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Leakage-Resilient Hardness vs Randomness

Authors Yanyi Liu, Rafael Pass

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Author Details

Yanyi Liu
  • Cornell Tech, New York, NY, USA
Rafael Pass
  • Tel-Aviv University, Israel
  • Cornell Tech, New York, NY, USA

Funding

  • Liu, Yanyi: Work done while visiting Tel-Aviv University.
  • Pass, Rafael: Supported in part by NSF Award CNS 2149305, NSF Award SATC-1704788, NSF Award RI-1703846, AFOSR Award FA9550-18-1-0267, a JP Morgan Faculty Award, and an Algorand Foundation grant (MEGA-ACE). This material is based upon work supported by DARPA under Agreement No. HR00110C0086. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the United States Government or DARPA.

Cite AsGet BibTex

Yanyi Liu and Rafael Pass. Leakage-Resilient Hardness vs Randomness. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 32:1-32:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.CCC.2023.32

Abstract

A central open problem in complexity theory concerns the question of whether all efficient randomized algorithms can be simulated by efficient deterministic algorithms. The celebrated "hardness v.s. randomness” paradigm pioneered by Blum-Micali (SIAM JoC’84), Yao (FOCS’84) and Nisan-Wigderson (JCSS’94) presents hardness assumptions under which e.g., prBPP = prP (so-called "high-end derandomization), or prBPP ⊆ prSUBEXP (so-called "low-end derandomization), and more generally, under which prBPP ⊆ prDTIME(𝒞) where 𝒞 is a "nice" class (closed under composition with a polynomial), but these hardness assumptions are not known to also be necessary for such derandomization. In this work, following the recent work by Chen and Tell (FOCS’21) that considers "almost-all-input" hardness of a function f (i.e., hardness of computing f on more than a finite number of inputs), we consider "almost-all-input" leakage-resilient hardness of a function f - that is, hardness of computing f(x) even given, say, √|x| bits of leakage of f(x). We show that leakage-resilient hardness characterizes derandomization of prBPP (i.e., gives a both necessary and sufficient condition for derandomization), both in the high-end and in the low-end setting. In more detail, we show that there exists a constant c such that for every function T, the following are equivalent: - prBPP ⊆ prDTIME(poly(T(poly(n)))); - Existence of a poly(T(poly(n)))-time computable function f :{0,1}ⁿ → {0,1}ⁿ that is almost-all-input leakage-resilient hard with respect to n^c-time probabilistic algorithms. As far as we know, this is the first assumption that characterizes derandomization in both the low-end and the high-end regime. Additionally, our characterization naturally extends also to derandomization of prMA, and also to average-case derandomization, by appropriately weakening the requirements on the function f. In particular, for the case of average-case (a.k.a. "effective") derandomization, we no longer require the function to be almost-all-input hard, but simply satisfy the more standard notion of average-case leakage-resilient hardness (w.r.t., every samplable distribution), whereas for derandomization of prMA, we instead consider leakage-resilience for relations.

Subject Classification

ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • Derandomization
  • Leakage-Resilient Hardness

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References

  1. Adi Akavia, Shafi Goldwasser, and Vinod Vaikuntanathan. Simultaneous hardcore bits and cryptography against memory attacks. In Theory of cryptography conference, pages 474-495. Springer, 2009. Google Scholar
  2. László Babai, Lance Fortnow, Noam Nisan, and Avi Wigderson. BPP has subexponential time simulations unless EXPTIME has publishable proofs. Computational Complexity, 3:307-318, 1993. Google Scholar
  3. Manuel Blum and Silvio Micali. How to generate cryptographically strong sequences of pseudo-random bits. SIAM Journal on Computing, 13(4):850-864, 1984. Google Scholar
  4. Zvika Brakerski and Yael Tauman Kalai. A parallel repetition theorem for leakage resilience. In Theory of Cryptography Conference, pages 248-265. Springer, 2012. Google Scholar
  5. Lijie Chen, Ron D Rothblum, Roei Tell, and Eylon Yogev. On exponential-time hypotheses, derandomization, and circuit lower bounds. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 13-23. IEEE, 2020. Google Scholar
  6. Lijie Chen and Roei Tell. Hardness vs randomness, revised: Uniform, non-black-box, and instance-wise. Electronic Colloquium on Computational Complexity, 2021. URL: https://eccc.weizmann.ac.il/report/2021/080/l.
  7. Don Coppersmith. Small solutions to polynomial equations, and low exponent rsa vulnerabilities. Journal of cryptology, 10(4):233-260, 1997. Google Scholar
  8. Stefan Dziembowski and Krzysztof Pietrzak. Leakage-resilient cryptography. In FOCS, pages 293-302, 2008. Google Scholar
  9. 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. Google Scholar
  10. Oded Goldreich. Two comments on targeted canonical derandomizers. In Electron. Colloquium Comput. Complex., volume 18, page 47, 2011. Google Scholar
  11. Shuichi Hirahara. Non-disjoint promise problems from meta-computational view of pseudorandom generator constructions. In 35th Computational Complexity Conference (CCC 2020). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2020. Google Scholar
  12. R Impagliazzo and A Wigderson. Randomness vs. time: de-randomization under a uniform assumption. In Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No. 98CB36280), pages 734-743. IEEE, 1998. Google Scholar
  13. 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
  14. Russell Impagliazzo and Avi Wigderson. P = BPP if e requires exponential circuits: Derandomizing the xor lemma. In STOC '97, pages 220-229, 1997. Google Scholar
  15. Yuval Ishai, Amit Sahai, and David Wagner. Private circuits: Securing hardware against probing attacks. In Annual International Cryptology Conference, pages 463-481. Springer, 2003. Google Scholar
  16. Valentine Kabanets. Easiness assumptions and hardness tests: Trading time for zero error. Journal of Computer and System Sciences, 63(2):236-252, 2001. Google Scholar
  17. Oliver Korten. Derandomization from time-space tradeoffs. In 37th Computational Complexity Conference (CCC 2022). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2022. Google Scholar
  18. Yanyi Liu and Rafael Pass. Characterizing derandomization through hardness of levin-kolmogorov complexity. In CCC, 2022. Google Scholar
  19. Yanyi Liu and Rafael Pass. Leakage-resilient hardness vs randomness. Electronic Colloquium on Computational Complexity, 2022. URL: https://eccc.weizmann.ac.il/report/2022/113/.
  20. Ueli M Maurer. Factoring with an oracle. In Workshop on the Theory and Application of of Cryptographic Techniques, pages 429-436. Springer, 1992. Google Scholar
  21. Silvio Micali and Leonid Reyzin. Physically observable cryptography. In Theory of Cryptography Conference, pages 278-296. Springer, 2004. Google Scholar
  22. Cody Murray and Ryan Williams. Circuit lower bounds for nondeterministic quasi-polytime: an easy witness lemma for np and nqp. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pages 890-901, 2018. Google Scholar
  23. Noam Nisan. Pseudorandom bits for constant depth circuits. Combinatorica, 11(1):63-70, 1991. Google Scholar
  24. Noam Nisan and Avi Wigderson. Hardness vs randomness. J. Comput. Syst. Sci., 49(2):149-167, 1994. Google Scholar
  25. Rafael Pass. Unprovability of leakage-resilient cryptography beyond the information-theoretic limit. In SCN, 2020. Google Scholar
  26. Ronald L Rivest and Adi Shamir. Efficient factoring based on partial information. In Workshop on the Theory and Application of of Cryptographic Techniques, pages 31-34. Springer, 1985. Google Scholar
  27. 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
  28. Roei Tell. Proving that prBPP= prP is as hard as proving that "almost NP" is not contained in P/poly. Information Processing Letters, 152:105841, 2019. Google Scholar
  29. Salil P Vadhan. Pseudorandomness. Foundations and Trendsregistered in Theoretical Computer Science, 7(1-3):1-336, 2012. Google Scholar
  30. Andrew Chi-Chih Yao. Theory and applications of trapdoor functions (extended abstract). In 23rd Annual Symposium on Foundations of Computer Science, Chicago, Illinois, USA, 3-5 November 1982, pages 80-91, 1982. Google Scholar
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