Abstract
A hitting-set generator is a deterministic algorithm that generates a set of strings such that this set intersects every dense set that is recognizable by a small circuit. A polynomial time hitting-set generator readily implies \(\mathcal{RP}=\mathcal{P}\), but it is not apparent what this implies for \(\mathcal{BPP}\). Nevertheless, Andreev et al. (ICALP’96, and JACM 1998) showed that a polynomial-time hitting-set generator implies the seemingly stronger conclusion \(\mathcal{BPP=P}\). We simplify and improve their (and later) constructions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Adleman, L.: Two theorems on random polynomial-time. In: 19th FOCS, pp. 75–83 (1978)
Andreev, A.E., Clementi, A.E.F., Rolim, J.D.P.: A new general derandomization method. Journal of the Association for Computing Machinery (J. of ACM) 45(1), 179–213 (1998); Hitting Sets Derandomize BPP. In: XXIII International Colloquium on Algorithms, Logic and Programming, ICALP 1996 (1996)
Andreev, A.E., Clementi, A.E.F., Rolim, J.D.P., Trevisan, L.: Weak Random Sources, Hitting Sets, and BPP Simulations. To appear in SICOMP (1997); Preliminary version in 38th FOCS, pp. 264–272 (1997)
Buhrman, H., Fortnow, L.: One-sided versus two-sided randomness. In: Proceedings of the 16th Symposium on Theoretical Aspects of Computer Science. LNCS, Springer, Berlin (1999)
Even, S., Selman, A.L., Yacobi, Y.: The Complexity of Promise Problems with Applications to Public-Key Cryptography. Inform. and Control 61, 159–173 (1984)
Goldreich, O., Zuckerman, D.: Another proof that BPP ⊆ PH (and more). In: Goldreich, O., et al.: Studies in Complexity and Cryptography. LNCS, vol. 6650, pp. 40–53. Springer, Heidelberg (1997)
Goldreich, O., Wigderson, A.: Improved derandomization of BPP using a hitting set generator. In: Hochbaum, D.S., Jansen, K., Rolim, J.D.P., Sinclair, A. (eds.) RANDOM 1999 and APPROX 1999. LNCS, vol. 1671, pp. 131–137. Springer, Heidelberg (1999)
Impagliazzo, R., Wigderson, A.: P=BPP unless E has Subexponential Circuits: Derandomizing the XOR Lemma. In: 29th STOC, pp. 220–229 (1997)
Lautemann, C.: BPP and the Polynomial Hierarchy. Information Processing Letters 17, 215–217 (1983)
Sipser, M.: A complexity-theoretic approach to randomness. In: 15th STOC, pp. 330–335 (1983)
Zuckerman, D.: Simulating BPP Using a General Weak Random Source. Algorithmica 16, 367–391 (1996)
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Goldreich, O., Vadhan, S., Wigderson, A. (2011). Simplified Derandomization of BPP Using a Hitting Set Generator. In: Goldreich, O. (eds) Studies in Complexity and Cryptography. Miscellanea on the Interplay between Randomness and Computation. Lecture Notes in Computer Science, vol 6650. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22670-0_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-22670-0_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22669-4
Online ISBN: 978-3-642-22670-0
eBook Packages: Computer ScienceComputer Science (R0)