A. Ta-Shma, "On Extracting Randomness from Weak Random Sources." In Proceedings of the 28th Annual ACM Symposium on Theory of Computing, 1996, pp. 276-285.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....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 ....

....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 is (log log n) O(1) Acknowledgements We thank Noga Alon, Nabil Kahale, Nick Pippenger, and Greg Plaxton ....

A. Ta-Shma, "On Extracting Randomness from Weak Random Sources." In Proceedings of the 28th Annual ACM Symposium on Theory of Computing, 1996, pp. 276-285.

....Improved Leftover Hash Lemma and its use in extractors have served as building blocks for several recent results. First, Saks, Srinivasan, and Zhou [SSZ] have improved our RP simulation to poly(n) time. Second, substantial progress on the BPP simulation question has been made by Ta Shma [Ta S], where an R O(log (k) R) algorithm is given for every fixed positive integer k. Third, ideas from this paper have been extended by the second author to give optimal extractors for constant rate sources, as well as randomness optimal samplers [Zu4] In an exciting new result, Andreev, ....

....of ffi sources with min entropy R fl for any fixed fl 0, efficient extractors which use O(log R) purely random bits to extract as many as (1 Gamma o(1) R fl bits, which are quasi random to within R Gamma Theta(1) It is easy to show that such extractors exist, non constructively. In [Ta S] it is shown that polylog(R) bits suffice to do this. Constructing a near optimal family of dispersers may be an interesting step in this direction. See [Nis] for a survey of some of the recent results in this area. Acknowledgements. We thank Avi Wigderson for helpful discussions, and the two ....

A. Ta-Shma, "On extracting randomness from weak random sources," Proc. 28th Annual ACM Symposium on Theory of Computing, 1996, pp. 276-285.

....if better explicit extractor constructions are found. We have stated our results in general terms so that new results on extractors will be immediately applicable. The current best known explicit constructions for extractors are due to Ta Shma, Zuckerman, Trevisan, and Raz, Reingold and Vadhan [Ta 96, Zuc96, Tre99, RRV99]. The extractors best suited for our purposes are the ones which can be constructed for any k, and with m = k, with the 6 smallest amount of additional randomness. We illustrate our results with the parameters obtained from Ta Shma s construction. Theorem 10 (Ta Shma) There is an explicit ....

....is an absolute constant, c is a constant depending only on , k = 1 2 (CND 2 d Deltap (x) Gamma 2 log(n) Gamma c 1 Gamma 1) and (n; k; d; m; are the parameters of an explicit extractor. 10 Theorem 18 follows by applying Theorem 19 with parameters obtained from Ta Shma s extractor [Ta 96]. Proof: Sketch) Consider a family of extractors with parameters n; k; m(k) Fix any n; k and let G = G n;k;m ; m = m(k) be the extractor with parameters n; m; k. Later we will fix k to be a specific value. Let A n;m = fxj Gamma(x) C[q(n) m Gamma c ]g, where C[t; l] fzjC t (z) lg, ....

A. Ta-Shma. On extracting randomness from weak random sources (extended abstract). In Proceedings of the 28th ACM Symposium on the Theory of Computing, pages 276--285, 1996.

.... of STOC 99 dreich [CG88] and Cohen and Wigderson [CW89] and finally by Zuckerman [Zuc90] who introduced the modern definition of weak random source and a construction of extractors (although the term extractor was coined later, in [NZ93] Improved constructions of extractors appeared in [NZ93, SZ94, TS96, Zuc96b]. Neither 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 al. SSZ98] ....

....They also 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 constuction 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] The literature on explicit construction of extractors and dispersers is vast and technically challenging. An excellent ....

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A. Ta-Shma. On extracting randomness from weak random sources. In Proceedings of the 28th ACM Symposium on Theory of Computing, pages 276--285, 1996.

....with a family of (n; n o(1) sources. The best one can hope to achieve is to have, for every 0, a simulation that works in polynomial time given a (n; n ) source. We will call such a simulation an entropy rate optimal simulation. Improved constructions of extractors appeared in [NZ93, SZ94, TS96, Zuc96b] but none of these constructions implies an entropy rate optimal simulation of randomized algorithms. Dispersers are objects similar to, but less powerful than, extractors. Randomized algorithms having one sided error probability can be simulated by using weak random sources and ....

.... Saks et al. SSZ98] give a construction of dispersers that implies an entropy rate optimal simulation of one sided error randomized 2 Reference Min entropy k Output length m Additional randomness t Type [GW97] n a n (a) O(a) Extractor [Zuc96b] k = n) m = 1 )k t = O(log n) Extractor [TS96] any k m = k t = O( log n) 9 ) Extractor [TS96] k = n 19 m = k 1 t = O(log n log log n) Extractor [SSZ98] k = n 19 m = k 1 t = O(log n) Disperser [TS98] any k m = k poly log n t = O(log n) Disperser This paper k = n 19 m = k 1 t = O(log n) Extractor any k m = k 1 t = ....

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A. Ta-Shma. On extracting randomness from weak random sources. In Proceedings of the 28th ACM Symposium on Theory of Computing, pages 276-285, 1996.

.... fact, most of the known explicit constructions satisfy the second requirement (and thus satisfy the first as well) These include almost all the explicit constructions of pseudorandom generators (e.g. NW88, Nis90] expanders (e.g. GG81, LPS86] dispersers (e.g. CW89, SSZ95] extractors (e.g. [NZ93, TaS96]) k wise independent sample spaces (e.g. Lub85, ABI86] small bias sample spaces (e.g. NN90, AGHP91] etc. On the other hand, explicit constructions obtained via using the method of conditional probabilities (e.g. Rag88, SZ96] or solving linear constraints (e.g. Sch92, KK94] typically ....

....sources. The best previously known constructions are due to Zuckerman [Zuc91, Zuc96] who achieved degree polylogarithmic in N if N = T O(1) and Srinivasan and Zuckerman [SZ94] whose construction works for N = 2 polylog(T ) but requires degree (log N) O(loglogN) In recent work, Ta Shma [TaS96] improved the degree of the construction in [SZ94] to (log N) O(log (k) log N) for any fixed integer k, where log (k) denotes the logarithm to the base 2 iterated k times. In this chapter, we give an improved construction of an (N; M;T ) disperser. Main Theorem: 8; 1 0, 9N 0 ( ....

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A. Ta-Shma. On extracting randomness from weak random sources. In Proc. of ACM Symposium on Theory of Computing, 1996.

....of theorem 1 stated here. A stronger version in which = 1=polylog(n) is necessary. The stronger version follows from our exact analysis, see section 7. 2 Table 1: Extracting a constant fraction: m = 1 )k for arbitrary 0 reference min entropy k seed length d [Zuc97] k = n) O(log n) [TS96] any k O(log 9 n) ISW00] k = 2 O( p log n) O(log n log log log n) RRV99b] any k O(log 2 n) Thm. 1 any k O(log n (log log n) 2 ) optimal any k O(log n) Table 2: Optimizing the seed length: d = O(log n) reference min entropy k output length m [Zuc97] k = n) 1 )k [Tre99] k ....

....Speaking informally, a strong extractor is an extractor in which the output distribution is independent of the seed 6 . In some applications of extractors it is bene cial to have the strong version. Most extractor constructions naturally lead to strong extractors, yet some (with examples being [TS96, ISW00] and the constructions of this paper) are not strong or dicult to analyze. We solve this diculty by giving a general explicit transformation which transforms any extractor into a strong extractor with essentially the same parameters. Exact details are given in section 8. 1.4 Technique High ....

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Amnon Ta-Shma. On extracting randomness from weak random sources (extended abstract) . In Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing, pages 276-285, Philadelphia, Pennsylvania, 22-24 May 1996.

....E#x; y# is # close to the uniform distribution on # m . E#x; y# denotes the node on the right hand side that is reached from x following y. Ideally, we would like to be able to effectively and efficiently build extractors. In fact, quite good extractors have been constructed in recent years [TS96], Zuc96] see also the survey paper [Nis96] However, in the best such extractors, the length of y is polylog in x and 1=# and this is too large for our purposes. In our setting, y will be part of the local random string, and thus, we would like to use an as short y as possible. On the other ....

A. Ta-Shma. On extracting randomness from weak random sources. In Proceedings of the 26th ACM Symposium on Theory of Computing, pages 276--285, 1996.

....optimal extractors (as in Definition 4.2) exist then there exists an RL Turing machine T such that: 1. The inputs for T are M and A. 2. T uses O(log n) random bits (and O(log n) space) 3. The output of T is a 2 Gammar error pseudo random sequence for M . Using the extractors of Ta Shma [13] (rather than optimal ones) it is possible to prove an unconditional version of Corollary 2.1 for r = p log n= poly log log n) instead of r = p log n) This is almost as good as the result obtained by optimal extractors and it can be somewhat improved using new constructions of extractors ....

....f0; 1g n . Denote by Un the uniform distribution over I n . For simplicity, we also denote by Un a random variable uniformly distributed over I n (we identify randomvariables and their distributions in a few additional places in this paper) We use the definition of extractors given in [13] (which is a variant of the original definition given in [8] Definition 4.1 (Extractors) A function E : I Theta I t 7 I m 0 is an (m; ffl) extractor if for any random variable Z over I with min entropy at least m, D(E(Z; U t ) Um 0 ) ffl: That is, the distribution of E(Z; U t ) ....

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A. Ta-Shma, On extracting randomness from weak random sources, Proc. 28th Ann. ACM Symp. on Theory of Computing, pp. 276-285, 1996.

....of E(x; y) is close to the uniform distribution on m . E(x; y) denotes the node on the right hand side that is reached from x following y. Ideally, we would like to be able to e ectively and eciently build extractors. In fact, quite good extractors have been constructed in recent years [TS96], Zuc97] Tre99] Vad99] see also the survey paper [Nis96] However, in the best such extractors, the length of y is polylog in the length of 1= and, since we need to have = 2 jxj) this is too large for our purposes. In our setting, y will be part of the local random string, and thus, ....

A. Ta-Shma. On extracting randomness from weak random sources. In Proceedings of the 26th ACM Symposium on Theory of Computing, pages 276-285, 1996.

....not surprising that extractors have attracted much attention. The main objective has been the construction of extractors with better parameters. In general, considering n and k as xed, we want to make d and small and m large. The best results have been established by Zuckerman [Zuc97] TaShma [TS96] Trevisan [Tre99] and Raz, Reingold, and Vadhan[RRV99] In the case in which k = n) Zuckerman has constructed an extractor with d = O(log(n) log( 1 ) and m = n) Ta Shma s extractor works for any k (of course, k n) and has d = poly(log(n) log( 1 ) and m = k. Trevisan s ....

A. Ta-Shma. On extracting randomness from weak random sources. In Proceedings of the 26th ACM Symposium on Theory of Computing, pages 276-285, 1996.

....we state are subject to improvement if better explicit extractor constructions are found. We have stated our results in general terms so that new results on extractors will be immediately applicable. The current best known explicit constructions for extractors are due to Ta Shma and Zuckerman [Ta 96,Zuc96]. The Ta Shma construction is more useful for our purposes. Theorem 2. Ta Shma) There is an explicit construction which for every n; n) and every m = m(n) n yields an extractor with parameters (n; m; log O(1) n= m; It is useful to compare this construction to the current ....

.... Sigma n , CD p (x) 2 log(jA Sigma n j) c log n. We extend the work of Buhrman and Fortnow [BF97] by getting nearly the bound of Sipser [Sip83] without the random string used by Sipser. However, our result only works for most strings in A. Using the extractor construction of Ta Shma [Ta 96] we get the following theorem. Theorem 6. For any set A 2 P, n) there is a polynomial p such that for all n and for all but a 2 fraction of the x 2 A Sigma n , CD p (x) log jA Sigma n j log O(1) n= We give a few extensions of this result. In one extension, we ....

A. Ta-Shma. On extracting randomness from weak random sources (extended abstract). In Proceedings of the 28th ACM Symposium on the Theory of Computing, pages 276285. ACM, New York, 1996.

.... have been the focus of much research in recent years (see the survey in [16] and have found applications in a wide range of areas, including simulating randomized algorithms with weak random sources [37, 27] explicit constructions of expanders, superconcentrators and sorting networks [35, 30], constructive leader election [38, 23] and several diverse applications in complexity theory [18, 2, 29, 26, 10] The dispersers we use in this paper are ones that work on sources with very small min entropy; constructions for this parameter range are in [27, 9] logarithmic degree) and [16, 32, ....

A. Ta-Shma. On extracting randomness from weak random sources (extended abstract). In Proceedings of the 28th Annual ACM Symposium on the Theory of Computing (STOC 96), pages 276--285, Philadelphia, Pennsylvania, 22--24 May 1996.

....our new constructions yield better results, e.g. plugging our new extractors into a previous construction we achieve the first explicit N superconcentrators of linear size and polyloglog(N ) depth. This paper is a combination of the paper On Extracting Randomness From Weak Random Sources [Ta 96] and the paper Refining Randomness: Why and How [Nis96] This work was supported by BSF grant 92 00043 and by a Wolfeson award administered by the Israeli Academy of Sciences. y Institute of computer science, Hebrew University, Jerusalem, Israel z Institute of computer science, Hebrew ....

Ta-Shma. On extracting randomness from weak random sources. In ACM Symposium on Theory of Computing (STOC), 1996.

....strings are fed into the algorithm, and then the majority rule is used to decide whether to accept or reject. The procedure that computes the sample space starting from the output of the source is independent of the algorithm that we want to derandomize. This simulation is basically equivalent [Z90, Z96, NZ96, SZ94, SSZ95, T96] to a bipartite graph G = V, W,E) having 2 r nodes in the left component V , 2 m nodes in the right component W , and degree d, and such that if we select a node v in the left component according to an (r, r # ) source and then a random neighbor of v, the induced distribution in W is ....

....space the set of its neighbors. If, for some fixed #, one could achieve d and r polynomial in m, then a polynomial time simulation of BPP would be possible, using an (r, r # ) source. However, the best present construction of extractors for fixed # 0 and r = poly(m) has d = n log (k) n [T96]. This implies a quasi polynomial time simulation of BPP. A polynomial time simulation of BPP, using weak random sources of min entropy r # for any fixed # 0, was one of the major open questions in the field. It is not di#cult to show that, to simulate RP by means of a weak random source, OR ....

<F3.742e+05> A. Ta-Shma,<F4.081e+05> On extracting randomness from weak random sources,<F3.815e+05> in Proc. 28th ACM Symposium on Theory of Computing, 1996, pp. 276--285.

....Such strings are fed into the algorithm and then the majority rule is used to decide whether to accept or reject. The procedure that computes the sample space starting from the output of the source is independent of the algorithm that we want to derandomize. This simulation is basically equivalent [Zuc90, Zuc96b, NZ96, SZ94, SSZ95, TS96] to a bipartite graph G = V; W;E) having 2 r nodes in the left component V , 2 m nodes in the right component W , degree d and such that if we select a node v in the left component according to an (r; r fl ) source, and then a random neighbour of v, the induced distribution in W is ....

....the set of its neighbours. If, for some fixed fl, one could achieve d and r polynomial in m, then a polynomial time simulation of BPP would be possible using an (r; r fl ) source. However, the best present construction of extractors for fixed fl 0 and r = poly(m) has d = n log (k) n [TS96]. This implies a quasi polynomial time simulation of BPP. A polynomial time simulation of BPP using weak random sources of min entropy r fl for any fixed fl 0 was one of the major open questions in the field. It is not difficult to show that to simulate RP by means of a weak random source, OR ....

A. Ta-Shma. On extracting randomness from weak random sources. In Proceedings of the 28th ACM Symposium on Theory of Computing, pages 276--285, 1996.

....we state are subject to improvement if better explicit extractor constructions are found. We have stated our results in general terms so that new results on extractors will be immediately applicable. The current best known explicit constructions for extractors are due to Ta Shma and Zuckerman [Ta 96,Zuc96]. Theorem 2. Ta Shma) There is an explicit construction which for every n; n) and every m = m(n) n yields an extractor with parameters (n; m; log O(1) n= m; We use Ta Shma s construction in this paper in order to obtain concrete bounds. It is useful to compare these ....

.... Sigma n , CD p (x) 2 log(jA Sigma n j) c log n. We extend the work of Buhrman and Fortnow [BF97] by getting nearly the bound of Sipser [Sip83] without the random string used by Sipser. However, our result only works for most strings in A. Using the extractor construction of Ta Shma [Ta 96] we get the following theorem. Theorem 6. For any set A 2 P, n) there is a polynomial p such that for all n and for all but a 2 fraction of the x 2 A Sigma n , CD p (x) log jA Sigma n j log O(1) n= We give a few extensions of this result. In one extension, we ....

A. Ta-Shma. On extracting randomness from weak random sources (extended abstract). In Proceedings of the 28th ACM Symposium on the Theory of Computing, pages 276--285. ACM, New York, 1996.

....is ffl close to the uniform distribution on Sigma m . E(x; y) denotes the node on the right hand side that is reached from x following y. Ideally, we would like to be able to effectively and efficiently build extractors. In fact, quite good extractors have been constructed in recent years [TS96], Zuc96] see also the survey paper [Nis96] However, in the best such extractors, the length of y is polylog in x and 1=ffl and this is too large for our purposes. In our setting, y will be part of the local random string, and thus, we would like to use an as short y as possible. On the other ....

A. Ta-Shma. On extracting randomness from weak random sources. In Proceedings of the 26th ACM Symposium on Theory of Computing, pages 276--285, 1996.

.... Formally, an (m; ffl) extractor is a function E : X Theta T Y if, for any random variable X with min entropy H1 (X) m, the output Y = E(X;T ) of the extractor satisfies kP Y Gamma PU k v ffl when T is chosen uniformly random from T and where PU denotes the uniform distribution over Y [TS96]. Using n = log jX j, t = log jT j, and s = log jYj, the best currently 64 Smooth Entropy known constructions of extractors can produce s = Omega Gamma m) almost uniform bits using t = O(log(n=ffl) truly random bits, from an n bit source with min entropy m = Omega Gamma n) Zuc96a] Another ....

.... Entropy known constructions of extractors can produce s = Omega Gamma m) almost uniform bits using t = O(log(n=ffl) truly random bits, from an n bit source with min entropy m = Omega Gamma n) Zuc96a] Another construction achieves s = m with t = poly(log(n=ffl) extracting all min entropy [TS96], but requiring a larger auxiliary random input. At the core, all extractor constructions known are based on entropy smoothing by variations of universal hashing (in the sense of Theorems 2.7 and 4.2) But since it requires at least n uniformly random bits to smooth an n bit source by universal ....

Amnon Ta-Shma, On extracting randomness from weak random sources, Proc. 28th Annual ACM Symposiumon Theory of Computing (STOC), 1996, pp. 276--285.

....progress on this problem is summarized in Table 1 for the case of constant error . Building on [32, 33] Nisan and Zuckerman [15] constructed an extractor with t = O(log 2 n) for high min entropy k = n) Srinivasan and Zuckerman [25] extended this solution to the case k = n 1=2 and TaShma [26] further extended it for any min entropy k. Also, Ta Shma was the rst to extract all the min entropy from the source. Zuckerman [34] showed a construction with t = O(log n) working for high min entropies k = n) 1 For a de nition see Subsection 1.4. 2 An extractor E is explicit if E can be ....

....Subsection 1.4. 2 An extractor E is explicit if E can be evaluated in polynomial time. required no. of no. of reference entropy truly random bits output bits Any k log n (1) k t (1) L. bound non explicit [17] n) O(log 2 n) k) 15] n) O(log n) k) 34] Any k polylog(n) m = k [26] Any k O(log 2 n= log k) k 1 [28] Any k O(log n) k= log n [20] Any k O(log n log k(log log k) 2 ) 1 )k Thm. 5 Any k O(log n log 2 k(log log k) 2 ) k t O(1) Thm. 5 Table 1: Milestones in building explicit extractors. The error is a constant. Departing from previous ....

A. Ta-Shma. On extracting randomness from weak random sources. In 28th STOC, pages 276-285, 1996.

....of output bits, as large as possible. Building on earlier work of Zuckerman [Zuc90, Zuc96] Nisan and Zuckerman [NZ96] built an extractor with t =O(log 2 n) when the entropy of the source k was high, k =W(n) Srinivasan and Zuckerman extended this solution to the case k = n 1=2 e and Ta Shma [TS96] further extended it for any entropy k. Also, TaShma was the first to extract all the entropy from the source. Zuckerman [Zuc97] showed a construction with t = O(logn) working for high entropies k =W(n) All of this work used hashing and k wise independence in various forms. Departing from ....

A. Ta-Shma. On extracting randomness from weak random sources. In Proceedings of the 28th Annual ACM Symposium on Theory of Computing, pages 276--285, 1996.

....described later) We simplify the SSZ combinatorial construction, giving a simpler analysis with better bounds. We use the new family of segmentations to build the disperser graph in much the same way as is done in [SSZ95] but we prove the correctness of the construction using the terminology of [Ta 96], which results in a simpler proof with tighter bounds. Theorem 1 For every constant fl 1 , ffl 2 Gamman fl and any k n, there is an explicit (K = 2 k ; ffl) disperser G = V = N = 2 n ] W = M = 2 k Gammapoly(log n;log( 1 ffl ) E) with degree D poly(n; 1 ffl ) ....

....[SSZ95] disperser has n Omega Gamma43 entropy loss, while our disperser has poly(log(n) log( 1 ffl ) entropy loss. The reason we achieve much smaller entropy loss is connected to the fact that we have good dispersers for any parameter k, and uses the existence of a good extractor presented in [Ta 96]. Yet, even the entropy loss we achieve is still away from the optimal one which is only log log( 1 ffl ) O(1) and reducing the min entropy loss to the optimal is an important open problem with many applications (e.g. for the construction of explicit a expanding graphs and depth 2 ....

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A. Ta-Shma. On extracting randomness from weak random sources. In STOC, 1996.

....Research Pearls in Theory of Computation [NW] we refine the construction in [WZ93] and get the upper bound by putting together a small number of dispersers. These dispersers are obtained by probabilistic arguments; the best explicit construction known uses dispersers constructed by TaShma [T96a], and gives N superconcentrators of size O(N(log N ) poly log log n ) see [T96b, Nis96] We also observe a connection in the opposite direction: every depth two superconcentrator contains many disjoint dispersers. Thus, lower bounds for dispersers imply lower bounds for depth two ....

A. Ta-Shma. On extracting randomness from weak random sources. In STOC 1996, pages 276--285, May 1996.

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Amnon Ta-Shma. On extracting randomness from weak random sources (extended abstract). In Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing, pages 276--285, Philadelphia, Pennsylvania, 22--24 May 1996.

No context found.

Amnon Ta-Shma. On extracting randomness from weak random sources (extended abstract). In Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing, pages 276--285, Philadelphia, Pennsylvania, 22--24 May 1996.

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