Journal Article10.1145/273865.273933
A new general derandomization method
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TL;DR: It is proved that if a logarithmic price quick hitting set generator exists then BPP = P, and the new derandomization method is based on a deterministic algorithm that constructs a discrepancy set for C, which depends on C.
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Abstract: We show that quick hitting set generators can replace quick pseudorandom generators to derandomize any probabilistic two-sided error algorithms. Up to now quick hitting set generators have been known as the general and uniform derandomization method for probabilistic one-sided error algorithms, while quick pseudorandom generators as the generators as the general and uniform method to derandomize probabilistic two-sided error algorithms.Our method is based on a deterministic algorithm that, given a Boolean circuit C and given access to a hitting set generator, constructs a discrepancy set for C. The main novelty is that the discrepancy set depends on C, so the new derandomization method is not uniform (i.e., not oblivious).The algorithm works in time exponential in k(p(n)) where k(*) is the price of the hitting set generator and p(*) is a polynomial function in the size of C. We thus prove that if a logarithmic price quick hitting set generator exists then BPP = P.
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Citations
Derandomizing polynomial identity tests means proving circuit lower bounds
Valentine Kabanets,Russell Impagliazzo +1 more
- 01 Dec 2004
TL;DR: If Permanent requires superpolynomial-size arithmetic circuits, then one can test in subexponential time whether a given arithmetic circuit of polynomially bounded degree computes an identically zero polynomial.
443
•Journal Article
Derandomizing Polynomial Identity Tests Means Proving Circuit Lower Bounds
TL;DR: In this paper, it was shown that derandomizing Polynomial Identity Testing is equivalent to proving arithmetic circuit lower bounds for NEXP, and that if one can test in polynomial time (or even non-deterministic subexponential time, infinitely often) whether a given arithmetic circuit over integers computes an identically zero poynomial, then either NEXP ⊄ P/poly or Permanent is not computable by polynomially-size arithmetic circuits.
342
In search of an easy witness: exponential time vs. probabilistic polynomial time
Russell Impagliazzo,Valentine Kabanets,Avi Wigderson +2 more
- 18 Jun 2001
TL;DR: A number of results are established relating the complexity of exponential-time and probabilistic polynomial-time complexity classes, including NEXP/spl sub/P/poly/spl hArr/NEXP=MA, which can be interpreted to say that no derandomization of MA is possible unless NEXP contains a hard Boolean function.
224
Simple extractors for all min-entropies and a new pseudorandom generator
Ronen Shaltiel,Christopher Umans +1 more
TL;DR: A simple, self-contained extractor construction that produces good extractors for all min-entropies and gives a quantitatively optimal hardness versus randomness tradeoff for AM and the first hardness amplification result for nondeterministic circuits.
203
Pseudo-random generators for all hardnesses
Christopher Umans
- 19 May 2002
TL;DR: A key element in this construction is an augmentation of the standard low-degree extension encoding that exploits the field structure of the underlying space in a new way to give a direct proof of the optimal hardness vs. randomness tradeoff.
161
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Manuel Blum,Silvio Micali +1 more
TL;DR: In this article, the authors give a set of conditions that allow one to generate 50-50 unpredictable bits, and present a general algorithmic scheme for constructing polynomial-time deterministic algorithms that stretch a short secret random input into a long sequence of unpredictable pseudo-random bits.
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Noam Nisan,Avi Wigderson +1 more
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Benny Chor,Oded Goldreich +1 more
TL;DR: A new model for weak random physical sources is presented that strictly generalizes previous models and provides a fruitful viewpoint on problems studied previously such as Extracting almost-perfect bits from sources of weak randomness.
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