Abstract
The classic Impagliazzo–Nisan–Wigderson (INW) pseudorandom generator (PRG) (STOC ‘94) for space-bounded computation uses a seed of length \(O(\log n \cdot \log (nw/\varepsilon )+\log d)\) to fool ordered branching programs of length n, width w, and alphabet size d to within error \(\varepsilon \). A series of works have shown that the analysis of the INW generator can be improved for the class of permutation branching programs or the more general regular branching programs, improving the \(O(\log ^2 n)\) dependence on the length n to \(O(\log n)\) or \(\tilde{O}(\log n)\). However, when also considering the dependence on the other parameters, these analyses still fall short of the optimal PRG seed length \(O(\log (nwd/\varepsilon ))\).
In this paper, we prove that any “spectral analysis” of the INW generator requires seed length
to fool ordered permutation branching programs of length n, width w, and alphabet size d to within error \(\varepsilon \). By “spectral analysis” we mean an analysis of the INW generator that relies only on the spectral expansion of the graphs used to construct the generator; this encompasses all prior analyses of the INW generator. Our lower bound matches the upper bound of Braverman–Rao–Raz–Yehudayoff (FOCS 2010, SICOMP 2014) for regular branching programs of alphabet size \(d=2\) except for a gap between their \(O(\log n \cdot \log \log n)\) term and our \(O(\log n \cdot \log \log \min \{n,d\})\) term. It also matches the upper bounds of Koucký–Nimbhorkar–Pudlák (STOC 2011), De (CCC 2011), and Steinke (ECCC 2012) for constant-width (\(w=O(1)\)) permutation branching programs of alphabet size \(d=2\) to within a constant factor.
To fool permutation branching programs in the measure of spectral norm, we prove that any spectral analysis of the INW generator requires a seed of length \(\varOmega (\log n\cdot \log \log n+\log n\cdot \log (1/\varepsilon )+\log d)\) when the width is at least polynomial in n (\(w=n^{\varOmega (1)}\)), matching the recent upper bound of Hoza–Pyne–Vadhan (ITCS ‘21) to within a constant factor.
Full Version [PV21a]: https://eccc.weizmann.ac.il/report/2021/108/.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Braverman, M., Cohen, G., Garg, S.: Hitting sets with near-optimal error for read-once branching programs. In: Diakonikolas, I., Kempe, D., Henzinger, M. (eds.) Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, 25–29 June 2018, pp. 353–362. ACM (2018)
Babai, L., Nisan, N., Szegedy, M.: Multiparty protocols, pseudorandom generators for logspace, and time-space trade-offs. J. Comput. Syst. Sci. 45(2), 204–232 (1992). Twenty-first Symposium on the Theory of Computing (Seattle, WA, 1989)
Braverman, M., Rao, A., Raz, R., Yehudayoff, A.: Pseudorandom generators for regular branching programs. In: FOCS [IEE10], pp. 40–47 (2010)
Brody, J., Verbin, E.: The coin problem and pseudorandomness for branching programs. In: FOCS [IEE10], pp. 30–39 (2010)
Cohen, G., Doron, D., Renard, O., Sberlo, O., Ta-Shma, A.: Error reduction for weighted PRGs against read once branching programs. In: Kabanets, V. (ed.) 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Dagstuhl, Germany, vol. 200, pp. 22:1–22:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
Chattopadhyay, E., Liao, J.-J.: Optimal error pseudodistributions for read-once branching programs. In: Saraf, S. (ed.) 35th Computational Complexity Conference, CCC 2020, Saarbrücken, Germany, 28–31 July 2020, (Virtual Conference). LIPIcs, vol. 169, pp. 25:1–25:27. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)
De, A: Pseudorandomness for permutation and regular branching programs. In: IEEE Conference on Computational Complexity, pp. 221–231. IEEE Computer Society (2011)
Goldreich, O., Wigderson, A.: Tiny families of functions with random properties: a quality-size trade-off for hashing. Random Struct. Algorithms 11(4), 315–343 (1997)
Haitner, I., Harnik, D., Reingold, O.: On the power of the randomized iterate. SIAM J. Comput. 40(6), 1486–1528 (2011)
Hoza, W.: Better pseudodistributions and derandomization for space-bounded computation. ECCC preprint TR21-019 (2021)
Hoza, W.M., Pyne, E., Vadhan, S.P.: Pseudorandom generators for unbounded-width permutation branching programs. In: Lee, J.R. (ed.) 12th Innovations in Theoretical Computer Science Conference, ITCS 2021, 6–8 January 2021, Virtual Conference. LIPIcs, vol. 185, pp. 7:1–7:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
Healy, A., Vadhan, S., Viola, E.: Using nondeterminism to amplify hardness. SIAM J. Comput. 35(4), 903–931 (electronic) (2006)
IEEE. 51th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2010, Las Vegas, Nevada, USA, 23–26 October 2010. IEEE Computer Society (2010)
Indyk, P.: Stable distributions, pseudorandom generators, embeddings, and data stream computation. J. ACM 53(3), 307–323 (2006)
Impagliazzo, R., Nisan, N., Wigderson, A.: Pseudorandomness for network algorithms. In: Proceedings of the Twenty-Sixth Annual ACM Symposium on the Theory of Computing, Montréal, Québec, Canada, 23–25 May 1994, pp. 356–364 (1994)
Koucký, M., Nimbhorkar, P., Pudlák, P.: Pseudorandom generators for group products: extended abstract. In: Fortnow, L., Vadhan, S.P. (eds.) STOC, pp. 263–272. ACM (2011)
Kaplan, E., Naor, M., Reingold, O.: Derandomized constructions of k-Wise (almost) independent permutations. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds.) APPROX/RANDOM -2005. LNCS, vol. 3624, pp. 354–365. Springer, Heidelberg (2005). https://doi.org/10.1007/11538462_30
Meka, R., Reingold, O., Tal, A.: Pseudorandom generators for width-3 branching programs. In: Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pp. 626–637. ACM (2019)
Nisan, N.: Pseudorandom generators for space-bounded computation. Combinatorica 12(4), 449–461 (1992)
Nisan, N., Zuckerman, D.: Randomness is linear in space. J. Comput. Syst. Sci. 52(1), 43–52 (1996)
Pyne, E., Vadhan, S.: Limitations of the Impagliazzo-Nisan-Wigderson Pseudorandom Generator against Permutation Branching Programs. ECCC preprint TR21-108 (2021)
Pyne, E., Vadhan, S.: Pseudodistributions that beat all pseudorandom generators (extended abstract). In: Kabanets, V (ed.) 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Dagstuhl, Germany, vol. 200, pp. 33:1–33:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
Reingold, O.: Undirected connectivity in log-space. J. ACM 55(4), 24, Article no. 17 (2008)
Rozenman, E., Vadhan, S.: Derandomized squaring of graphs. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds.) APPROX/RANDOM 2005. LNCS, vol. 3624, pp. 436–447. Springer, Heidelberg (2005). https://doi.org/10.1007/11538462_37
Sivakumar, D.: Algorithmic derandomization via complexity theory. In: Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing, (electronic), pp. 619–626. ACM, New York (2002)
Steinke, T.: Pseudorandomness for permutation branching programs without the group theory. Technical Report TR12-083, Electronic Colloquium on Computational Complexity (ECCC), July 2012
Acknowledgements
We thank Ronen Shaltiel for asking a question at ITCS 2021 that prompted us to write this paper. S.V. thanks Omer Reingold and Luca Trevisan for discussions many years ago that provided some of the ideas in this paper, in particular the tensor product construction used in the proof of Theorem 4 and the probabilistic existence proof in Theorem 3.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Pyne, E., Vadhan, S. (2021). Limitations of the Impagliazzo–Nisan–Wigderson Pseudorandom Generator Against Permutation Branching Programs. In: Chen, CY., Hon, WK., Hung, LJ., Lee, CW. (eds) Computing and Combinatorics. COCOON 2021. Lecture Notes in Computer Science(), vol 13025. Springer, Cham. https://doi.org/10.1007/978-3-030-89543-3_1
Download citation
DOI: https://doi.org/10.1007/978-3-030-89543-3_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-89542-6
Online ISBN: 978-3-030-89543-3
eBook Packages: Computer ScienceComputer Science (R0)