Srinivasan, A., Zuckerman, D.: Computing with Very Weak Random Sources. SIAM Journal on Computing 28:4 (1999) 14531459  Home/Search   Document Details and Download   Summary   Related Articles   Check

....poly(n) compute h i (x) Thus, investing enough true randomness, namely the amount needed to select a random member of H, one can extract something statistically close to a truly random string from the randomness in a given distribution X. Much work has been done in developing this area (e.g. [46, 32, 59, 64, 45, 61, 50, 49]) In particular, it turns out that one can extract almost all the randomness in X by investing very few truly random bits (i.e. having small H) We will use the following eciently constructible families of strong extractors developed by [59, 49] Theorem 4 ( 59, 49] For any n, m and such ....

.... developing this area (e.g. 46, 32, 59, 64, 45, 61, 50, 49] In particular, it turns out that one can extract almost all the randomness in X by investing very few truly random bits (i.e. having small H) We will use the following eciently constructible families of strong extractors developed by [59, 49]. Theorem 4 ( 59, 49] For any n, m and such that n m 2 log(1= there exist ecient strong (m; extractor families H = fh i : f0; 1g g 43 1. k = m 2 log(1= O(1) and d = 4(m log(1= O(logn) 59] 2. k = 1 )m O(log(1= and d = O(log n log(1= 8 const. 0) 49] ....

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A. Srinivasan, D. Zuckerman. Computing with Very Weak Random Sources. In SIAM J. on Computing, 28(4):1433-1459, 1999.

....uniform distribution even when the random function h is revealed. Perhaps the best known example of a strong extractor is given in the Leftover Hash Lemma of [13] where standard 2 universal hash families are shown to be strong extractors. Much work has been done in developing this area (e.g. [24, 26, 18]) In particular, it turns out that one can extract almost all the randomness in X by investing very few truly random bits (i.e. having small H) The intuition behind our construction is as follows. Notice that after the adversary observes (n ) bits of the input (no matter how it chose those ....

....u) must be chosen uniformly at random (and then possibly observed by the adversary) Our most important requirement is that the hash function in the strong extractor family be describable by a very short random string. This requirement is met by the strong extractor of Srinivasan and Zuckerman [24] using the hash families of Naor and Naor [17] Their results can be summarized as follows: Lemma 1 ( 24] For any and t =2, there exists a family H of hash functions mapping f0; 1g to a range f0; 1g , where k = 2t, such that the following holds: A random member of H can be ....

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A. Srinivasan, D. Zuckerman. Computing with Very Weak Random Sources. In Proc. of FOCS, pp. 264-275, 1994.

.... rather than dispersers (unlike [Ta 98] The disadvantage of the extractors of [GW97] described in Figure 2 is that they only use a small number of truly random bits when the source min entropy k is very close to the input length n (e.g. k = n polylog(n) There are also extractors given in [GW97, SZ98] which extract all of the min entropy, but these use a small number of truly random bits only when the source min entropy is very small (e.g. k = polylog(n) and these extractors are further discussed in the context of entropy loss. Plugging the second extractor of Theorem 1 into a construction ....

....of [Ta 98] have entropy losses of polylog n. The extractor of [GW97] is actually better than Figure 2 indicates; it is a strong extractor with an entropy loss of n k O(log(1= though this is only interesting when k is very close to n) In addition, the tiny families of hash functions of [SZ98] give strong extractors with d = O(k log n) and entropy loss 2 log(1= O(1) these have optimal entropy loss but are only interesting when k is very small (e.g. k = polylog n) as d is linear in k. The fact that d here does not explicitly depend on is not a contradiction to the lower bound ....

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Aravind Srinivasan and David Zuckerman. Computing with very weak random sources. To appear in SIAM Journal on Computing, 1998. Preliminary version in FOCS `94.

.... decreasing the error probability of randomized algorithms (deterministic amplification, oblivious sampling) and the construction of graphs with random properties (superconcentrators, expanders) More information about applications of extractors can be found in the work of Zuckerman [Zuc91, SZ94, WZ95, NZ95, Zuc96a, Zuc96b] and Nisan [Nis96] 4.4 Spoiling Knowledge We now turn to further characterizations of smooth entropy and to lower bounds in terms of R enyi entropy. Corollary 4.1 shows that R enyi entropy of order 2 is a lower bound for the smooth entropy of a distribution. As ....

.... using almost universal hash functions based on almost k wise independent random variables that can be constructed efficiently [AGHP92] Such functions g : X Y can be described with about 5 log jYj instead of log jX j bits and can replace universal hash functions in privacy amplification [GW96, SZ94] 5.4.6 Discussion Our results show that unconditional security can be based on assumptions about the adversary s available memory. In essence, such a system exploits the capacity gap between fast communication and mass storage technology. We discuss a few implications of this fact. First of ....

Aravind Srinivasan and David Zuckerman, Computing with very weak random sources, Preprint available from the authors, preliminary version presented at 35th FOCS (1994), 1994.

.... [NZ96] Much of the motivation for research on extractors comes from work done on somewhat random sources [SV86, CG88, Vaz87b, VV85, Vaz84, Vaz87a] There have been a number of papers giving explicit constructions of dispersers and extractors, with a steady improvement in the parameters [Zuc96, NZ96, WZ95, GW97, SZ98, SSZ98, NT98, TS98, Tre98]. Most of the work on extractors is based on techniques such as k wise independence, the Leftover hash lemma [ILL89] and various forms of composition. A new approach to constructing extractors was recently initiated by Trevisan [Tre98] who discovered that the Nisan Wigderson pseudorandom ....

....extractors of [GW97] described in Figure 1 is that they only use a small number of truly random bits when the source min entropy k is very close to the input length n (e.g. k = n Gamma polylog(n) whereas ours uses O(log n) random bits for any min entropy. There are also extractors given in [GW97, SZ98] which extract all of the min entropy, but these use a small number of truly random bits only when the source min entropy is very small (e.g. k = polylog(n) and these extractors are better discussed later in the context of entropy loss. Plugging our second extractor into a construction of ....

[Article contains additional citation context not shown here]

Aravind Srinivasan and David Zuckerman. Computing with very weak random sources. To appear in SIAM Journal on Computing, 1998. Preliminary version in FOCS `94.

.... and Cohen and Wigderson [CW89] and finally by Zuckerman [Zuc90] who introduced the modern definition (based on min entropy) of weak random sources and a construction of extractors (although the term extractor was coined later, in [NZ93] Improved constructions of extractors appeared in [NZ93, SZ94, TS96, Zuc96b] None of these constructions implies an optimal simulation of randomized algorithms. Dispersers are objects similar to, but less powerful than, extractors. Randomized algorithm having one sided error probability can be simulated by using weak random sources and dispersers. Saks et ....

....yield expander graphs, as discovered by Wigderson and Zuckerman [WZ93] that in turn have applications to superconcentrators, sorting in rounds, and routing in optical networks. Constructions of expanders via constructions of extractors and the Wigderson Zuckerman connection appeared in [NZ93, SZ94, TS96] among others. Extractors can also be used to give simple proofs of certain complexity theoretic results [GZ97] and to prove certain hardness of approximation results [Zuc96a] An excellent introduction to extractors and their applications is given by a recent survey by Nisan [Nis96] see ....

A. Srinivasan and D. Zuckerman. Computing with very weak random sources. In 264--275, 1994. 16

....(pseudo )min entropy divided by its length) is at least ffi =2, where P is hard to compute correctly on more than a 1 Gamma ffi fraction of inputs. This means that if P has constant average case hardness, it suffices to use a good extractor for constant entropy rate, such as those in [Zuc96, SZ99, Zuc97] Remark 16 It is natural to ask whether similar ideas can be used to directly construct BMY type pseudorandom generators from mild hardness. Specifically, consider a modification of the BMY construction [BM84, Yao82] of pseudorandom generators from strong (i.e. very hard on average) ....

Aravind Srinivasan and David Zuckerman. Computing with very weak random sources. SIAM J. Comput., 28(4):1433--1459 (electronic), 1999.

....s = O(log n) 2.1. 2 Previous work and Applications Extractors were first defined and constructed by Nisan and Zuckerman [NZ96] This followed a large body of work in the late 1980 s, regarding weak notions of randomness [SV86, VV85, CG88, Zuc90] Improved constructions of extractors appeared in [WZ95, GW97, SZ98, Zuc97, NT98]. A new approach to constructing extractors was recently initiated by Trevisan [Tre99] and then [RRV99a] who uses the Nisan Wigderson pseudo random generator [NW94] for constructing extractors (these constructions will be discussed in detail in subsection 2.1.3) Besides the natural ....

Aravind Srinivasan and David Zuckerman. Computing with very weak random sources. To appear in SIAM Journal on Computing, 1998. Preliminary version in FOCS `94.

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A. Srinivasan and D. Zuckerman. Computing with very weak random sources. SIAM Journal on Computing, 28:1433-1459, 1999.

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A. Srinivasan and D. Zuckerman. Computing with very weak random sources. SIAM Journal on Computing, 28:1433--1459, 1999.

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A. Srinivasan, D. Zuckerman, Computing with very weak random sources, SIAM Journal on Computing 28 (1999) 1433--1459.

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A. Srinivasan and D. Zuckerman. Computing with very weak random sources. SIAM Journal on Computing, 28:1433--1459, 1999.

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A. Srinivasan and D. Zuckerman. Computing with very weak random sources. SIAM Journal on Computing, 28:1433--1459, 1999.

....jf Gamma1 (0)j jf Gamma1 (1)j, so jf Gamma1 (0)j 2 n Gamma1 . Then any ffi source outputting only values in f Gamma1 (0) contradicts the claim about f . Given this, is there any hope in using a ffi source Idea: add a small number of truly random bits. Building on earlier work [NZ96, SZ94, WZ95, Zuc96], in [Zuc97] I constructed an efficient extractor E : f0; 1g n Theta f0; 1g O(logn) f0; 1g :99ffin : This has the property that if x is output according to any ffi source and y is uniformly random, then E(x; y) is close to random. The small amount of randomness may be eliminated by ....

A. Srinivasan and D. Zuckerman. Computing with very weak random sources. SIAM Journal on Computing. To appear. Preliminary version in Proceedings of the 35th Annual IEEE Symposium on Foundations of Computer Science, pages 264--275, 1994.

....k. Thus, the following theorem is immediate. Theorem 5.4 For all k and n, there are efficiently constructible wide sense nonblocking generalized n connectors of size O (n 1 1=k ) and depth k. 7 6 Subsequent Work Subsequent to this work, the n o(1) factors have been improved twice [SZ, TS96], by constructing stronger extractors and applying our methods. In the most recent improvement, Ta Shma [TS96] obtained expanders where the n o(1) factors are exp( log log n) O(1) Hence all the applications have these new n o(1) factors and the depth of the linear sized superconcentrator ....

A. Srinivasan and D. Zuckerman, "Computing with Very Weak Random Sources." In Proceedings of the 35th Annual IEEE Symposium on Foundations of Computer Science, 1994, pp. 264-275. To appear in SIAM Journal on Computing.

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Srinivasan, A., Zuckerman, D.: Computing with Very Weak Random Sources. SIAM Journal on Computing 28:4 (1999) 14531459

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A. Srinivasan and D. Zuckerman, Computing with very weak random sources, in Proceedings IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1994, pp. 264--275.

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A. Srinivasan and D. Zuckerman. Computing with very weak random sources. In Proceedings of the 35th Annual IEEE Symposium on the Foundations of Computer Science, 1994.

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A. Srinivasan and D. Zuckerman, Computing with very weak random sources, in Proceedings, 35th Annual IEEE Symposium on the Foundations of Computer Science, IEEE, 1994."

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A. Srinivasan and D. Zuckerman. Computing with very weak random sources. SIAM Journal on Computing, 28(4):1433-1459, August 1999.

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A. Srinivasan and D. Zuckerman, "Computing with Very Weak Random Sources", 35th FOCS, pp. 264--275, 1994.

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A. Srinivasan and D. Zuckerman. Computing with very weak random sources. SIAM Journal on Computing, 28(4):1433--1459, Aug. 1999.

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A. Srinivasan and D. Zuckerman. Computing with very weak random sources. SIAM Journal on Computing, 28:1433--1459, 1999.

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A. Srinivasan and D. Zuckerman. Computing with very weak random sources. SIAM Journal on Computing, 28(4):1433--1459, 1999.

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A. Srinivasan, D. Zuckerman. Computing with Very Weak Random Sources. In Proc. of FOCS, pp. 264--275, 1994.

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