Abstract
Hardness amplification results show that for every Boolean function f, there exists a Boolean function Amp(f) such that if every size s circuit computes f correctly on at most a 1 − δ fraction of inputs, then every size s′ circuit computes Amp(f) correctly on at most a \({1/2+\epsilon}\) fraction of inputs. All hardness amplification results in the literature suffer from “size loss” meaning that \({s' \leq \epsilon \cdot s}\). We show that proofs using “non-uniform reductions” must suffer from such size loss.
A reduction is an oracle circuit \({R^{(\cdot)}}\) which given oracle access to any function D that computes Amp(f) correctly on a \({1/2+\epsilon}\) fraction of inputs, computes f correctly on a 1 − δ fraction of inputs. A non-uniform reduction is allowed to also receive a short advice string that may depend on both f and D. The well-known connection between hardness amplification and list-decodable error-correcting codes implies that reductions showing hardness amplification cannot be uniform for \({\epsilon < 1/4}\). We show that every non-uniform reduction must make at least \({\Omega(1/\epsilon)}\) queries to its oracle, which implies size loss. Our result is the first lower bound that applies to non-uniform reductions that are adaptive, whereas previous bounds by Shaltiel & Viola (SICOMP 2010) applied only to non-adaptive reductions. We also prove similar bounds for a stronger notion of “function-specific” reductions in which the reduction is only required to work for a specific function f.
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References
A. Akavia, O. Goldreich, S. Goldwasser & D. Moshkovitz (2006). On basing one-way functions on NP-hardness. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing, 701–710.
A. Atserias (2006). Distinguishing SAT from Polynomial-Size Circuits, through Black-Box Queries. In IEEE Conference on Computational Complexity, 88–95.
Babai L., Fortnow L., Nisan N., Wigderson A. (1993) BPP Has Subexponential Time Simulations Unless EXPTIME has Publishable Proofs. Computational Complexity 3: 307–318
A. Bogdanov & M. Safra (2007). Hardness Amplification for Errorless Heuristics. In Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, 418–426.
Bogdanov A., Trevisan L. (2006) OnWorst-Case to Average-Case Reductions for NP Problems. SIAM J. Comput. 36(4): 1119–1159
Feigenbaum J., Fortnow L. (1993) Random-Self-Reducibility of Complete Sets. SIAM J. Comput. 22(5): 994–1005
O. Goldreich, N. Nisan & A. Wigderson (2011). On Yao’s XORLemma. In Studies in Complexity and Cryptography, Oded Goldreich, editor, volume 6650 of Lecture Notes in Computer Science, 273–301. Springer. ISBN 978-3-642-22669-4.
S. Goldwasser, D. Gutfreund, A. Healy, T. Kaufman & G. N. Rothblum (2007). Verifying and decoding in constant depth. In Proceedings of the 39th Annual ACM Symposium on Theory of Computing, 440–449.
Gopalan P., Guruswami V. (2011) Hardness amplification within NP against deterministic algorithms. J. Comput. Syst. Sci. 77(1): 107–121
Guruswami V., Kabanets V. (2008) Hardness Amplification via Space-Efficient Direct Products. Computational Complexity 17(4): 475–500
D. Gutfreund & G. Rothblum (2008). The Complexity of Local List Decoding. In Proceedings of the 12th Intl. Workshop on Randomization and Computation.
Gutfreund D., Shaltiel R., Ta-Shma A. (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
D. Gutfreund & A. Ta-Shma (2007). Worst-Case to Average-Case Reductions Revisited. In Proceedings of the 11th Intl. Workshop on Randomization and Computation, 569–583.
D. Gutfreund & S. P. Vadhan (2008). Limitations of Hardness vs. Randomness under Uniform Reductions. In Proceedings of the 12th Intl. Workshop on Randomization and Computation, 469–482.
Healy A., Vadhan S.P., Viola E. (2006) Using Nondeterminism to Amplify Hardness. SIAM J. Comput. 35(4): 903–931
R. Impagliazzo (1995). Hard-Core Distributions for Somewhat Hard Problems. In Proceedings of the 36th Annual IEEE Symposium on Foundations of Computer Science, 538–545.
Impagliazzo R., Jaiswal R., Kabanets V. (2009a) Approximate List-Decoding of Direct Product Codes and Uniform Hardness Amplification. SIAM J. Comput. 39(2): 564–605
Impagliazzo R., Jaiswal R., Kabanets V. (2009b) Chernoff-Type Direct Product Theorems. J. Cryptology 22(1): 75–92
Impagliazzo R., Jaiswal R., Kabanets V., Wigderson A. (2010) Uniform Direct Product Theorems: Simplified, Optimized, and Derandomized. SIAM J. Comput. 39(4): 1637–1665
R. Impagliazzo & A. Wigderson (1997). P = BPP if E Requires Exponential Circuits: Derandomizing the XOR Lemma. In Proceedings of the 29th Annual ACM Symposium on Theory of Computing, 220–229.
Impagliazzo R., Wigderson A. (2001) Randomness vs Time: Derandomization under a Uniform Assumption. J. Comput. Syst. Sci. 63(4): 672–688
Klivans A., Servedio R.A. (2003) Boosting and Hard-Core Sets. Machine Learning 53(3): 217–238
Levin L.A. (1987) One-way functions and pseudorandom generators. Combinatorica 7(4): 357–363
R. Lipton (1991). New Directions in Testing. In Proceedings of DIMACS Workshop on Distributed Computing and Cryptography, volume 2, 191–202. ACM/AMS.
Lu C.-J., Tsai S.-C., Wu H.-L. (2008) On the Complexity of Hardness Amplification. IEEE Transactions on Information Theory 54(10): 4575–4586
Lu C.-J., Tsai S.-C., Wu H.-L. (2011) Complexity of Hard-Core Set Proofs. Computational Complexity 20(1): 145–171
O’Donnell R. (2004) Hardness amplification within NP. J. Comput. Syst. Sci. 69(1): 68–94
Raz R. (1998) A Parallel Repetition Theorem. SIAM J. Comput. 27(3): 763–803
O. Reingold, L. Trevisan & S. P. Vadhan (2004). Notions of Reducibility between Cryptographic Primitives. In Proceedings of the 1st Theory of Cryptography Conference, 1–20.
Shaltiel R., Umans C. (2005) Simple extractors for all minentropies and a new pseudorandom generator. J. ACM 52(2): 172–216
Shaltiel R., Viola E. (2010) Hardness Amplification Proofs Require Majority. SIAM J. Comput. 39(7): 3122–3154
Sudan M., Trevisan L., Vadhan S.P. (2001) Pseudorandom Generators without the XOR Lemma. J. Comput. Syst. Sci. 62(2): 236–266
L. Trevisan (2003). List-Decoding Using The XOR Lemma. In Proceedings of the 44th Symposium on Foundations of Computer Science, 126–135.
L. Trevisan (2004). Some applications of coding theory in computational complexity. In Complexity of computations and proofs, volume 13 of Quad. Mat., 347–424. Dept. Math., Seconda Univ. Napoli, Caserta.
L. Trevisan (2005). On uniform amplification of hardness in NP. In Proceedings of the 37th Annual ACM Symposium on Theory of Computing, 31–38.
Trevisan L., Vadhan S. (2007) Pseudorandomness and Average-Case Complexity Via Uniform Reductions. Computational Complexity 16(4): 331–364
Viola E. (2005a) The complexity of constructing pseudorandom generators from hard functions. Computational Complexity 13(3-4): 147–188
E. Viola (2005b). On Constructing Parallel Pseudorandom Generators from One-Way Functions. In IEEE Conference on Computational Complexity, 183–197.
T. Watson (2011). Query Complexity in Errorless Hardness Amplification. In Proceedings of the 15th Intl. Workshop on Randomization and Computation, 688–699.
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Artemenko, S., Shaltiel, R. Lower Bounds on the Query Complexity of Non-uniform and Adaptive Reductions Showing Hardness Amplification. comput. complex. 23, 43–83 (2014). https://doi.org/10.1007/s00037-012-0056-2
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DOI: https://doi.org/10.1007/s00037-012-0056-2