On the Complexity of Hard-Core Set Constructions

  • Chi-Jen Lu
  • Shi-Chun Tsai
  • Hsin-Lung Wu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4596)

Abstract

We study a fundamental result of Impagliazzo (FOCS’95) known as the hard-core set lemma. Consider any function f:{0,1}n →{0,1} which is “mildly-hard”, in the sense that any circuit of size s must disagree with f on at least δ fraction of inputs. Then the hard-core set lemma says that f must have a hard-core set H of density δ on which it is “extremely hard”, in the sense that any circuit of size

s’= \( {O} {(s/({{1}\over{\varepsilon^2}}\log(\frac{1}{\varepsilon\delta})))}\) must disagree with f on at least (1 − ε)/2 fraction of inputs from H.

There are three issues of the lemma which we would like to address: the loss of circuit size, the need of non-uniformity, and its inapplicability to a low-level complexity class. We introduce two models of hard-core set constructions, a strongly black-box one and a weakly black-box one, and show that those issues are unavoidable in such models.

First, we show that in any strongly black-box construction, one can only prove the hardness of a hard-core set for smaller circuits of size at most \(s'=O(s/(\frac{1}{\varepsilon^2}log\frac{1}{\delta}))\). Next, we show that any weakly black-box construction must be inherently non-uniform — to have a hard-core set for a class G of functions, we need to start from the assumption that f is hard against a non-uniform complexity class with \(\Omega(\frac{1}{\varepsilon}log|G|)\) bits of advice. Finally, we show that weakly black-box constructions in general cannot be realized in a low-level complexity class such as AC0[p] — the assumption that f is hard for AC0[p] is not sufficient to guarantee the existence of a hard-core set.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alon, N., Spencer, J.: The probabilistic method, 2nd edn. Wiley-Interscience, New York (2000)MATHGoogle Scholar
  2. 2.
    Babai, L., Fortnow, L., Nisan, N., Wigderson, A.: BPP has subexponential time simulations unless EXPTIME has publishable proofs. Computational Complexity 3(4), 307–318 (1993)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Goldmann, M., Håstad, J., Razborov, A.: Majority gates vs. general weighted threshold gates. Computational Complexity 2, 277–300 (1992)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Hajnal, A., Maass, W., Pudlák, P., Szegedy, M., Turán, G.: Threshold circuits of bounded depth. In: Proceedings of the 28th Annual IEEE Symposium on Foundations of Computer Science, pp. 99–110 (1987)Google Scholar
  5. 5.
    Healy, A., Vadhan, S., Viola, E.: Using nondeterminism to amplify hardness. SIAM Journal on Computing 35(4), 903–931 (2006)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Holenstein, T.: Key agreement from weak bit agreement. In: Proceedings of the 37th ACM Symposium on Theory of Computing, pp. 664–673. ACM Press, New York (2005)Google Scholar
  7. 7.
    Impagliazzo, R.: Hard-core distributions for somewhat hard problems. In: Proceedings of the 36th Annual IEEE Symposium on Foundations of Computer Science, pp. 538–545. IEEE Computer Society Press, Los Alamitos (1995)Google Scholar
  8. 8.
    Impagliazzo, R., Wigderson, A.: P=BPP if E requires exponential circuits: Derandomizing the XOR lemma. In: Proceedings of the 29th ACM Symposium on Theory of Computing, pp. 220–229. ACM Press, New York (1997)Google Scholar
  9. 9.
    Klivans, A., Servedio, R.A.: Boosting and hard-core sets. Machine Learning 51(3), 217–238 (2003)MATHCrossRefGoogle Scholar
  10. 10.
    Lu, C.-J., Tsai, S.-C., Wu, H.-L.: On the complexity of hardness amplification. In: Proceedings of the 20th Annual IEEE Conference on Computational Complexity, pp. 170–182. IEEE Computer Society Press, Los Alamitos (2005)Google Scholar
  11. 11.
    O’Donnell, R.: Hardness amplification within NP. In: Proceedings of the 34th ACM Symposium on Theory of Computing, pp. 751–760. ACM Press, New York (2002)Google Scholar
  12. 12.
    Sudan, M., Trevisan, L., Vadhan, S.: Pseudorandom generators without the XOR lemma. Journal of Computer and System Sciences 62(2), 236–266 (2001)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Smolensky, R.: Algebraic methods in the theory of lower bounds for Boolean circuit complexity. In: Proceedings of the 19th ACM Symposium on Theory of Computing, pp. 77–82. ACM Press, New York (1987)Google Scholar
  14. 14.
    Szegedy, M.: Algebraic methods in lower bounds for computational models with limited communication. Ph.D. thesis, University of Chicago (1989)Google Scholar
  15. 15.
    Tarui, J.: Degree complexity of boolean functions and its applications to relativized separations. In: Proceedings of the 6th Annual IEEE Conference on Structure in Complexity Theory, pp. 382–390. IEEE Computer Society Press, Los Alamitos (1991)CrossRefGoogle Scholar
  16. 16.
    Trevisan, L.: List decoding using the XOR lemma. In: Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, pp. 126–135. IEEE Computer Society Press, Los Alamitos (2003)CrossRefGoogle Scholar
  17. 17.
    Trevisan, L.: On uniform amplification of hardness in NP. In: Proceedings of the 37th ACM Symposium on Theory of Computing, pp. 31–38. ACM Press, New York (2005)Google Scholar
  18. 18.
    Viola, E.: The Complexity of Hardness Amplification and Derandomization. Ph.D. thesis, Harvard University (2006)Google Scholar
  19. 19.
    Yao, A.: Theory and applications of trapdoor functions. In: Proceedings of the 23rd Annual IEEE Symposium on Foundations of Computer Science, pp. 80–91. IEEE Computer Society Press, Los Alamitos (1982)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Chi-Jen Lu
    • 1
  • Shi-Chun Tsai
    • 2
  • Hsin-Lung Wu
    • 2
  1. 1.Institute of Information Science, Academia Sinica, TaipeiTaiwan
  2. 2.Department of Computer Science, National Chiao-Tung University, HsinchuTaiwan

Personalised recommendations