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On Hardness Assumptions Needed for "Extreme High-End" PRGs and Fast Derandomization

Authors Ronen Shaltiel, Emanuele Viola

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Ronen Shaltiel
  • Department of computer science, University of Haifa, Israel
Emanuele Viola
  • Khoury College of Computer Sciences, Northeastern University, Boston, MA, USA


We thank anonymous referees for very helpful comments and suggestions.

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Ronen Shaltiel and Emanuele Viola. On Hardness Assumptions Needed for "Extreme High-End" PRGs and Fast Derandomization. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 116:1-116:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


The hardness vs. randomness paradigm aims to explicitly construct pseudorandom generators G:{0,1}^r → {0,1}^m that fool circuits of size m, assuming the existence of explicit hard functions. A "high-end PRG" with seed length r = O(log m) (implying BPP=P) was achieved in a seminal work of Impagliazzo and Wigderson (STOC 1997), assuming the high-end hardness assumption: there exist constants 0 < β < 1 < B, and functions computable in time 2^{B ⋅ n} that cannot be computed by circuits of size 2^{β ⋅ n}. Recently, motivated by fast derandomization of randomized algorithms, Doron et al. (FOCS 2020) and Chen and Tell (STOC 2021), construct "extreme high-end PRGs" with seed length r = (1+o(1))⋅ log m, under qualitatively stronger assumptions. We study whether extreme high-end PRGs can be constructed from the corresponding hardness assumption in which β = 1-o(1) and B = 1+o(1), which we call the extreme high-end hardness assumption. We give a partial negative answer: - The construction of Doron et al. composes a PEG (pseudo-entropy generator) with an extractor. The PEG is constructed starting from a function that is hard for MA-type circuits. We show that black-box PEG constructions from the extreme high-end hardness assumption must have large seed length (and so cannot be used to obtain extreme high-end PRGs by applying an extractor). To prove this, we establish a new property of (general) black-box PRG constructions from hard functions: it is possible to fix many output bits of the construction while fixing few bits of the hard function. This property distinguishes PRG constructions from typical extractor constructions, and this may explain why it is difficult to design PRG constructions. - The construction of Chen and Tell composes two PRGs: G₁:{0,1}^{(1+o(1)) ⋅ log m} → {0,1}^{r₂ = m^{Ω(1)}} and G₂:{0,1}^{r₂} → {0,1}^m. The first PRG is constructed from the extreme high-end hardness assumption, and the second PRG needs to run in time m^{1+o(1)}, and is constructed assuming one way functions. We show that in black-box proofs of hardness amplification to 1/2+1/m, reductions must make Ω(m) queries, even in the extreme high-end. Known PRG constructions from hard functions are black-box and use (or imply) hardness amplification, and so cannot be used to construct a PRG G₂ from the extreme high-end hardness assumption. The new feature of our hardness amplification result is that it applies even to the extreme high-end setting of parameters, whereas past work does not. Our techniques also improve recent lower bounds of Ron-Zewi, Shaltiel and Varma (ITCS 2021) on the number of queries of local list-decoding algorithms.

Subject Classification

ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
  • Complexity Theory
  • Derandomization
  • Pseudorandom generators
  • Black-box proofs


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  1. E. Allender, H. Buhrman, M. Koucký, D. van Melkebeek, and D. Ronneburger. Power from random strings. SIAM J. Comput., 35(6):1467-1493, 2006. URL:
  2. B. Applebaum, S. Artemenko, R. Shaltiel, and G. Yang. Incompressible functions, relative-error extractors, and the power of nondeterministic reductions. Computational Complexity, 25(2):349-418, 2016. URL:
  3. S. Artemenko, R. Impagliazzo, V. Kabanets, and R. Shaltiel. Pseudorandomness when the odds are against you. In 31st Conference on Computational Complexity, pages 9:1-9:35, 2016. URL:
  4. S. Artemenko and R. Shaltiel. Lower bounds on the query complexity of non-uniform and adaptive reductions showing hardness amplification. Computational Complexity, 23(1):43-83, 2014. URL:
  5. L. Babai, L. Fortnow, N. Nisan, and A. Wigderson. Bpp has subexponential time simulations unless exptime has publishable proofs. Computational Complexity, 3:307-318, 1993. Google Scholar
  6. B. Barak, R. Shaltiel, and A. Wigderson. Computational analogues of entropy. In RANDOM-APPROX, pages 200-215, 2003. URL:
  7. M. Blum and S. Micali. How to generate cryptographically strong sequences of pseudo-random bits. SIAM Journal on Computing, 13(4):850-864, November 1984. Google Scholar
  8. M. L. Carmosino, J. Gao, R. Impagliazzo, I. Mihajlin, R. Paturi, and S. Schneider. Nondeterministic extensions of the strong exponential time hypothesis and consequences for non-reducibility. In Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science, ITCS, pages 261-270, 2016. URL:
  9. M. L. Carmosino, R. Impagliazzo, V. Kabanets, and A. Kolokolova. Learning algorithms from natural proofs. In Ran Raz, editor, 31st Conference on Computational Complexity, CCC, volume 50 of LIPIcs, pages 10:1-10:24. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL:
  10. L. Chen and R. Tell. Hardness vs randomness, revised: Uniform, non-black-box, and instance-wise. Electron. Colloquium Comput. Complex., 28:80, 2021. URL:
  11. L. Chen and R. Tell. Simple and fast derandomization from very hard functions: eliminating randomness at almost no cost. In STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 283-291. ACM, 2021. URL:
  12. D. Doron, D. Moshkovitz, J. Oh, and D. Zuckerman. Nearly optimal pseudorandomness from hardness. In 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS, pages 1057-1068. IEEE, 2020. URL:
  13. O. Goldreich and A. Wigderson. On derandomizing algorithms that err extremely rarely. In Symposium on Theory of Computing, STOC, pages 109-118. ACM, 2014. URL:
  14. S. Goldwasser and S. Micali. Probabilistic encryption. Journal of Computer and System Sciences, 28(2):270-299, April 1984. Preliminary version appeared in STOC' 82. Google Scholar
  15. A. Grinberg, R. Shaltiel, and E. Viola. Indistinguishability by adaptive procedures with advice, and lower bounds on hardness amplification proofs. In 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS, pages 956-966, 2018. URL:
  16. J. Håstad, R. Impagliazzo, L. A. Levin, and M. Luby. A pseudorandom generator from any one-way function. SIAM J. Comput., 28(4):1364-1396, 1999. Google Scholar
  17. S. Hirahara. Non-black-box worst-case to average-case reductions within NP. In 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS, pages 247-258, 2018. URL:
  18. R. Impagliazzo. Hard-core distributions for somewhat hard problems. In FOCS, pages 538-545, 1995. Google Scholar
  19. R. Impagliazzo, R. Shaltiel, and A. Wigderson. Near-optimal conversion of hardness into pseudo-randomness. In FOCS, pages 181-190, 1999. URL:
  20. R. Impagliazzo, R. Shaltiel, and A. Wigderson. Reducing the seed length in the nisan-wigderson generator. Combinatorica, 26(6):647-681, 2006. URL:
  21. R. Impagliazzo and A. Wigderson. P = BPP if E requires exponential circuits: Derandomizing the XOR lemma. In STOC, pages 220-229, 1997. Google Scholar
  22. A. Klivans and D. van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. SIAM J. Comput., 31(5):1501-1526, 2002. URL:
  23. P. Bro Miltersen and N. V. Vinodchandran. Derandomizing arthur-merlin games using hitting sets. Computational Complexity, 14(3):256-279, 2005. URL:
  24. N. Nisan and A. Wigderson. Hardness vs. randomness. JCSS: Journal of Computer and System Sciences, 49, 1994. Google Scholar
  25. J. Radhakrishnan and A. Ta-Shma. Bounds for dispersers, extractors, and depth-two superconcentrators. SIAM Journal on Discrete Mathematics, 13(1):2-24, 2000. Google Scholar
  26. N. Ron-Zewi, R. Shaltiel, and Nithin Varma. Query complexity lower bounds for local list-decoding and hard-core predicates (even for small rate and huge lists). In 12th Innovations in Theoretical Computer Science Conference, ITCS, volume 185 of LIPIcs, pages 33:1-33:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL:
  27. R. Shaltiel and C. Umans. Simple extractors for all min-entropies and a new pseudorandom generator. J. ACM, 52(2):172-216, 2005. URL:
  28. R. Shaltiel and C. Umans. Pseudorandomness for approximate counting and sampling. Computational Complexity, 15(4):298-341, 2006. URL:
  29. R. Shaltiel and E. Viola. Hardness amplification proofs require majority. SIAM J. Comput., 39(7):3122-3154, 2010. URL:
  30. R. Shaltiel and E. Viola. On hardness assumptions needed for "extreme high-end" prgs and fast derandomization. Electron. Colloquium Comput. Complex., page 43, 2021. Google Scholar
  31. M. Sudan, L. Trevisan, and S. P. Vadhan. Pseudorandom generators without the xor lemma. J. Comput. Syst. Sci., 62(2):236-266, 2001. Google Scholar
  32. A. Ta-Shma and D. Zuckerman. Extractor codes. IEEE Trans. Information Theory, 50(12):3015-3025, 2004. Google Scholar
  33. R. Tell. How to find water in the ocean: A survey on quantified derandomization. Electron. Colloquium Comput. Complex., page 120, 2021. URL:
  34. L. Trevisan. Extractors and pseudorandom generators. Journal of the ACM, 48, 2001. Google Scholar
  35. C. Umans. Pseudo-random generators for all hardnesses. Journal of Computer and System Sciences, 67:419-440, 2003. Google Scholar
  36. C. Umans. Reconstructive dispersers and hitting set generators. Algorithmica, 55(1):134-156, 2009. URL:
  37. E. Viola. The complexity of constructing pseudorandom generators from hard functions. Computational Complexity, 13(3-4):147-188, 2005. URL:
  38. E. Viola. The Complexity of Hardness Amplification and Derandomization. PhD thesis, Harvard University, 2006. Google Scholar
  39. A. C. Yao. Theory and applications of trapdoor functions (extended abstract). In FOCS, pages 80-91, 1982. Google Scholar
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