Document Open Access Logo

Derandomization with Minimal Memory Footprint

Authors Dean Doron , Roei Tell

PDF

File

Thumbnail PDF
PDF
LIPIcs.CCC.2023.11.pdf
  • Filesize: 0.8 MB
  • 15 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity theory and logic
  • Theory of computation → Pseudorandomness and derandomization
  • Theory of computation → Error-correcting codes
Keywords
  • derandomization
  • space-bounded computation
  • catalytic space

Metrics

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

Abstract

Existing proofs that deduce BPL = 𝐋 from circuit lower bounds convert randomized algorithms into deterministic algorithms with large constant overhead in space. We study space-bounded derandomization with minimal footprint, and ask what is the minimal possible space overhead for derandomization. We show that BPSPACE[S] ⊆ DSPACE[c ⋅ S] for c ≈ 2, assuming space-efficient cryptographic PRGs, and, either: (1) lower bounds against bounded-space algorithms with advice, or: (2) lower bounds against certain uniform compression algorithms. Under additional assumptions regarding the power of catalytic computation, in a new setting of parameters that was not studied before, we are even able to get c ≈ 1.
Our results are constructive: Given a candidate hard function (and a candidate cryptographic PRG) we show how to transform the randomized algorithm into an efficient deterministic one. This follows from new PRGs and targeted PRGs for space-bounded algorithms, which we combine with novel space-efficient evaluation methods. A central ingredient in all our constructions is hardness amplification reductions in logspace-uniform TC⁰, that were not known before.

Cite As Get BibTex

Dean Doron and Roei Tell. Derandomization with Minimal Memory Footprint. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.CCC.2023.11

Author Details

Dean Doron
  • Ben-Gurion University of the Negev, Beer-Sheva, Israel
Roei Tell
  • The Institute for Advanced Study at Princeton, NJ, USA
  • DIMACS Center at Rutgers University, Piscataway, NJ, USA

Funding

The second author is supported in part by the National Science Foundation under grant number CCF-1900460.

Acknowledgements

We are grateful to Avi Wigderson for several useful conversations regarding the gap between double space blow-up and single space blow-up. We thank Lijie Chen for suggesting the idea of using catalytic space to save on complexity, early in this work, and for pointing out a gap in a previous version of the proof of Theorem 2. We also thank Ian Mertz for several useful conversations exploring the abilities of catalytic space. We are very grateful to an anonymous reviewer for a careful read of this paper and for many useful comments. Part of this work was done while the second author was visiting the Simons Institute for the Theory of Computing.

References

  1. Benny Applebaum. Cryptography in constant parallel time. Information Security and Cryptography. Springer, 2014. Google Scholar
  2. Benny Applebaum, Yuval Ishai, and Eyal Kushilevitz. Cryptography in NC⁰. SIAM Journal on Computing, 36(4):845-888, 2006. Google Scholar
  3. Benny Applebaum and Pavel Raykov. Fast pseudorandom functions based on expander graphs. In Theory of Cryptography. Part I, volume 9985, pages 27-56. Springer, 2016. Google Scholar
  4. Sanjeev Arora and Boaz Barak. Computational complexity: A modern approach. Cambridge University Press, Cambridge, 2009. Google Scholar
  5. Allan Borodin, Stephen Cook, and Nicholas Pippenger. Parallel computation for well-endowed rings and space-bounded probabilistic machines. Information and Control, 58(1-3):113-136, 1983. Google Scholar
  6. Harry Buhrman, Richard Cleve, Michal Koucký, Bruno Loff, and Florian Speelman. Computing with a full memory: catalytic space. In Proc. 46th Annual ACM Symposium on Theory of Computing (STOC), pages 857-866, 2014. Google Scholar
  7. Lijie Chen. Non-deterministic quasi-polynomial time is average-case hard for ACC circuits. In Proc. 60th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2019. Google Scholar
  8. Lijie Chen, Xin Lyu, and R. Ryan Williams. Almost-everywhere circuit lower bounds from non-trivial derandomization. In Proc. 61st Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2020. Google Scholar
  9. Lijie Chen, Ron D. Rothblum, and Roei Tell. Unstructured hardness to average-case randomness. In Proc. 63rd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 429-437. IEEE, 2022. Google Scholar
  10. Lijie Chen and Roei Tell. Hardness vs. randomness, revised: uniform, non-black-box, and instance-wise. In Proc. 62nd Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2021. Google Scholar
  11. Lijie Chen and Roei Tell. Simple and fast derandomization from very hard functions: Eliminating randomness at almost no cost. In Proc. 53st Annual ACM Symposium on Theory of Computing (STOC), 2021. Google Scholar
  12. Lijie Chen and Roei Tell. When arthur has neither random coins nor time to spare: Superfast derandomization of proof systems. Electronic Colloquium on Computational Complexity: ECCC, 2022. Google Scholar
  13. Dean Doron, Dana Moshkovitz, Justin Oh, and David Zuckerman. Nearly optimal pseudorandomness from hardness. In 61st Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2020. Google Scholar
  14. Dean Doron and Roei Tell. Derandomization with minimal memory footprint. Electronic Colloquium on Computational Complexity: ECCC, 30:036, 2023. Google Scholar
  15. Oded Goldreich. Computational complexity: A conceptual perspective. Cambridge University Press, New York, NY, USA, 2008. Google Scholar
  16. Oded Goldreich. Candidate one-way functions based on expander graphs. In Studies in Complexity and Cryptography. Miscellanea on the Interplay between Randomness and Computation, pages 76-87. Springer, 2011. Google Scholar
  17. Oded Goldreich. In a world of P = BPP. In Studies in Complexity and Cryptography. Miscellanea on the Interplay Randomness and Computation, pages 191-232. Springer, 2011. Google Scholar
  18. Oded Goldreich and Avi Wigderson. Derandomization that is rarely wrong from short advice that is typically good. In Proc. 6th International Workshop on Randomization and Approximation Techniques in Computer Science (RANDOM), pages 209-223, 2002. Google Scholar
  19. Shafi Goldwasser, Dan Gutfreund, Alexander Healy, Tali Kaufman, and Guy N. Rothblum. Verifying and decoding in constant depth. In Proc. 39th Annual ACM Symposium on Theory of Computing (STOC), pages 440-449, 2007. Google Scholar
  20. Aryeh Grinberg, Ronen Shaltiel, and Emanuele Viola. Indistinguishability by adaptive procedures with advice, and lower bounds on hardness amplification proofs. In Proc. 59th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 956-966, 2018. Google Scholar
  21. Venkatesan Guruswami and Valentine Kabanets. Hardness amplification via space-efficient direct products. Computational Complexity, 17(4):475-500, 2008. Google Scholar
  22. Venkatesan Guruswami, Christopher Umans, and Salil Vadhan. Unbalanced expanders and randomness extractors from Parvaresh-Vardy codes. Journal of the ACM, 56(4):Art. 20, 34, 2009. Google Scholar
  23. William M. Hoza. Better pseudodistributions and derandomization for space-bounded computation. In Proc. 25th International Workshop on Randomization and Approximation Techniques in Computer Science (RANDOM), pages 28:1-28:23, 2021. Google Scholar
  24. Russel Impagliazzo and Avi Wigderson. Randomness vs. time: Derandomization under a uniform assumption. Journal of Computer and System Sciences, 63(4):672-688, 2001. Google Scholar
  25. 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. Google Scholar
  26. Russell Impagliazzo and Avi Wigderson. P = BPP if E requires exponential circuits: derandomizing the XOR lemma. In Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, STOC '97, page 220–229. Association for Computing Machinery, 1997. Google Scholar
  27. Yuval Ishai, Eyal Kushilevitz, Rafail Ostrovsky, and Amit Sahai. Cryptography with constant computational overhead. In Proc. 40th Annual ACM Symposium on Theory of Computing (STOC), pages 433-442, 2008. Google Scholar
  28. Valentine Kabanets and Russell Impagliazzo. Derandomizing polynomial identity tests means proving circuit lower bounds. Computational Complexity, 13(1-2):1-46, 2004. Google Scholar
  29. Itay Kalev and Amnon Ta-Shma. Unbalanced expanders from multiplicity codes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022), volume 245 of Leibniz International Proceedings in Informatics (LIPIcs), pages 12:1-12:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. Google Scholar
  30. Jeff Kinne, Dieter van Melkebeek, and Ronen Shaltiel. Pseudorandom generators, typically-correct derandomization, and circuit lower bounds. Computational Complexity, 21(1):3-61, 2012. Google Scholar
  31. Adam 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. Google Scholar
  32. Yanyi Liu and Rafael Pass. Characterizing derandomization through hardness of Levin-Kolmogorov complexity. In 37th Computational Complexity Conference (CCC 2022), volume 234 of Leibniz International Proceedings in Informatics (LIPIcs), pages 35:1-35:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. Google Scholar
  33. Yanyi Liu and Rafael Pass. Leakage-resilient hardness vs. randomness. Electronic Colloquium on Computational Complexity: ECCC, 30:113, 2022. Google Scholar
  34. Cody Murray and R. Ryan Williams. Circuit lower bounds for nondeterministic quasi-polytime: An easy witness lemma for NP and NQP. In Proc. 50th Annual ACM Symposium on Theory of Computing (STOC), 2018. Google Scholar
  35. Noam Nisan. Pseudorandom generators for space-bounded computation. Combinatorica, 12(4):449-461, 1992. Google Scholar
  36. Noam Nisan. RL ⊆ SC. Computational Complexity, 4(1):1-11, 1994. Google Scholar
  37. Noam Nisan and Avi Wigderson. Hardness vs. randomness. Journal of Computer and System Sciences, 49(2):149-167, 1994. Google Scholar
  38. Hanlin Ren and Rahul Santhanam. Hardness of KT characterizes parallel cryptography. In 36th Computational Complexity Conference (CCC 2021), volume 200 of Leibniz International Proceedings in Informatics (LIPIcs), pages 35:1-35:58. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  39. Michael E. Saks and Shiyu Zhou. BP_HSPACE(S) ⊆ DSPACE(S^2/3). Journal of Computer and System Sciences, 58(2):376-403, 1999. Google Scholar
  40. Walter J. Savitch. Relationships between nondeterministic and deterministic tape complexities. Journal of Computer and System Sciences, 4(2):177-192, 1970. Google Scholar
  41. Ronen Shaltiel and Emanuele Viola. Hardness amplification proofs require majority. In Proc. 40th Annual ACM Symposium on Theory of Computing (STOC), pages 589-598, 2008. Google Scholar
  42. Daniel A. Spielman. Linear-time encodable and decodable error-correcting codes. toit, 42(6):1723-1731, 1996. Codes and complexity. Google Scholar
  43. Madhu Sudan, Luca Trevisan, and Salil Vadhan. Pseudorandom generators without the XOR lemma. Journal of Computer and System Sciences, 62(2):236-266, 2001. Google Scholar
  44. Roei Tell. Proving that prBPP = P is as hard as proving that "almost NP" is not contained in P/poly. Information Processing Letters, 152:105841, 2019. Google Scholar
  45. Luca Trevisan and Salil Vadhan. Pseudorandomness and average-case complexity via uniform reductions. ç, 16(4):331-364, 2007. Google Scholar
  46. Emanuele Viola. Hardness vs. randomness within alternating time. In çc18th, pages 53-69, 2003. Google Scholar
  47. R. Ryan Williams. Non-uniform acc circuit lower bounds. In 2011 IEEE 26th Annual Conference on Computational Complexity, pages 115-125, 2011. 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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact