• DOI:10.1145/258533.258590
• Corpus ID: 18921599

P = BPP if E requires exponential circuits: derandomizing the XOR lemma

@inproceedings{Impagliazzo1997PB,
  title={P = BPP if E requires exponential circuits: derandomizing the XOR lemma},
  author={Russell Impagliazzo and Avi Wigderson},
  booktitle={Symposium on the Theory of Computing},
  year={1997}
}

• R. Impagliazzo, A. Wigderson
• Published in
  Symposium on the Theory of…
  4 May 1997
• Mathematics

Russell Impagliazzo* Avi Wigdersont Department of Computer Science Institute of Computer Science University of California Hebrew University San Diego, CA 91097-0114 Jerusalem, Israel russell@cs .ucsd. edu avi@cs .huj i. ac. il Yao showed that the XOR of independent random instances of a somewhat hard Boolean problem becomes almost completely unpredictable. In this paper we show that, in non-uniform settings, total independence is not necessary for this result to hold. We give a pseudo-random generator which produces n instances of a problem for which the analog of the XOR lemma holds. Combining this generator with the results of [25, 6] gives substantially improved results for hardness vs randomness tradeoffs. In particular, we show that if any problem in E = DTIAl E(2°t”j) has circuit complexity 2Q(”), then P = BPP. Our generator is a combination of two known ones the random walks on expander graphs of [1, 10, 19] and the nearly disjoint subsets generator of [23, 25]. The quality of the generator is proved via a new proof of the XOR lemma which may be useful for other direct product results. *Research supported by NSF YI Award CCR-92s70979, Sloan Research Fellowship BR-3311, grant #93025 of the joint US-Czechoslovak Science and Technology Program, and USA-Israel BSF Grant 92-00043 tWork pmtly done while visiting the Institute for Advanced Study, Princeton, N. J. 08540 and Princeton University. Research supported the Sloan Foundation, American-Israeli BSF grant 92-00106, and the Wolfson Research Awards, administered by the Israel Academy of Sciences.