinod Vaikuntanathan. Zero-knowledge proofs of proximity. In Anna R. Karlin, editor, 9th Innovations in Theoretical Computer Science Conference, ITCS 2018, January 11-14, 2018, Cambridge, MA, USA, volume 94 of LIPIcs, pages 19:1-19:20. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018. URL: http://dx.doi.org/10.4230/LIPIcs.ITCS.2018.19.
  10. Zvika Brakerski, Justin Holmgren, and Yael Tauman Kalai. Non-interactive delegation and batch NP verification from standard computational assumptions. In Hamed Hatami, Pierre McKenzie, and Valerie King, editors, Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, pages 474-482. ACM, 2017. URL: http://dx.doi.org/10.1145/3055399.3055497.
  11. Alessandro Chiesa and Tom Gur. Proofs of proximity for distribution testing. In Anna R. Karlin, editor, 9th Innovations in Theoretical Computer Science Conference, ITCS 2018, January 11-14, 2018, Cambridge, MA, USA, volume 94 of LIPIcs, pages 53:1-53:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018. URL: http://dx.doi.org/10.4230/LIPIcs.ITCS.2018.53.
  12. Funda Ergün, Ravi Kumar, and Ronitt Rubinfeld. Fast approximate probabilistically checkable proofs. Inf. Comput., 189(2):135-159, 2004. URL: http://dx.doi.org/10.1016/j.ic.2003.09.005.
  13. Uriel Feige, Shafi Goldwasser, László Lovász, Shmuel Safra, and Mario Szegedy. Interactive proofs and the hardness of approximating cliques. J. ACM, 43(2):268-292, 1996. URL: http://dx.doi.org/10.1145/226643.226652.
  14. Eldar Fischer, Yonatan Goldhirsh, and Oded Lachish. Partial tests, universal tests and decomposability. In Moni Naor, editor, Innovations in Theoretical Computer Science, ITCS'14, Princeton, NJ, USA, January 12-14, 2014, pages 483-500. ACM, 2014. URL: http://dx.doi.org/10.1145/2554797.2554841.
  15. Lance Fortnow, John Rompel, and Michael Sipser. On the power of multi-prover interactive protocols. Theor. Comput. Sci., 134(2):545-557, 1994. URL: http://dx.doi.org/10.1016/0304-3975(94)90251-8.
  16. Martin Fürer, Oded Goldreich, Yishay Mansour, Michael Sipser, and Stathis Zachos. On completeness and soundness in interactive proof systems. Advances in Computing Research, 5:429-442, 1989. Google Scholar
  17. Oded Goldreich. Modern cryptography, probabilistic proofs and pseudorandomness, volume 17 of Algorithms and Combinatorics. Springer-Verlag, 1999. Google Scholar
  18. Oded Goldreich. Overview of the doubly-efficient interactive proof systems of RRR. Electronic Colloquium on Computational Complexity (ECCC), 24:102, 2017. URL: https://eccc.weizmann.ac.il/report/2017/102.
  19. Oded Goldreich, Tom Gur, and Ron D. Rothblum. Proofs of proximity for context-free languages and read-once branching programs - (extended abstract). In Magnús M. Halldórsson, Kazuo Iwama, Naoki Kobayashi, and Bettina Speckmann, editors, Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part I, volume 9134 of Lecture Notes in Computer Science, pages 666-677. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-662-47672-7_54.
  20. Oded Goldreich and Johan Håstad. On the complexity of interactive proofs with bounded communication. Inf. Process. Lett., 67(4):205-214, 1998. URL: http://dx.doi.org/10.1016/S0020-0190(98)00116-1.
  21. Oded Goldreich, Silvio Micali, and Avi Wigderson. Proofs that yield nothing but their validity for all languages in NP have zero-knowledge proof systems. J. ACM, 38(3):691-729, 1991. URL: http://dx.doi.org/10.1145/116825.116852.
  22. Oded Goldreich, Salil P. Vadhan, and Avi Wigderson. On interactive proofs with a laconic prover. Computational Complexity, 11(1-2):1-53, 2002. URL: http://dx.doi.org/10.1007/s00037-002-0169-0.
  23. Shafi Goldwasser, Yael Tauman Kalai, and Guy N. Rothblum. Delegating computation: interactive proofs for muggles. In Cynthia Dwork, editor, Proceedings of the 40th Annual ACM Symposium on Theory of Computing, Victoria, British Columbia, Canada, May 17-20, 2008, pages 113-122. ACM, 2008. URL: http://dx.doi.org/10.1145/1374376.1374396.
  24. Shafi Goldwasser, Silvio Micali, and Charles Rackoff. The knowledge complexity of interactive proof systems. SIAM J. Comput., 18(1):186-208, 1989. URL: http://dx.doi.org/10.1137/0218012.
  25. Tom Gur and Ron D. Rothblum. Non-interactive proofs of proximity. In Tim Roughgarden, editor, Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science, ITCS 2015, Rehovot, Israel, January 11-13, 2015, pages 133-142. ACM, 2015. URL: http://dx.doi.org/10.1145/2688073.2688079.
  26. Tom Gur and Ron D. Rothblum. A hierarchy theorem for interactive proofs of proximity. In Christos H. Papadimitriou, editor, 8th Innovations in Theoretical Computer Science Conference, ITCS 2017, January 9-11, 2017, Berkeley, CA, USA, volume 67 of LIPIcs, pages 39:1-39:43. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.ITCS.2017.39.
  27. Yael Tauman Kalai and Ran Raz. Probabilistically checkable arguments. In Shai Halevi, editor, Advances in Cryptology - CRYPTO 2009, 29th Annual International Cryptology Conference, Santa Barbara, CA, USA, August 16-20, 2009. Proceedings, volume 5677 of Lecture Notes in Computer Science, pages 143-159. Springer, 2009. URL: http://dx.doi.org/10.1007/978-3-642-03356-8_9.
  28. Yael Tauman Kalai and Ron D. Rothblum. Arguments of proximity - [extended abstract]. In Rosario Gennaro and Matthew Robshaw, editors, Advances in Cryptology - CRYPTO 2015 - 35th Annual Cryptology Conference, Santa Barbara, CA, USA, August 16-20, 2015, Proceedings, Part II, volume 9216 of Lecture Notes in Computer Science, pages 422-442. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-662-48000-7_21.
  29. Carsten Lund, Lance Fortnow, Howard J. Karloff, and Noam Nisan. Algebraic methods for interactive proof systems. J. ACM, 39(4):859-868, 1992. URL: http://dx.doi.org/10.1145/146585.146605.
  30. Omer Reingold, Guy N. Rothblum, and Ron Rothblum. Efficient batch verification for UP. Electronic Colloquium on Computational Complexity (ECCC), 25:22, 2018. URL: https://eccc.weizmann.ac.il/report/2018/022.
  31. Omer Reingold, Guy N. Rothblum, and Ron D. Rothblum. Constant-round interactive proofs for delegating computation. In Daniel Wichs and Yishay Mansour, editors, Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, June 18-21, 2016, pages 49-62. ACM, 2016. URL: http://dx.doi.org/10.1145/2897518.2897652.
  32. Guy N. Rothblum, Salil P. Vadhan, and Avi Wigderson. Interactive proofs of proximity: delegating computation in sublinear time. In Dan Boneh, Tim Roughgarden, and Joan Feigenbaum, editors, Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 793-802. ACM, 2013. URL: http://dx.doi.org/10.1145/2488608.2488709.
  33. Adi Shamir. IP = PSPACE. J. ACM, 39(4):869-877, 1992. URL: http://dx.doi.org/10.1145/146585.146609.
  34. L. Valiant and V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47:85-93, 1986. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact