EXTRACTORS FOR CIRCUIT SOURCES

We obtain the first deterministic extractors for sources generated (or sampled) by small circuits of bounded depth. Our main results are (1) we extract k(k/nd)(O(1)) bits with exponentially small error from n-bit sources of min-entropy k that are generated by functions f : {0, 1}(l) -> {0, 1}(n), where each output bit depends on <= d input bits. In particular, we extract from NC0 sources, corresponding to d = O(1); (2) we extract k(k/n(1+gamma))(O(1)) bits with superpolynomially small error from n-bit sources of min-entropy k that are generated by poly(n)-size AC(0) circuits, for any gamma > 0. As our starting point, we revisit the connection by Trevisan and Vadhan [IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, Los Alamitos, CA, 2000, pp. 32-42] between circuit lower bounds and extractors for sources generated by circuits. We note that such extractors (with very weak parameters) are equivalent to lower bounds for generating distributions [E. Viola, SIAM J. Comput., 41 (2012), pp. 191-218; S. Lovett and E. Viola, Comput. Complexity, 21 (2012), pp. 245-256]. Building on those bounds, we prove that the sources in (1) and (2) are (close to) a convex combination of high-entropy "bit-block" sources. Introduced here, such sources are a special case of affine ones. As extractors for (1) and (2) one can use the extractor for low-weight affine sources by Rao [IEEE Conference on Computational Complexity, IEEE Computer Society, Los Alamitos, CA, 2009, pp. 95-101]. Along the way, we exhibit an explicit boolean function b : {0, 1}(n) -> {0, 1} such that poly(n)-size AC(0) circuits cannot generate the distribution (Y, b(Y)), solving a problem about the complexity of distributions. Independently, De and Watson [ACM Trans. Comput. Theory, 4 (2012), 3] obtain a result similar to (1) in the special case d = o(lg n).