Leakage Resilience, Targeted Pseudorandom Generators, and Mild Derandomization of Arthur-Merlin Protocols

Authors Dieter van Melkebeek, Nicollas Mocelin Sdroievski



PDF
Thumbnail PDF

File

LIPIcs.FSTTCS.2023.29.pdf
  • Filesize: 0.74 MB
  • 22 pages

Document Identifiers

Author Details

Dieter van Melkebeek
  • University of Wisconsin-Madison, WI, USA
Nicollas Mocelin Sdroievski
  • University of Wisconsin-Madison, WI, USA

Cite As Get BibTex

Dieter van Melkebeek and Nicollas Mocelin Sdroievski. Leakage Resilience, Targeted Pseudorandom Generators, and Mild Derandomization of Arthur-Merlin Protocols. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 29:1-29:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.FSTTCS.2023.29

Abstract

Many derandomization results for probabilistic decision processes have been ported to the setting of Arthur-Merlin protocols. Whereas the ultimate goal in the first setting consists of efficient simulations on deterministic machines (BPP vs. P problem), in the second setting it is efficient simulations on nondeterministic machines (AM vs. NP problem). Two notable exceptions that have not yet been ported from the first to the second setting are the equivalence between whitebox derandomization and leakage resilience (Liu and Pass, 2023), and the equivalence between whitebox derandomization and targeted pseudorandom generators (Goldreich, 2011). We develop both equivalences for mild derandomizations of Arthur-Merlin protocols, i.e., simulations on Σ₂-machines. Our techniques also apply to natural simulation models that are intermediate between nondeterministic machines and Σ₂-machines.

Subject Classification

ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • Hardness versus randomness tradeoff
  • leakage resilience
  • Arthur-Merlin protocol
  • targeted hitting set generator

Metrics

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

References

  1. Leonard Adleman. Two theorems on random polynomial time. In Symposium on Foundations of Computer Science (FOCS), pages 75-83, 1978. URL: https://doi.org/10.1109/SFCS.1978.37.
  2. Barış Aydınlıoğlu and Dieter van Melkebeek. Nondeterministic circuit lower bounds from mildly derandomizing Arthur-Merlin games. Computational Complexity, 26(1):79-118, 2017. URL: https://doi.org/10.1007/s00037-014-0095-y.
  3. László Babai, Lance Fortnow, Noam Nisan, and Avi Wigderson. BPP has subexponential time simulations unless EXPTIME has publishable proofs. Computational Complexity, 3(4):307-318, 1993. URL: https://doi.org/10.1007/BF01275486.
  4. Venkatesan T. Chakaravarthy and Sambuddha Roy. Arthur and Merlin as oracles. Computational Complexity, 20(3):505-558, 2011. URL: https://doi.org/10.1007/s00037-011-0015-3.
  5. Lijie Chen and Roei Tell. Hardness vs randomness, revised: Uniform, non-black-box, and instance-wise. In Symposium on Foundations of Computer Science (FOCS), pages 125-136, 2022. URL: https://doi.org/10.1109/FOCS52979.2021.00021.
  6. Joan Feigenbaum and Lance Fortnow. Random-self-reducibility of complete sets. SIAM Journal on Computing, 22(5):994-1005, 1993. URL: https://doi.org/10.1137/0222061.
  7. Martin Furer, Oded Goldreich, Yishay Mansour, Michael Sipser, and Stahis Zachos. On completeness and soundness in interactive proof systems. Advances in Computing Research, 5:429-442, 1989. Google Scholar
  8. Oded Goldreich. In a world of P=BPP. In Studies in Complexity and Cryptography. Miscellanea on the Interplay between Randomness and Computation, pages 191-232. Springer, 2011. Part of the Lecture Notes in Computer Science book series (LNCS, volume 6650). URL: https://doi.org/10.1007/978-3-642-22670-0_20.
  9. Oded Goldreich. Two comments on targeted canonical derandomizers. In Computational Complexity and Property Testing: On the Interplay Between Randomness and Computation, pages 24-35. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-43662-9_4.
  10. Dan Gutfreund, Ronen Shaltiel, and Amnon Ta-Shma. Uniform hardness versus randomness tradeoffs for Arthur-Merlin games. Computational Complexity, 12(3):85-130, 2003. URL: https://doi.org/10.1007/s00037-003-0178-7.
  11. Russell Impagliazzo, Valentine Kabanets, and Avi Wigderson. In search of an easy witness: Exponential time vs. probabilistic polynomial time. Journal of Computer and System Sciences, 65(4):672-694, 2002. URL: https://doi.org/10.1016/S0022-0000(02)00024-7.
  12. Russell Impagliazzo and Avi Wigderson. P = BPP if E requires exponential circuits: Derandomizing the XOR lemma. In Symposium on Theory of Computing (STOC), page 220–229, 1997. URL: https://doi.org/10.1145/258533.258590.
  13. Adam R. Klivans and Dieter van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. SIAM Journal on Computing, 31(5):1501-1526, 2002. URL: https://doi.org/10.1137/S0097539700389652.
  14. Yanyi Liu and Rafael Pass. Characterizing derandomization through hardness of Levin-Kolmogorov complexity. In Computational Complexity Conference (CCC), volume 234, pages 35:1-35:17, 2022. URL: https://doi.org/10.4230/LIPIcs.CCC.2022.35.
  15. Yanyi Liu and Rafael Pass. Leakage-Resilient Hardness vs Randomness. In Computational Complexity Conference (CCC), volume 264, pages 32:1-32:20, 2023. URL: https://doi.org/10.4230/LIPIcs.CCC.2023.32.
  16. Peter Bro Miltersen and N. V. Vinodchandran. Derandomizing Arthur-Merlin games using hitting sets. Computational Complexity, 14(3):256-279, 2005. URL: https://doi.org/10.1007/s00037-005-0197-7.
  17. Noam Nisan and Avi Wigderson. Hardness vs randomness. Journal of Computer and System Sciences, 49(2):149-167, 1994. URL: https://doi.org/10.1016/S0022-0000(05)80043-1.
  18. Ronen Shaltiel and Christopher Umans. Simple extractors for all min-entropies and a new pseudorandom generator. Journal of the ACM, 52(2):172-216, 2005. URL: https://doi.org/10.1145/1059513.1059516.
  19. Ronen Shaltiel and Christopher Umans. Pseudorandomness for approximate counting and sampling. Computational Complexity, 15(4):298-341, 2006. URL: https://doi.org/10.1007/s00037-007-0218-9.
  20. Ronen Shaltiel and Christopher Umans. Low-end uniform hardness versus randomness tradeoffs for AM. SIAM Journal on Computing, 39(3):1006-1037, 2009. URL: https://doi.org/10.1137/070698348.
  21. Madhu Sudan, Luca Trevisan, and Salil Vadhan. Pseudorandom Generators without the XOR Lemma. Journal of Computer and System Sciences, 62(2):236-266, 2001. URL: https://doi.org/10.1006/jcss.2000.1730.
  22. Christopher Umans. Pseudo-random generators for all hardnesses. Journal of Computer and System Sciences, 67(2):419-440, 2003. URL: https://doi.org/10.1016/S0022-0000(03)00046-1.
  23. Dieter van Melkebeek and Nicollas Mocelin Sdroievski. Instance-wise hardness versus randomness tradeoffs for Arthur-Merlin protocols. In Computational Complexity Conference (CCC), volume 264, pages 17:1-17:36, 2023. URL: https://doi.org/10.4230/LIPIcs.CCC.2023.17.
  24. R. Ryan Williams. Natural proofs versus derandomization. SIAM Journal on Computing, 45(2):497-529, 2016. URL: https://doi.org/10.1137/130938219.
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