Some consequences of the existence of pseudorandom generators
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
A pseudo-random bit generator based on elliptic logarithms
Proceedings on Advances in cryptology---CRYPTO '86
Small-bias probability spaces: efficient constructions and applications
STOC '90 Proceedings of the twenty-second annual ACM symposium on Theory of computing
On deterministic approximation of DNF
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
Journal of Computer and System Sciences
Improved pseudorandom generators for depth 2 circuits
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Bounded Independence Fools Halfspaces
SIAM Journal on Computing
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We explicitly construct, for every integer n and ε 0, a family of functions (psuedo-random bit generators) fn,ε:{0,1}nε → {0,1}n with the following property: for a random seed, the pseudorandom output "looks random" to any polynomial size, constant depth, unbounded fan-in circuit. Moreover, the functions fn,ε themselves can be computed by uniform polynomial size, constant depth circuits. Some (interrelated) consequences of this result are given below. 1) Deterministic simulation of probabilistic algorithms. The constant depth analogues of the probabilistic complexity classes RP and BPP are contained in the deterministic complexity classes DSPACE(nε) and DTIME(2nε) for any ε 0. 2) Making probabilistic constructions deterministic. Some probablistic constructions of structures that elude explicit constructions can be simulated in the above complexity classes. 3) Approximate counting. The number of satisfying assignments to a (CNF or DNF) formula, if not too small, can be arbitrarily approximated in DSPACE(nε) and DTIME(2nε), for any ε 0. We also present two results for the special case of depth 2 circuits. They deal, respectively, with finding a satisfying assignment and approximately counting the number of assignments. For example, for 3-CNF formulas with a fixed fraction of satisfying assignmemts, both tasks can be performed in polynomial time!