Abstract
Yao’s XOR lemma states that for every function \(f:\{0,1\}^k \rightarrow \{0,1\}\), if f has hardness 2/3 for P/poly (meaning that for every circuit C in P/poly, \(\Pr[C(X)=f(X)] \le 2/3\) on a uniform input X), then the task of computing \(f(X_1) \oplus \ldots \oplus f(X_t)\) for sufficiently large t has hardness \(\frac{1}{2} + \epsilon\) for P/poly.
Known proofs of this lemma cannot achieve \(\epsilon=\frac{1}{k^{\omega(1)}}\), and even for \(\epsilon=\frac{1}{k}\), we do not know how to replace P/poly by AC0[parity] (the class of constant depth circuits with the gates {and, or, not, parity} of unbounded fan-in).
Grinberg, Shaltiel and Viola (FOCS 2018) (building on a sequence of earlier works) showed that these limitations cannot be circumvented by black-box reductions. Namely, by reductions \({\rm Red}^{(\cdot)}\) that given oracle access to a function D that violates the conclusion of Yao’s XOR lemma, implement a circuit that violates the assumption of Yao’s XOR lemma.
There are a few known reductions in the related literature on worst-case to average-case reductions that are non-black-box. Specifically, the reductions of Gutfreund, Shaltiel and Ta-Shma (Computational Complexity 2007) and Hirahara (FOCS 2018)) are “class reductions” that are only guaranteed to succeed when given oracle access to an oracle D from some efficient class of algorithms. These works seem to circumvent some black-box impossibility results.
In this paper, we extend the previous limitations of Grinberg, Shaltiel and Viola to several types of class reductions, giving evidence that class reductions cannot yield the desired improvements in Yao’s XOR lemma. To the best of our knowledge, this is the first limitation on reductions for hardness amplification that applies to class reductions.
Our technique imitates the previous lower bounds for black-box reductions, replacing the inefficient oracle used in that proof, with an efficient one that is based on limited independence, and developing tools to deal with the technical difficulties that arise following this replacement.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Miklós Ajtai (1983). \(\sum^{1}\)1-Formulae on finite structures. Ann. Pure Appl. Log. 24(1), 1–48. URL https://doi.org/10.1016/0168-0072(83)90038-6.
Adi Akavia, Oded Goldreich, Shafi Goldwasser & Dana Moshkovitz (2006). On basing one-way functions on NP-hardness. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing, 701–710. URL https://doi.org/10.1145/1132516.1132614.
Benny Applebaum, Sergei Artemenko, Ronen Shaltiel & Guang Yang (2016). Incompressible Functions, Relative-Error Extractors, and the Power of Nondeterministic Reductions. Computational Complexity 25(2), 349–418. URL https://doi.org/10.1007/s00037-016-0128-9.
Sergei Artemenko & Ronen Shaltiel (2014). Lower Bounds on the Query Complexity of Non-uniform and Adaptive Reductions Showing Hardness Amplification. Computational Complexity 23(1), 43–83. URL https://doi.org/10.1007/s00037-012-0056-2.
Albert Atserias (2006). Distinguishing SAT from Polynomial-Size Circuits, through Black-Box Queries. In 21st Annual IEEE Conference on Computational Complexity, 88–95. URL https://doi.org/10.1109/CCC.2006.17.
Andrej Bogdanov & Luca Trevisan (2006). On Worst-Case to Average-Case Reductions for NP Problems. SIAM J. Comput. 36(4), 1119–1159. URL https://doi.org/10.1137/S0097539705446974.
Bill Fefferman, Ronen Shaltiel, Christopher Umans & Emanuele Viola (2013). On beating the hybrid argument. Theory of Computing 9, 809–843.
Joan Feigenbaum & Lance Fortnow (1993). Random-Self- Reducibility of Complete Sets. SIAM J. Comput. 22(5), 994–1005. URL https://doi.org/10.1137/0222061.
Oded Goldreich & Hugo Krawczyk (1996). On the Composition of Zero-Knowledge Proof Systems. SIAM J. Comput. 25(1), 169–192. URL https://doi.org/10.1137/S0097539791220688.
Oded Goldreich, Noam Nisan & Avi Wigderson (2011). On Yao’s XOR lemma. In Studies in Complexity and Cryptography. Miscellanea on the Interplay between Randomness and Computation, Oded Goldreich, editor, volume 6650 of Lecture Notes in Computer Science, 273–301. Springer. URL https://doi.org/10.1007/978-3-642-22670-0_23.
Aryeh Grinberg, Ronen Shaltiel & Emanuele Viola (2018). Indistinguishability by Adaptive Procedures with Advice, and Lower Bounds on Hardness Amplification Proofs. In 59th IEEE Annual Symposium on Foundations of Computer Science, 956–966. URL https://doi.org/10.1109/FOCS.2018.00094.
Dan Gutfreund (2006). Worst-Case Vs. Algorithmic Average-Case Complexity in the Polynomial-Time Hierarchy. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX and 10th International Workshop on Randomization and Computation, RANDOM, volume 4110 of Lecture Notes in Computer Science, 386–397. URL https://doi.org/10.1007/11830924_36.
Dan Gutfreund & Guy N. Rothblum (2008). The Complexity of Local List Decoding. In Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques, 11th International Workshop, APPROX, and 12th International Workshop, RANDOM, volume 5171 of Lecture Notes in Computer Science, 455–468. URL https://doi.org/10.1007/978-3-540-85363-3_36.
Dan Gutfreund, Ronen Shaltiel & Amnon Ta-Shma (2007). If NP Languages are Hard on the Worst-Case, Then it is Easy to Find Their Hard Instances. Computational Complexity 16(4), 412–441. URL https://doi.org/10.1007/s00037-007-0235-8.
Dan Gutfreund & Amnon Ta-Shma (2007). Worst-Case to Average- Case Reductions Revisited. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, 10th International Workshop, APPROX, and 11th International Workshop, RANDOM, volume 4627 of Lecture Notes in Computer Science, 569–583. URL https://doi.org/10.1007/978-3-540-74208-1_41.
Dan Gutfreund & Salil P. Vadhan (2008). Limitations of Hardness vs. Randomness under Uniform Reductions. In Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques, 11th International Workshop, APPROX, and 12th International Workshop, RANDOM, volume 5171 of Lecture Notes in Computer Science, 469–482. URL https://doi.org/10.1007/978-3-540-85363-3_37.
Dan Gutfreund & Emanuele Viola (2004). Fooling Parity Tests with Parity Gates. In Approximation, Randomization, and Combinatorial Optimization, Algorithms and Techniques, 7th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX, and 8th International Workshop on Randomization and Computation, RANDOM, volume 3122 of Lecture Notes in Computer Science, 381–392. URL https://doi.org/10.1007/978-3-540-27821-4_34.
Shuichi Hirahara (2018). Non-Black-Box Worst-Case to Average- Case Reductions within NP. In 59th IEEE Annual Symposium on Foundations of Computer Science, 247–258. URL https://doi.org/10.1109/FOCS.2018.00032.
Russell Impagliazzo (1995). Hard-Core Distributions for Somewhat Hard Problems. In 36th Annual Symposium on Foundations of Computer Science, 538–545. URL https://doi.org/10.1109/SFCS.1995.492584.
Russell Impagliazzo & Steven Rudich (1989). Limits on the Provable Consequences of One-Way Permutations. In Proceedings of the 21st Annual ACM Symposium on Theory of Computing, 44–61. URL https://doi.org/10.1145/73007.73012.
Russell Impagliazzo & Avi Wigderson (1997). P = BPP if E Requires Exponential Circuits: Derandomizing the XOR Lemma. In Proceedings of the Twenty-Ninth Annual ACM Symposium on the Theory of Computing, 220–229. URL https://doi.org/10.1145/258533.258590.
Adam R. Klivans & Rocco A. Servedio (2003). Boosting and Hard-Core Set Construction. Machine Learning 51(3), 217–238. URL https://doi.org/10.1023/A:1022949332276.
Nutan Limaye, Karteek Sreenivasaiah, Srikanth Srinivasan, Utkarsh Tripathi & S. Venkitesh (2019). A fixed-depth sizehierarchy theorem for AC0[⊕] via the coin problem. In Proceedings of the 51st Annual ACM Symposium on Theory of Computing, 442–453. URL https://doi.org/10.1145/3313276.3316339.
Chi-Jen Lu, Shi-Chun Tsai & Hsin-Lung Wu (2008). On the Complexity of Hardness Amplification. IEEE Trans. Information Theory 54(10), 4575–4586. URL https://doi.org/10.1109/TIT.2008.928988.
Igor Carboni Oliveira, Rahul Santhanam & Srikanth Srinivasan (2019). Parity Helps to Compute Majority. In 34th Computational Complexity Conference, volume 137, 23:1–23:17. URL https://doi.org/10.4230/LIPIcs.CCC.2019.23.
Alexander Razborov (1987). Lower bounds on the dimension of schemes of bounded depth in a complete basis containing the logical addition function. Akademiya Nauk SSSR. Matematicheskie Zametki 41(4), 598–607. English translation in Mathematical Notes of the Academy of Sci. of the USSR, 41(4):333-338, 1987.
Omer Reingold, Luca Trevisan & Salil P. Vadhan (2004). Notions of Reducibility between Cryptographic Primitives. In Theory of Cryptography, First Theory of Cryptography Conference, TCC, volume 2951 of Lecture Notes in Computer Science, 1–20. URL https://doi.org/10.1007/978-3-540-24638-1_1.
Ronen Shaltiel (2020). Is It Possible to Improve Yao’s XOR Lemma Using Reductions That Exploit the Efficiency of Their Oracle? In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2020, August 17-19, 2020, Virtual Conference, Jaroslaw Byrka & Raghu Meka, editors, volume 176 of LIPIcs, 10:1–10:20. Schloss Dagstuhl - Leibniz- Zentrum für Informatik. URL https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.10.
Ronen Shaltiel & Emanuele Viola (2010). Hardness Amplification Proofs Require Majority. SIAM J. Comput. 39(7), 3122–3154. URL https://doi.org/10.1137/080735096.
Roman Smolensky (1987). Algebraic Methods in the Theory of Lower Bounds for Boolean Circuit Complexity. In Proceedings of the 19th Annual ACM Symposium on Theory of Computing, 77–82. URL https://doi.org/10.1145/28395.28404.
Madhu Sudan, Luca Trevisan & Salil P. Vadhan (2001). Pseudorandom Generators without the XOR Lemma. J. Comput. Syst. Sci. 62(2), 236–266. URL https://doi.org/10.1006/jcss.2000.1730.
Luca Trevisan & Salil P. Vadhan (2007). Pseudorandomness and Average-Case Complexity Via Uniform Reductions. Computational Complexity 16(4), 331–364. URL https://doi.org/10.1007/s00037-007-0233-x.
Emanuele Viola (2003). Hardness vs. Randomness within Alternating Time. In 18th Annual IEEE Conference on Computational Complexity, 53. URL https://doi.org/10.1109/CCC.2003.1214410.
Emanuele Viola (2006). The Complexity of Hardness Amplification and Derandomization. Ph.D. thesis, Harvard University.
Acknowledgements
A preliminary version of this paper appeared in RANDOM 2020 (Shaltiel 2020). We are grateful to Emanuele Viola for very helpful discussions, and to anonymous referees for excellent comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Shaltiel, R. Is it possible to improve Yao’s XOR lemma using reductions that exploit the efficiency of their oracle?. comput. complex. 32, 5 (2023). https://doi.org/10.1007/s00037-023-00238-9
Received:
Published:
DOI: https://doi.org/10.1007/s00037-023-00238-9