N. Nisan. Extracting randomness: How and why, a survey. IEEE Conference on Computational Complexity, pages 44--58, 1996.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....we extract Of course, the catch is that we do not lose the extra randomness, so the extractor is still very useful. On the other hand, the last two extractors are much more randomness ecient (provided m is large enough) For more information on these topics, see the excellent survey articles [44, 45]. 2.8 Deterministic Extractors and t wise Independent Functions In the previous section we saw that we can extract almost all the randomness from any distribution of min entropy m by investing very few extra truly random bits. On the other hand, it would be very desirable not to invest any ....

N. Nisan. Extracting Randomness: How and Why, A survey. In IEEE Conference on Computational Complexity, pp. 44-58, 1996.

....are unknown a priori, therefore we need algorithms which can extract truly random bits from arbitrary sources. The basic result of the theory of extractors states that all min entropy can be extracted asymptotically by randomized extractors. An introduction to early results can be found in [7], while the basic idea of polynomial time construction is given in [8] This extractor is universal in the sense that the same extraction algorithm can be used for all distributions with the same min entropy. In accordance, we call an extraction algorithm non universal if the algorithm has to be ....

....by its output dimensions. This requirement of uniformity is met if and only if the preimage partitioning of the EXT mapping is uniform. Linear affine hash functions guarantee uniformity, however, cryptographic hashing algorithms do not. The mother of all extractors as it is called in [7] is a technique for randomized extraction, when a hash function is selected randomly from a family H = Phi h : f0; 1g Psi of hash functions. Random selection of a hash mapping to a fixed input p.d. or a random selection of an input p.d. to a fixed mapping are essentially dual ....

NISAN, N. : Extracting Randomness: How and Why. A survey, Proc. of the 11th IEEE Conference on Computational Complexity, May 1996.

....[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 also [NTS98] In Table 1 we summarize the best known constructions, for di#erent combination of the parameters, and we state the parameters of two special cases of our construction. 1.3 Our Result The extractors constructed in this paper work for any min entropy k = n##42 , extracts a ....

N. Nisan. Extracting randomness: How and why. In Proceedings of the 11th IEEE Conference on Computational Complexity, pages 44--58, 1996.

....fixed polynomial, fooling space S computations. As mentioned earlier, this construction together with the naive deterministic simulation, yields Theorem 3.5. At the center of their construction is a combinatorial construction called an extractor. A thorough survey of extractors is given by Nisan [33]; we content ourselves with a short discussion. Extractors first arose in the following context. Suppose we have a random source that outputs bits that are faulty ; a k bit string produced by the source may have some biased bits, or dependencies among its bits. We would like to find a mapping ....

.... it possible to convert the algorithm to one that still works in polynomial time, but is robust in the sense that it works properly with any # source These two questions have been studied extensively in the context of poly time computation and there are very strong results for both of them (see [33] for a survey) One can ask the same questions in the context of space bounded computation. Here there is essentially nothing non trivial known. All of the methods known in the poly time case (for either problem) involve generating a large set of random bits and using the same bits repeatedly ....

N. Nisan. Extracting randomness: How and why. In 11th Annual Conference on Computational Complexity, 1996.

....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 referees for their detailed and helpful suggestions. ....

N. Nisan, "Extracting Randomness: How and Why", Proc. IEEE Conference on Computational Complexity (formerly "Structure in Complexity Theory"), 1996, pp. 44--58.

....we extract Of course, the catch is that we do not lose the extra randomness, so the extractor is still very useful. On the other hand, the last two extractors are much more randomness ecient (provided m is large enough) For more information on these topics, see the excellent survey articles [44, 45]. 2.8 Deterministic Extractors and t wise Independent Functions In the previous section we saw that we can extract almost all the randomness from any distribution of min entropy m by investing very few extra truly random bits. On the other hand, it would be very desirable not to invest any ....

N. Nisan. Extracting Randomness: How and Why, A survey. In IEEE Conference on Computational Complexity, pp. 44-58, 1996.

....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 and accessible introduction is given by a recent survey by Nisan [Nis96] (see also [NTS98] In this paper we show that pseudorandom generator constructions of a certain kind are extractors. Using our connection and some new ideas we describe constructions of extractors that improve or subsume all the previously known constructions and that are exceedingly simpler ....

N. Nisan. Extracting randomness: How and why. In Proceedings of the 11th IEEE Conference on Computational Complexity, pages 44-- 58, 1996.

....are simple and admit a short self contained description. 1 Introduction An extractor is an algorithm that converts a weak source of randomness into an almost uniform distribution by using a small number of additional truly random bits. Extractors have several important applications (see e.g. Nis96] In this paper we show that pseudorandom generator constructions of a certain kind are extractors. Using our connection and some new ideas we describe constructions of extractors that improve most previously known constructions and that are simpler than previous ones. 1.1 De nitions We now ....

....[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 also [NTS98] and [Gol99] for a broader perspective) In Table 1 we summarize the parameters of the previous best constructions, and we state two special cases of the parameters arising in our construction. 1.3 Our Result The extractors constructed in this paper work for any min entropy ....

N. Nisan. Extracting randomness: How and why. In Proceedings of the 11th IEEE Conference on Computational Complexity, pages 44-58, 1996.

....amplification, on the other hand, is to build a type of bipartite expander called dispersers [Sip88] or its variant called extractors [NZ93] The application of this approach gives extremely high success probability using only a moderate number of random bits. We refer the reader to [Nis96] for a survey on this method. With respect to polynomial time randomized computation, there are two fundamental approaches to derandomization: the method of conditional probabilities [ES73, Spe87, Rag88] and constructing small sized sample spaces [KW84, Lub85, ABI86, AKS87, NN90] In the former ....

....researchers, and provided much of the motivation for the search for good dispersers. The details of these improvements can be derived from the corresponding original papers by plugging in our construction, and will not be given here. We refer the reader to a comprehensive survey paper by Nisan [Nis96]. 3.1.1 The Equivalence of RP and Strong RP Definition 3.1.1 Random polynomial time (RP) is the set of languages L f0; 1g such that there is a deterministic polynomial time Turing machine ML ( Delta; Delta) for which x 2 L Pr[ML (x; y) accepts] 1=2; and x 62 L Pr[ML (x; y) accepts] ....

N. Nisan. Extracting Randomness: How and Why. In Proc. of 11th Annual Conference on Computational Complexity, pp. 44-58, 1996.

....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 hand, since we allow the use of a global ....

N. Nisan. Extracting randomness: how and why. Asurvey. InProceedings of the 11th Computational Complexity Conference, pages 44--58, 1996.

....obtained by analyzing the expected error (i.e. the 1 norm) c) Pseudorandom sets for combinatorial rectangles. One major goal in derandomization is to efficiently construct a discrete structure (e.g. constant degree expanders (Lubotzky, Phillips and Sarnak[27] dispersers and extractors (Nisan [31]) hash function families (Carter and Wegman [10] that is usually easily shown to exist by a probabilistic argument. Converting randomized algorithms to deterministic ones is one of the many applications of such results. Another application is that a random construction (notably of hashing ....

N. Nisan. Extracting Randomness: How and Why. In Proc. IEEE Conference on Computational Complexity (formerly "Structure in Complexity Theory"), pages 44--58, 1996.

..... 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, we would like to use an as short y as possible. We will ....

N. Nisan. Extracting randomness: how and why. A survey. In Proceedings of the 11th Structure in Complexity Theory Conference, pages 44-58, 1996.

....Extractors have many applications in theoretical computer science. The most celebrated ones are simulation of randomized algorithms using defective random sources, oblivious sampling, proofs of hardness results and conversion of probabilistic existence proofs to explicit constructions. Refer to [12] for a survey about extractors. Our main interest is in the connection of extractors to deterministic amplification. It turns out that black box amplifiers are equivalent to weak extractors, 3 , a slightly weaker notion than extractors. An (l; r; k) bipartite graph is called a (ffi; ffl) weak ....

....of y. Lemma 2.8 M is an (l; r; k) black box simulator if and only if for all r the graph GM;r is an (l; r; k) bipartite graph. Furthermore, M is a (ffi; ffl) amplifier iff for all r the graph GM;r is a (ffi; ffl) weak extractor. The proof of this lemma is fairly simple, and can be found in [12]. Weak extractors were introduced by Cohen and Wigderson in [3] they called them majority dispersers) and they lie between the two more familiar notions of extractors and dispersers. Definition 2.9 An (l; r; k) bipartite graph is called a (ffi; ffl) extractor if for every subset U V 1 , ....

N. Nisan. Extracting randomness: How and why, a survey. In Proceedings of the 11 th Annual IEEE Conference on Computational Complexity, pages 44--58, 1996.

....the fact that G 0 is an extractor with the required parameters. Let us therefore explore other possibilities. Hash functions are a key ingredient in many of the known constructions of extractors. The basic technical tool in these constructions is the leftover hash lemma (see [ILL89] Nis96] We shall use a variant of it. First we need some de nitions. De nition 3. Let H be a family of functions h : f0; 1g n f0; 1g m . We say that H has collision error if for all x 1 6= x 2 , P rob h2H (h(x 1 ) h(x 2 ) 1 )2 m . De nition 4. The collision probability of a ....

N. Nisan. Extracting randomness: how and why. A survey. In Proceedings of the 11th Structure in Complexity Theory Conference, pages 44-58, 1996.

....property that for any large enough collection S of nodes of the left, the process of randomly choosing a neighbor of a node in S generates a close to uniform distribution of the nodes on the right. We give more details in Section 2.1. We also recommend the excellent survey on extractors by Nisan [Nis96]. Extractors get their name from their ability to iextractj randomness from a distribution on strings. One would expect some connections with Kolmogorov complexity, one of the best known measures of irandomnessj. We show how to use extractors to achieve this connection in time bounded Kolmogorov ....

....we can nd a string y that captures most of the randomness of x using only a small additional number of bits. 2 Preliminaries We start by giving some basic denitions of Kolmogorov complexity and to introduce the notion of an extractor graph. We follow the presentation in Nisan s survey paper [Nis96]. 2.1 Extractors An extractor can be thought of as a bipartite graph, whose rst color class is larger than the second color class. By convention we think of the rst color class as being on the left, and the second on the right. The vertices on the left side are all the strings of length n, so ....

[Article contains additional citation context not shown here]

N. Nisan. Extracting randomness: How and why (a survey). In Proceedings of the 11th IEEE Conference on Computational Complexity, pages 4458. IEEE, New York, 1996.

....than 2 m 2 vertices on the right side, there is at least one edge joining V # and W # . This construction is somewhat easier to obtain, and indeed Saks, Srinivasan, and Zhou [SSZ95] give a disperser with d = poly(n) for any constant # 0, allowing for a polynomial time simulation of RP. See [N96] for a complete survey on extractors, dispersers, and weak random sources. Pseudorandom generators and hitting sets. A more ambitious goal than simulating BPP with weak random sources is the deterministic simulation of BPP. Research on this subject tries to isolate reasonable complexity ....

<F3.742e+05> N. Nisan,<F4.081e+05> Extracting randomness: How and why,<F3.815e+05> in Proc. 11th IEEE Conference on Computational Complexity, 1996, pp. 44--58.

....obtained by analyzing the expected error (i.e. the 1 norm) c) Pseudorandom sets for combinatorial rectangles. One major goal in derandomization is to efficiently construct a discrete structure (e.g. constant degree expanders (Lubotzky, Phillips and Sarnak[26] dispersers and extractors (Nisan [29]) hash function families (Carter and Wegman [9] that is usually easily shown to exist by a probabilistic argument. Converting randomized algorithms to deterministic ones is one of the many applications of such results. Another application is that a random construction (notably of hashing ....

N. Nisan. Extracting Randomness: How and Why. In Proc. IEEE Conference on Computational Complexity (formerly "Structure in Complexity Theory"), pages 44--58, 1996.

....in derandomization, as 2 several randomized algorithms are robust to such small changes in the probabilities. For instance, another major derandomization approach, which uses dispersers and extractors that we do not define here, also uses small bias spaces: see, e.g. the survey of Nisan [23]. A third key approach to derandomization, the method of conditional probabilities, can be described informally as follows. Suppose a randomized algorithm chooses n random bits X = X 1 ; Xn ) and outputs a function f( X ) Typically, by analysis we can show that E(f( X) is ....

N. Nisan, Extracting Randomness: How and Why, in Proc. IEEE Conference on Computational Complexity (formerly "Structure in Complexity Theory"), pages 44--58, 1996.

....W of more than 2 m =2 vertices on the right side, there is at least one edge joining V 0 and W 0 . This construction is somewhat easier to obtain, and Saks et al. SSZ95] give indeed a disperser with d = poly(n) for any constant fl 0, allowing for a polynomial time simulation of RP. See [Nis96] for a complete survey on extractors, dispersers, and weak random sources. Pseudorandom Generators and Hitting Sets A more ambitious goal than simulating BPP with weak random sources is the deterministic simulation of BPP. Research on this subject tries to isolate reasonable complexity ....

....in polynomial time using weak random sources of r bits and min entropy r fl for any fl 0. The main novelty in our result has been the use of dispersers in a context where extractors seemed to be necessary. Extractors have other applications besides the use of weak random sources (see, e.g. [Nis96]) It could be the case that techniques similar to ours can give stronger results or simplified proofs in these other applications as well. It remains an open question whether it is possible, for any fl 0 and any m, to efficiently construct a (2 r ; 2 m ; d; r fl ; 1=7) extractor with r ....

N. Nisan. Extracting randomness: How and why. In Proceedings of the 11th IEEE Conference on Computational Complexity, pages 44--58, 1996.

....property that for any large enough collection S of nodes of the left, the process of randomly choosing a neighbor of a node in S generates a close to uniform distribution of the nodes on the right. We give more details in Section 2.2. We also recommend the excellent survey on extractors by Nisan [Nis96]. Extractors get their name from their ability to extract randomness from a distribution on strings. One would expect some connections with Kolmogorov complexity, one of the best known measures of randomness . We show how to use extractors to achieve this connection in time bounded ....

....p complexity we can find a string y that captures most of the randomness of x using only a small additional number of bits. 2 Preliminaries We start by giving some basic definitions that are needed to introduce the notion of an extractor graph. We follow the presentation in Nisan s survey paper [Nis96]. An extractor can be thought of as a bipartite graph, whose first color class is larger than the second color class. By convention we think of the first color class as being on the left, and the second on the right. The vertices on the left side are all the strings of length n, so the first color ....

[Article contains additional citation context not shown here]

N. Nisan. Extracting randomness: How and why (a survey). In Proceedings of the 11th IEEE Conference on Computational Complexity, pages 44--58. IEEE, New York, 1996.

....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 hand, since we allow the use of a global ....

N. Nisan. Extracting randomness: how and why. A survey. In Proceedings of the 11th Computational Complexity Conference, pages 44--58, 1996.

....of Hastad [Has97] only imply the somewhat weaker consequence that ZPP=NP) It is likely that more applications of extractors will be found in the future. Nisan remarks that extractors exhibit some of the most random like properties of explicitly constructed combinatorial structures [Nis96] The literature on explicit construction of extractors and dispersers is vast and technically challenging. An excellent and accessible introduction is given by a recent survey by Nisan [Nis96] see also [NTS98] In Table 1 we summarize the best known constructions, for different combination of ....

.... exhibit some of the most random like properties of explicitly constructed combinatorial structures [Nis96] The literature on explicit construction of extractors and dispersers is vast and technically challenging. An excellent and accessible introduction is given by a recent survey by Nisan [Nis96] see also [NTS98] In Table 1 we summarize the best known constructions, for different combination of the parameters, and we state the parameters of (a special case of) our construction. Our Main Result. In this paper we introduce a new approach to constructing extractors. An application of ....

N. Nisan. Extracting randomness: How and why. In Proceedings of the 11th IEEE Conference on Computational Complexity, pages 44--58, 1996.

....W 0 W of more than 2 m =2 vertices on the right side, there is at least one edge joining V and W . This construction is somewhat easier to obtain, and Saks et al. SSZ95] give indeed a disperser with d = poly(n) for any constant fl 0, allowing for a polynomial time simulation of RP. See [Nis96] for a very complete and updated survey on extractors, dispersers, and weak random sources. Pseudorandom Generators and Hitting Sets A more ambitious goal than simulating BPP with weak random sources is the deterministic simulation of BPP. Research on this subject tries to isolate reasonable ....

....algorithms in polynomial time using weak random sources with the smallest possible min entropy. The main novelty in our result has been the use of dispersers in a context where extractors seemed to be necessary. Extractors have other applications besides the use of weak random sources (see e.g. [Nis96]) It could be case that techniques similar to ours can give stronger results or simplified proofs in these other applications as well. We also emphasize that our simulation runs in NC and, furthermore, it is possible to give an NC construction of the SSZ dispersers [SSZ97] This implies that our ....

N. Nisan. Extracting randomness: How and why. In Proceedings of the 11th IEEE Conference on Computational Complexity, pages 44--58, 1996.

....researchers, and provided much of the motivation for the search for good dispersers. The details of these improvements can be derived from the corresponding original papers by plugging in our construction, and will not be given here. We refer the reader to a comprehensive survey paper by Nisan [Nis96]. 1.1 The Equivalence of RP and Strong RP Definition 1.1 Random polynomial time (RP) is the set of languages L f0; 1g such that there is a deterministic polynomial time Turing machine ML ( Delta; Delta) for which x 2 L P r[ML (x; y) accepts] 1=2; and x 62 L P r[ML (x; y) accepts] ....

N. Nisan, "Extracting Randomness: How and Why", Proc. IEEE Conference on Computational Complexity (formerly "Structure in Complexity Theory"), 1996, pp. 44--58.

.... having one sided random (i.e. hitting) properties has turned out to be more efficient than that of combinatorial objects having two sided random (i.e. discrepancy) properties (for a survey of these cases see Appendix C of [10] This is for instance the case for extractors and OR dispersers ([13]) Another case in which one sided randomness seems to be easier to achieve is in the case of small linear subspaces of f0; 1g n [4] It is indeed possible to construct small hitting sets for this class of subsets (and, so, for the corresponding characteristic functions) that imply some ....

....exists having circuit complexity 2 Omega Gamma n) then P = BPP. 4 Related Results and Conclusion De randomization is not the only research direction that has been explored about the use of randomness in computation. An alternative approach deals with the use of weak sources of randomness (see [13]) Even in this case there is a difference between one sided pseudorandom structures and two sided pseudorandom structures and Theorem 3.1 is a useful tool [5] A natural question to ask is whether the results described in this column (or at least part of them) extend to parallel and ....

Nisan N. (1996), "Extracting randomness: How and why", In Proceedings of the 11th Annual IEEE Conference on Computational Complexity, pp. 44-58.

....we start by giving the formal background. The notions of Kolmogorov complexity that we use in this chapter are described in Chapter 2; here we give the de nitions pertaining to extractors. The de nitions presented here are inspired by the presentation given in Nisan s excellent survey paper [Nis96]. An extractor can be thought of as a bipartite graph, whose rst color class is larger than the second color class. By convention we think of the rst color class as being on the left, and the second on the right. The vertices on the left side are all the strings of length n, so the rst color ....

N. Nisan. Extracting randomness: How and why (a survey). In Proceedings of the 11th IEEE Conference on Computational Complexity, pages 4458, 1996.

....property that for any large enough collection S of nodes of the left, the process of randomly choosing a neighbor of a node in S generates a close to uniform distribution of the nodes on the right. We give more details in Section 2.2. We also recommend the excellent survey on extractors by Nisan [Nis96]. University of Chicago, Department of Computer Science, 1100 E. 58th St. Chicago, IL 60637. Email: fortnow cs.uchicago.edu. URL: http: www.cs.uchicago.edu fortnow. Supported in part by NSF grant CCR 92 53582 and the Fulbright scholar program. y University of Chicago, Department of ....

....p complexity we can find a string y that captures most of the randomness of x using only a small additional number of bits. 2 Preliminaries We start by giving some basic definitions that are needed to introduce the notion of an extractor graph. We follow the presentation in Nisan s survey paper [Nis96]. An extractor can be thought of as a bipartite graph, whose first color class is larger than the second color class. By convention we think of the first color class as being on the left, and the second on the right. The vertices on the left side are all the strings of length n, so the first color ....

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N. Nisan. Extracting randomness: How and why (a survey). In Proceedings of the 11th IEEE Conference on Computational Complexity, pages 44--58. IEEE, New York, 1996.

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N. Nisan. Extracting randomness: How and why, a survey. IEEE Conference on Computational Complexity, pages 44--58, 1996.

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N. Nisan. Extracting randomness: How and why. In Proceedings of the 11th IEEE Conference on Computational Complexity, pages 44--58, 1996.

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N. Nisan. Extracting randomness: How and why, a survey. IEEE Conference on Computational Complexity, pages 44-58, 1996.

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N. Nisan. Extracting randomness: how and why. A survey. In Proceedings of the 11th Structure in Complexity Theory Conference, pages 44--58, 1996.

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