Constructions Of Extractors Research Articles

An Improved Analysis of Linear Mergers

Mergers are procedures that, with the aid of a short random string, transform k (possibly dependent) random sources into a single random source, in a way that ensures that if one of the input sources has min-entropy rate ? then the output has min-entropy rate close to ?. Mergers were first introduced by Ta-Shma [28th STOC, pp. 276-285, 1996] and have proven to be a very useful tool in explicit constructions of extractors and condensers. In this work we present a new analysis of the merger construction of Lu et al [35th STOC, pp. 602-611, 2003]. We prove that the merger's output is close to a distribution with min-entropy rate of at least $$\frac{6}{11}\delta$$ . We show that the distance from this distribution is polynomially related to the number of additional random bits that were used by the merger (i.e., its seed). We are also able to prove a bound of $$\frac{4}{7}\delta$$ on the min-entropy rate at the cost of increasing the statistical error. Both results are improvements to the previous known lower bound of $$\frac{1}{2}\delta$$ (however, in the $$\frac{1}{2}\delta$$ result the error decreases exponentially in the length of the seed). To obtain our results we deviate from the usual linear algebra methods that were used by Lu et al and introduce techniques from additive number theory.

Lossless Condensers, Unbalanced Expanders, And Extractors

Trevisan showed that many pseudorandom generator constructions give rise to constructions of explicit extractors. We show how to use such constructions to obtain explicit lossless condensers. A lossless condenser is a probabilistic map using only O(logn) additional random bits that maps n bits strings to poly(logK) bit strings, such that any source with support size K is mapped almost injectively to the smaller domain. Our construction remains the best lossless condenser to date. By composing our condenser with previous extractors, we obtain new, improved extractors. For small enough min-entropies our extractors can output all of the randomness with only O(logn) bits. We also obtain a new disperser that works for every entropy loss, uses an O(logn) bit seed, and has only O(logn) entropy loss. This is the best disperser construction to date, and yields other applications. Finally, our lossless condenser can be viewed as an unbalanced bipartite graph with strong expansion properties.

Extracting Randomness via Repeated Condensing

Extractors (as defined by Nisan and Zuckerman) are procedures that use a small number of truly random bits (called the seed) to extract many (almost) truly random bits from arbitrary distributions as long as distributions have sufficient (min)-entropy. A natural weakening of an extractor is a condenser, whose output distribution has a higher entropy rate than the input distribution (without losing much of the initial entropy). An extractor can be viewed as an ultimate condenser because it outputs a distribution with the maximal entropy rate. In this paper we construct explicit condensers with short seed length. The condenser constructions combine (variants of or more efficient versions of) ideas from several works, including the block extraction scheme of [N. Nisan and D. Zuckerman, J. Comput. System Sci., 52 (1996), pp. 43-52], the observation made in [A. Srinivasan and D. Zuckerman, SIAM J. Comput., 28 (1999), pp. 1433-1459; N. Nisan and A. Ta-Shma, J. Comput. System Sci., 58 (1999), pp. 148-173] that a failure of the block extraction scheme is also useful, the recursive win-win case analysis of [R. Impagliazzo, R. Shaltiel, and A. Wigderson, Near-optimal conversion of hardness into pseudo-randomness, in Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science, IEEE, Los Alamitos, CA, 1999, pp. 181-190; R. Impagliazzo, R. Shaltiel, and A. Wigderson, Extractors and pseudo-random generators with optimal seed length, in Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, ACM, New York, 2000, pp. 1-10], and the error correction of random sources used in [L. Trevisan, J. ACM, 48 (2001), pp. 860-879]. As a by-product (via repeated iterating of condensers), we obtain new extractor constructions. The new extractors give significant qualitative improvements over previous ones for sources of arbitrary min-entropy; they are nearly optimal simultaneously in the two main parameters of seed length and output length. Specifically, our extractors can make any one of these two parameters optimal (up to a constant factor) only at a polylogarithmic loss in the other. Previous constructions require polynomial loss in both cases for general sources. We also give a simple reduction converting standard extractors (which are good for an average seed) into ones (which are good for most seeds), with essentially the same parameters. With this reduction, all the above improvements apply to strong extractors as well.

Progress in Computational Complexity Theory

We briefly survey a number of important recent achievements in Theoretical Computer Science (TCS), especially Computational Complexity Theory. We will discuss the PCP Theorem, its implications to inapproximability on combinatorial optimization problems; space bounded computations, especially deterministic logspace algorithm for undirected graph connectivity problem; deterministic polynomial-time primality test; lattice complexity, worst-case to average-case reductions; pseudorandomness and extractor constructions; and Valiant's new theory of holographic algorithms and reductions.

Simple extractors for all min-entropies and a new pseudorandom generator

A “randomness extractor” is an algorithm that given a sample from a distribution with sufficiently high min-entropy and a short random seed produces an output that is statistically indistinguishable from uniform. (Min-entropy is a measure of the amount of randomness in a distribution.) We present a simple, self-contained extractor construction that produces good extractors for all min-entropies. Our construction is algebraic and builds on a new polynomial-based approach introduced by Ta-Shma et al. [2001b]. Using our improvements, we obtain, for example, an extractor with output length m = k /(log n ) O (1/α) and seed length (1 + α)log n for an arbitrary 0 < α ≤ 1, where n is the input length, and k is the min-entropy of the input distribution.A “pseudorandom generator” is an algorithm that given a short random seed produces a long output that is computationally indistinguishable from uniform. Our technique also gives a new way to construct pseudorandom generators from functions that require large circuits. Our pseudorandom generator construction is not based on the Nisan-Wigderson generator [Nisan and Wigderson 1994], and turns worst-case hardness directly into pseudorandomness. The parameters of our generator match those in Impagliazzo and Wigderson [1997] and Sudan et al. [2001] and in particular are strong enough to obtain a new proof that P = BPP if E requires exponential size circuits.Our construction also gives the following improvements over previous work:---We construct an optimal “hitting set generator” that stretches O (log n ) random bits into s Ω(1) pseudorandom bits when given a function on log n bits that requires circuits of size s . This yields a quantitatively optimal hardness versus randomness tradeoff for both RP and BPP and solves an open problem raised in Impagliazzo et al. [1999].---We give the first construction of pseudorandom generators that fool nondeterministic circuits when given a function that requires large nondeterministic circuits. This technique also give a quantitatively optimal hardness versus randomness tradeoff for AM and the first hardness amplification result for nondeterministic circuits.

Constructing Locally Computable Extractors and Cryptosystems in the Bounded-Storage Model

We consider the problem of constructing randomness extractors that are locally computable; that is, read only a small number of bits from their input. As recently shown by Lu locally computable extractors directly yield secure private-key cryptosystems in Maurer’s bounded-storage model. We suggest a general “sample-then-extract” approach to constructing locally computable extractors: use essentially any randomness-efficient sampler to select bits from the input and then apply any extractor to the selected bits. Plugging in known sampler and extractor constructions, we obtain locally computable extractors, and hence cryptosystems in the bounded-storage model, whose parameters improve upon previous constructions. We also provide lower bounds showing that the parameters we achieve are nearly optimal. The correctness of the sample-then-extract approach follows from a fundamental lemma of Nisan and Zuckerman, which states that sampling bits from a weak random source roughly preserves the min-entropy rate. We also present a refinement of this lemma, showing that the min-entropy rate is preserved up to an arbitrarily small additive loss, whereas the original lemma loses a logarithmic factor.

On the distribution of the number of roots of polynomials and explicit weak designs

AbstractWeak designs were defined in R. Raz, O. Reingold, and S. Vadhan [Extracting all the randomness and reducing the error in Trevisan's extractors, Proc 31st ACM Symp Theory of Computing, Atlanta, GA, May 1999, to appear in J Comput System Sci Special Issue on STOC 99] and are used there in constructions of extractors. Roughly speaking, a weak design is a collection of subsets satisfying some near‐disjointness properties. Constructions of weak designs with certain parameters are given in Raz et al. These constructions are explicit in the sense that they require time and space polynomial in the number of subsets. However, the constructions require time and space polynomial in the number of subsets even when needed to output only one specific subset out of the collection. Hence, the constructions are not explicit in a stronger sense. In this work we provide constructions of weak designs (with parameters similar to the ones of Raz et al.) that can be carried out in space logarithmic in the number of subsets. Moreover, our constructions are explicit even in a stronger sense: Given an index to a subset, we output the specified subset in time and space polynomial in the size of the index. Using our constructions, we obtain extractors similar to some of the ones given in Raz et al. in terms of parameters, and that can be evaluated in logarithmic space. Our main construction is algebraic. In order to prove the properties of weak designs, we prove some algebro‐combinatorial lemmas that may be interesting in their own right. These lemmas regard the number of roots of polynomials over finite fields. In particular, we prove that the number of polynomials (over any finite field) with k roots, vanishes exponentially in k. In other words, we prove that the number of roots of a random polynomial is not only bounded by its degree (a well‐known fact), but, furthermore, it is concentrated exponentially around its expectation (which is 1). Our lemmas are proved by algebro‐combinatorial arguments. The main lemma is also proved by a probabilistic argument. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 235–263, 2003

Extracting all the Randomness and Reducing the Error in Trevisan's Extractors

We give explicit constructions of extractors which work for a source of any min-entropy on strings of length n. These extractors can extract any constant fraction of the min-entropy using O(log 2 n) additional random bits, and can extract all the min-entropy using O(log 3 n) additional random bits. Both of these constructions use fewer truly random bits than any previous construction which works for all min-entropies and extracts a constant fraction of the min-entropy. We then improve our second construction and show that we can reduce the entropy loss to 2 log(1/ ε)+ O(1) bits, while still using O(log 3 n) truly random bits (where entropy loss is defined as [(source min−entropy)+(#truly random bits used)−(#output bits)], and ε is the statistical difference from uniform achieved). This entropy loss is optimal up to a constant additive term. Our extractors are obtained by observing that a weaker notion of “combinatorial design” suffices for the Nisan–Wigderson pseudorandom generator, which underlies the recent extractor of Trevisan. We give near-optimal constructions of such “weak designs” which achieve much better parameters than possible with the notion of designs used by Nisan–Wigderson and Trevisan. We also show how to improve our constructions (and Trevisan's construction) when the required statistical difference ε from the uniform distribution is relatively small. This improvement is obtained by using multilinear error-correcting codes over finite fields, rather than the arbitrary error-correcting codes used by Trevisan.

Extractors and pseudorandom generators

We introduce a new approach to constructing extractors. Extractors are algorithms that transform a “weakly random” distribution into an almost uniform distribution. Explicit constructions of extractors have a variety of important applications, and tend to be very difficult to obtain.We demonstrate an unsuspected connection between extractors and pseudorandom generators. In fact, we show that every pseudorandom generator of a certain kind is an extractor.A pseudorandom generator construction due to Impagliazzo and Wigderson, once reinterpreted via our connection, is already an extractor that beats most known constructions and solves an important open question. We also show that, using the simpler Nisan--Wigderson generator and standard error-correcting codes, one can build even better extractors with the additional advantage that both the construction and the analysis are simple and admit a short self-contained description.

Expanders That Beat the Eigenvalue Bound: Explicit Construction and Applications

and 0 < δ < 1, we construct graphs on n nodes such that every two sets of size share an edge, having essentially optimal maximum degree . Using known and new reductions from these graphs, we derive new explicit constructions of: 1. A k round sorting algorithm using comparisons. 2. A k round selection algorithm using comparisons. 3. A depth 2 superconcentrator of size . 4. A depth k wide-sense nonblocking generalized connector of size . All of these results improve on previous constructions by factors of , and are optimal to within factors of . These results are based on an improvement to the extractor construction of Nisan & Zuckerman: our algorithm extracts an asymptotically optimal number of random bits from a defective random source using a small additional number of truly random bits.

Randomness‐optimal oblivious sampling

We present the first efficient oblivious sampler that uses an optimal number of random bits, up to an arbitrary constant factor bigger than 1. Specifically, for any α>0, it uses (1+α)(m+log γ−1) random bits to output d=poly(ϵ−1, log γ−1, m) sample points z1,…,zd∈{0, 1}m such that for any function f: {0, 1}m→[0, 1], Pr[|(1/d)∑i=1df(zi)−Ef|≤ϵ]≥1−γ. Our proof is based on an improved extractor construction. An extractor is a procedure which takes as input the output of a defective random source and a small number of truly random bits, and outputs a nearly random string. We present the first optimal extractor, up to constant factors, for defective random sources with constant entropy rate. We give applications to constructive leader election and reducing randomness in interactive proofs. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 11, 345–367 (1997)