Weak random sources, hitting sets, and BPP simulations

We show how to simulate any BPP algorithm in polynomial time using a weak random source of min-entropy r/sup /spl gamma// for any /spl gamma/>0. This follows from a more general result about sampling with weak random sources. Our result matches an information-theoretic lower bound and solves a question that has been open for some years. The previous best results were a polynomial time simulation of RP (Saks et al., 1995) and a n(log/sup (k)/n)-time simulation of BPP for fixed k (Ta-Shma, 1996). Departing significantly from previous related works, we do not use extractors; instead we use the OR-disperser of (Saks et al., 1995) in combination with a tricky use of hitting sets borrowed from Andreev et al. (1996). Of independent interest is our new (simplified) proof of the main result of Andreev et al., (1996). Our proof also gives some new hardness/randomness trade-offs for parallel classes.

[1]  Noam Nisan,et al.  Randomness is Linear in Space , 1996, J. Comput. Syst. Sci..

[2]  Aravind Srinivasan,et al.  Explicit dispersers with polylog degree , 1995, STOC '95.

[3]  Yacov Yacobi,et al.  The Complexity of Promise Problems with Applications to Public-Key Cryptography , 1984, Inf. Control..

[4]  Manuel Blum,et al.  How to generate cryptographically strong sequences of pseudo random bits , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[5]  Noam Nisan.  Extracting randomness: how and why , 1996 .

[6]  Oded Goldreich,et al.  A Sample of Samplers - A Computational Perspective on Sampling (survey) , 1997, Electron. Colloquium Comput. Complex..

[7]  Noam Nisan.  Using hard problems to create pseudorandom generators , 1992, ACM Distinguished Dissertations.

[8]  José D. P. Rolim,et al.  Worst-Case Hardness Suffices for Derandomization: A New Method for Hardness-Randomness Trade-offs , 1997, Theoretical Computer Science.

[9]  Vijay V. Vazirani,et al.  Random polynomial time is equal to slightly-random polynomial time , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[10]  Mihir Bellare,et al.  Randomness-efficient oblivious sampling , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[11]  Andrew Chi-Chih Yao,et al.  Theory and application of trapdoor functions , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[12]  Rajeev Motwani,et al.  Randomized Algorithms , 1995, SIGA.

[13]  José D. P. Rolim,et al.  Worst-Case Hardness Suffices for Derandomization: A New Method for Hardness-Randomness Trade-offs , 1997, Theor. Comput. Sci..

[14]  Umesh Vazirani,et al.  Efficiency Considerations in Using Semi-random Sources (Extended Abstract) , 1987, STOC 1987.

[15]  Oded Goldreich,et al.  Unbiased Bits from Sources of Weak Randomness and Probabilistic Communication Complexity , 1988, SIAM J. Comput..

[16]  Amnon Ta-Shma,et al.  On Extracting Randomness From Weak Random Sources , 1995, Electron. Colloquium Comput. Complex..

[17]  Avi Wigderson,et al.  P = BPP if E requires exponential circuits: derandomizing the XOR lemma , 1997, STOC '97.

[18]  Noam Nisan,et al.  BPP has subexponential time simulations unless EXPTIME has publishable proofs , 1991, [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference.

[19]  V. Rich.  Personal communication , 1989, Nature.

[20]  Noam Nisan,et al.  Hardness vs. randomness , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[21]  Miklos Santha,et al.  Generating Quasi-random Sequences from Semi-random Sources , 1986, J. Comput. Syst. Sci..

[22]  Miklos Santha,et al.  Generating Quasi-Random Sequences from Slightly-Random Sources (Extended Abstract) , 1984, FOCS.

[23]  Aravind Srinivasan,et al.  Computing with very weak random sources , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[24]  Avi Wigderson,et al.  Dispersers, deterministic amplification, and weak random sources , 1989, 30th Annual Symposium on Foundations of Computer Science.

[25]  David Zuckerman,et al.  General weak random sources , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[26]  José D. P. Rolim,et al.  Weak Random Sources, Hitting Sets, and BPP Simulations , 1999, SIAM J. Comput..

[27]  Noam Nisan,et al.  Extracting randomness: how and why. A survey , 1996, Proceedings of Computational Complexity (Formerly Structure in Complexity Theory).

[28]  Ming Li,et al.  Kolmogorov Complexity and its Applications , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[29]  José D. P. Rolim,et al.  Hitting Sets Derandomize BPP , 1996, ICALP.

[30]  José D. P. Rolim,et al.  A new general derandomization method , 1998, JACM.

[31]  Michael Sipser,et al.  A complexity theoretic approach to randomness , 1983, STOC.

[32]  David Zuckerman,et al.  Randomness-optimal sampling, extractors, and constructive leader election , 1996, STOC '96.

[33]  Clemens Lautemann,et al.  BPP and the Polynomial Hierarchy , 1983, Inf. Process. Lett..