U. Vazirani, Efficiency considerations in using semi-random sources, in Proceedings, 19th Annual ACM Symposium on the Theory of Computing, ACM, 1987," pp. 160#168.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....h. Still, for sufficiently long messages, it will be faster to use the bucket hashing technique from the following section. We comment that there are many other well known techniques for universal hashing, such as the linear congruential hash (modulo a prime) 10] the shift register hash [31], or the Toeplitz matrix hash [18] 3 Bucket Hashing Let X = X1. X be a string, partitioned into n words. To hash X using bucket hashing we will scatter the words of X into N buckets, then XOR the contents of each bucket, and then concatenate the bucket contents. Some ways of scattering the ....

U. Vazirani, Efficiency considerations in using semi-random sources, Proceedings of the Nine- teenth Annual ACM Symposium on Theory of Computing, ACM Press, 1987, pp. 160-168.

....will be discussed in more detail under the heading Entropy loss. Previous work. Dispersers were first defined by Sipser [Sip88] and extractors were first defined by Nisan and Zuckerman [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 ....

Umesh V. Vazirani. Efficiency considerations in using semi-random sources (extended abstract). In Proceedings of the Nineteenth Annual ACM Symposium on Theory of Computing, pages 160--168, New York City, 25--27 May 1987.

....ff: Simulations must work for all such sources; we do not want to assume any knowledge about the source, except that it is semi random with the value of the parameter ff being known. They proved that it is impossible to extract even a single almost random bit from one such source (so Vazirani [Va1, Va3] used two independent sources) In light of this result, one might give up hope for simulating randomized algorithms with one semi random source. Nevertheless, VV] and [Va2] showed how to efficiently simulate all algorithms in RP and BPP with one semi random source, for any constant ff 0. Note ....

U. Vazirani, "Efficiency Considerations in Using Semi-Random Sources," Proc. 19th ACM Symposium on Theory of Computing, 1987, pp. 160-168.

....as a black box simulation. Applying this framework, efficient simulations of RP and BPP under various models of weak sources have been extensively studied. For example, Santha and Vazirani [SV86] studied the class of weak sources called slightly random sources , which was further examined in [VV85, Vaz86, Vaz87a, Vaz87b]. A more general model PRB sources is considered later by Chor and Goldreich [CG88] In [Zuc90, Zuc91] Zuckerman introduced the model of ffi sources which generalizes all the previous models. Let D be a probability distribution on a set X . The min entropy of D is defined to be the maximum ....

U. Vazirani. Efficiency Considerations in Using Semi-Random Sources. In Proc. of ACM Symposium on Theory of Computing, pp. 160-168, 1987.

....set, we write x X to indicate that x is selected according to X . Previous work Dispersers were first defined by Sipser [Sip88] and extractors were first defined by Nisan and Zuckerman [NZ96] Much of the motivation for research on extractors comes from work done on somewhat random sources [SV86, CG88, Vaz87b, VV85, Vaz84, Vaz87a, CW89]. 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, Zuc97, Ta 98, Tre99] Most of the work on extractors is based on techniques such as k wise independence, the ....

Umesh V. Vazirani. Efficiency considerations in using semi-random sources (extended abstract). In Proceedings of the Nineteenth Annual ACM Symposium on Theory of Computing, pages 160--168, New York City, 25--27 May 1987.

.... by Sipser [Sip88] while extractors were defined by Nisan and Zuckerman [NZ93] The roots of the research on extractors lie mostly in the work on somewhat random sources done in the late 1980 s, 5 by Vazirani, Santha and Vazirani, Vazirani and Vazirani, Chor and Goldreich, and others [SV86, Vaz87a, Vaz86, Vaz87b, VV85, CG88] The direct development of the constructions and applications of extractors and dispersers came first in papers written by Zuckerman in the early 1990 s [Zuc90, Zuc91] and then in a sequence of papers by various authors [NZ93, WZ93, SZ94, SSZ95, Zuc93, Zuc] The ....

....should try to produce physical source where this property is as strong as possible, while computer scientists should aim to rely on a property which is as week as possible. Several models of somewhat random sources were considered in the literature. Santha and Vazirani [SV86] Vazirani [Vaz87a, Vaz86, Vaz87b] and Vazirani and Vazirani [VV85] studied a class of sources, called slightly random sources , and showed that BPP can be simulated given such a source. Chor and Goldreich [CG88] generalized this model, and showed that BPP can be simulated even using the more general source. Many ....

U. Vazirani. Efficiency considerations in using semi-random sources. In Proceedings of the 19th Annual ACM Symposium on the Theory of Computing, ACM, pages 160--168, 1987.

....pseudo random generator. A quasi perfect prg can be used to decrease the probability of error of a BPP or RP algorithm from a constant 1=2 to an exponentially small amount, using only a constant factor more random bits. Until now, no prg was proven to be quasi perfect, although Vazirani [V] conjectured that a specific prg, related to the shift register prg presented here, is quasi perfect. Another recent direction of research has been to theoretically explain the practical success of simple prg s like the linear congruential generator. Bach [B] and Karloff and Raghavan [KR] have ....

....probability of error can be reduced to 2 Gammak using O(r k) random bits. The second class of quasi random prg s can be based on any family of universal hash functions ( CW] In particular, we prove that modified versions of both the linear congruential and shift register generators (from [V]) are quasi perfect. The modification is that only the r Gamma O(k) least significant bits of the generator s output on an r bit seed are used; the remaining bits are replaced by truly random bits. This method uses O(r k 2 ) random bits to reduce the error probability to 2 Gammak . The ....

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U. Vazirani, "Efficiency Considerations in Using Semi-Random Sources," 19th STOC, 1987.

....on the effective possibility for an algorithm to access truly random bits. Since it is questionable whether truly random sources really exist, much research has been devoted in the last decade to find weaker notions of randomness that are still sufficient to run BPP algorithms in polynomial time [VV85, SV86, Vaz86, Vaz87, CG88, Zuc90]. Several definitions of weak random source have been proposed in the literature, the most general being the following [CG88, Zuc90] for fl 0, an (r; r fl ) source is a random source that An extended abstract of this paper appears in the Proceedings of the 38 th IEEE Symposium on ....

U. Vazirani. Efficiency considerations in using semi-random sources. In Proceedings of the 19th ACM Symposium on Theory of Computing, pages 160--168, 1987.

....will be discussed in more detail under the heading Entropy loss. Previous work. Dispersers were first defined by Sipser [Sip88] and extractors were first defined by Nisan and Zuckerman [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 ....

Umesh V. Vazirani. Efficiency considerations in using semi-random sources (extended abstract). In Proceedings of the Nineteenth Annual ACM Symposium on Theory of Computing, pages 160--168, New York City, 25--27 May 1987.

.... ffi outputs bits X 1 X 2 : XR , such that for all i R and for all x 1 ; x i 2 f0; 1g, ffi Pr[X i = x i jX 1 = x 1 ; X i Gamma1 = x i Gamma1 ] 1 Gamma ffi: They proved that it is impossible to extract even a single almost random bit from one such source (so Vazirani [Va2, Va3] showed how to extract almost random bits from two independent sources) In light of this result, one might give up hope for simulating randomized algorithms with one semi random source. Nevertheless, VV] and [Va1] showed how to simulate RP and BPP with one semi random source. Chor and Goldreich ....

U. Vazirani, "Efficiency Considerations in Using Semi-Random Sources," 19th ACM Symposium on Theory of Computing, 1987.

....of a randomized procedure. One of the main applications of explicit constructions of such objects is the simulation of randomized algorithms using weak random sources. This research area originates from early work by Vazirani and Vazirani [VV85] Santha and Vazirani [SV86] Vazirani [Vaz86, Vaz87] Chor and Goldreich [CG88] and Cohen and Wigderson [CW89] who defined increasingly general models of weak random sources. The recognition of min entropy as the right parameter to measure the amount of algorithmically usable randomness in a source is due to Zuckerman [Zuc90] Extractors allow ....

U. Vazirani. Efficiency considerations in using semi-random sources. In Proceedings of the 19th ACM Symposium on Theory of Computing, pages 160--168, 1987.

....the effective possibility for an algorithm to access truly random bits. Since it is even questionable whether truly random sources really exist, much research has been devoted in the last decade to find weaker notions of randomness that are still sufficient to run BPP algorithms in polynomial time [VV85, SV86, Vaz86, Vaz87, CG88]. Several definitions of weak random source have been proposed in the literature, the most general being the following [CG88] a source has min entropy r fl for fl 0 if it outputs a string in f0; 1g r and no string has probability of being output larger than 2 r fl . An ....

U. Vazirani. Efficiency considerations in using semi-random sources. In Proceedings of the 19th ACM Symposium on Theory of Computing, pages 160--168, 1987.

....but a fraction of points in f0; 1g m with nonzero probability. Previous work. Dispersers were first defined by Sipser [Sip88] and extractors were first defined by Nisan and Zuckerman [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, Zuc97, TS98b, Tre98] Most of the work on extractors is based on techniques such as k wise independence, the ....

Umesh V. Vazirani. Efficiency considerations in using semi-random sources (extended abstract). In Proceedings of the Nineteenth Annual ACM Symposium on Theory of Computing, pages 160--168, New York City, 25--27 May 1987.

....as a black box simulation. Applying this framework, efficient simulations of RP and BPP under various models of weak sources have been extensively studied. For example, Santha and Vazirani [SV86] studied the class of weak sources called slightly random sources , which was further examined in [VV85, Vaz86, Vaz87a, Vaz87b]. A more general model PRB sources is considered later by Chor and Goldreich [CG88] In [Zuc90, Zuc91] Zuckerman introduced the model of ffi sources which generalizes all the previous models. Let D be a probability distribution on a set X. The min entropy of D is defined to be the maximum real ....

U. Vazirani, "Efficiency Considerations in Using Semi-Random Sources," Proc. ACM Symposium on Theory of Computing, 1987, pp. 160--168.

....are indeed independent. We will prove this with the help of the following Lemma: Lemma: If for some m 0, the family of vectors f a 1 ; am g is linearly independent then a 1 M; amM are independent random variables. Proof of Lemma: It follows from Vazirani s Parity Lemma [Vaz87] when applied to the uniform distribution. 2 It follows from the Lemma that the random variables X S 1 and X S 2 are independent when the family of four vectors f a 1 ; b 1 ; a 2 ; b 2 g is linearly independent. Thus we have to consider the covariances of only those random variables X S 1 ....

U. Vazirani. Efficiency considerations in using semi-random sources. In Proc. 19th ACM Symposium on Theory of Computing, pages 160--168, ACM, New York City, 1987.

....they simply hit all but a fraction of points in f0; 1g m . Previous work. Dispersers were first defined by Sipser [Sip88] and extractors were first defined by Nisan and Zuckerman [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, TS98b, Tre98] Most of the work on extractors is based on techniques such as k wise independence, the Leftover ....

Umesh V. Vazirani. Efficiency considerations in using semi-random sources (extended abstract). In Proceedings of the Nineteenth Annual ACM Symposium on Theory of Computing, pages 160--168, New York City, 25--27 May 1987.

.... these notes is due to Impagliazzo and Zuckerman [17] Our description is based on expositions in Wigderson s notes [38] and Schoning s book [31] Using expander graphs a similar result is proved in [11] The problem of simulating BPP algorithms with weak random sources was initiated and studied in [28, 35, 36]. More recently, the results of these papers have been substantially improved in [39, 34] 4 Randomization and Interaction Recall that the polynomial hierarchy is built up by setting up a Prover Challenger game, with a polynomial time verifier acting as referee. There is another way in which NP ....

U. Vazirani. Efficiency considerations in using semi-random sources. 19th ACM STOC, 1987.

....function h. Still, for sufficiently long messages, it will be faster to use the bucket hashing technique from the following section. We comment that there are many other well known techniques for universal hashing, such as the linear congruential hash (modulo a prime) 10] the shift register hash [31], or the Toeplitz matrix hash [18] It seems that these and other techniques we have reviewed would not have a software speed competitive with bucket hashing (assuming the message to be hashed is sufficiently long to make bucket) 3 Bucket Hashing Let X = X 1 : X n be a string, partitioned ....

U. Vazirani, Efficiency considerations in using semi-random sources, Proceedings of the Nineteenth Annual ACM Symposium on Theory of Computing, ACM Press, 1987, pp. 160--168.

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U. Vazirani, Efficiency considerations in using semi-random sources, in Proceedings, 19th Annual ACM Symposium on the Theory of Computing, ACM, 1987," pp. 160#168.

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U.V. Vazirani, "Efficiency Considerations in Using Semi-random Sources", Proc. 19th ACM Symp. on Theory of Computing, 1987, pp. 160-168.

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