N. Nisan and A. Wigderson, Hardness vs randomness, J. Comput. Syst. Sci. 49, 149--167, 1994. Home/Search Document Not in Database Summary Related Articles Check |
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Random Access to Advice Strings - And Collapsing Results (Correct)
....circuits of size 2 n## must err on an asymptotically 50 of inputs against parity. To attack this problem, the decision tree point of view was first introduced in [Cai86] This approach in terms of inapproximability has been found most fruitful in the beautiful work of Nisan and Wigderson [Nis91, NW94] on pseudorandom generators. Adapting Hastad s proof to the decision tree model, one can prove the following. Lemma 1. For any depth d 1 Boolean circuit C on z 1 , z L , with bottom fan in at most t, Pr[ DC(C # ) t ] size(C) t where # is a random restriction with the ....
N. Nisan and A. Wigderson, Hardness vs randomness, J. Comput. Syst. Sci. 49, 149-- 167, 1994.
A Note on Deterministic Approximate Counting for k-DNF - Trevisan (2002) (Correct)
....k DNF is satisfied by at least a 1 2 fraction of assignments. in time O(n log(1 #) They also give sub exponential time algorithm for the counting problem for functions computed by constant depth circuits (CNF and DNF are depth 2 circuits) Nisan [Nis91] and Nisan and Wigderson [NW94] construct a pseudorandom generator that fools constant depth circuits and that has poly logarithmic seed length. As a consequence, they achieve n (log n) O(1) time algorithms for the counting and satisfiability problems for constant depth circuits. Luby, Velickovic and Wigderson [LVW93] ....
....constant depth circuits and that has poly logarithmic seed length. As a consequence, they achieve n (log n) O(1) time algorithms for the counting and satisfiability problems for constant depth circuits. Luby, Velickovic and Wigderson [LVW93] optimize the constructions of Nisan and Wigderson [Nis91, NW94] to the case of depth 2 circuits, thus solving the counting and satisfiability problem in time n O( log n) for general CNF and DNF. Luby and Velickovic [LV91] show how to reduce arbitrary CNF and DNF to formula in a simplified format, and show that the counting and satisfiability problems can ....
Noam Nisan and Avi Wigderson. Hardness vs randomness. Journal of Computer and System Sciences, 49(2):149--167, October 1994.
Extracting all the Randomness and Reducing the Error in.. - Raz, Reingold, Vadhan (1999) (36 citations) (Correct)
....additional random bits as a catalyst. A large body of work has focused on giving explicit constructions of extractors, as such constructions have a wide variety of applications. A recent breakthrough was made by Luca Trevisan [Tre99] who discovered that the Nisan Wigderson pseudorandom generator [NW94], previously only used in a computational setting, could be used to construct extractors. For certain settings of the parameters, Trevisan s extractor is optimal and improves on previous constructions. More explicitly, Trevisan s extractor improves over previous constructions in the case of ....
....In particular, we give the de nitions of both combinatorial designs and weak designs. Designs. The main combinatorial objects underlying the Nisan Wigderson pseudorandom generator and subsequently Trevisan s extractors are collections of sets with small pairwise intersections. Following [NW94], we will refer to these as designs, but in the combinatorics literature, they are often called packings (cf. AS00, Sec. 4.7] De nition 5 For 2 N and 1, a family of sets S 1 ; Sm [d] is an ( design if 1. For all i, jS i j = 2. For all i 6= j, jS i S j j log . ....
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Noam Nisan and Avi Wigderson. Hardness vs randomness. Journal of Computer and System Sciences, 49(2):149-167, October 1994.
Constant-Round Coin-Tossing With a Man in the Middle or Realizing.. - Barak (2002) (17 citations) (Correct)
....as a list of elements, satis es the pseudorandomness and evasiveness properties. Note that for our purposes it is enough to use a pseudorandom generator that runs in time that is exponential in its seed length. Such generators exist under weaker conditions than the ones stated in the theorem [NW94, IW97] 2.1.4 Cryptographic Primitives Commitment Schemes. We let Comm denote a computational (i.e. perfectly binding) commitment scheme. That is, we denote a commitment to x by Comm(x) Comm(x;Um ) Note that we assume for simplicity that the commitment scheme Comm is non interactive, as the ....
Noam Nisan and Avi Wigderson. Hardness vs randomness. Journal of Computer and System Sciences, 49(2):149-167, October 1994.
Extracting All the Randomness from a Weakly Random Source - Vadhan (1998) (6 citations) (Correct)
....use fewer truly random bits than any previous construction which works for all min entropies and extracts a constant fraction of the min entropy. The extractors are obtained by observing that a weaker notion of combinatorial design suffices for the Nisan Wigderson pseudorandom generator [NW94], which underlies the recent extractor of Trevisan [Tre98] 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. 1 Introduction Roughly speaking, an extractor is a function ....
....of additional random bits as a catalyst. A large body of work has focused on giving explicit constructions of extractors, as such constructions have a number of applications. A recent breakthrough was made by Luca Trevisan [Tre98] who discovered that the Nisan Wigderson pseudorandom generator [NW94], previously only used in a computational setting, could be used to construct extractors. Trevisan s extractor improves on most previous constructions and is optimal for certain settings of the parameters. However, when one wants to extract all (or most) of the randomness from the weakly random ....
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Noam Nisan and Avi Wigderson. Hardness vs randomness. Journal of Computer and System Sciences, 49(2):149--167, October 1994.
Are Bitvectors Optimal? - Buhrman, Miltersen, Radhakrishnan.. (Correct)
....use are related to those constructed by Erdos, Frankl and Furedi [12] in their study of r cover free family of sets. Similar set systems have been used to great advantage in the construction of pseudorandom generators and extractors, starting with the paper of Nisan [29] and Nisan and Wigderson [30] (for recent applications see [34, 33, 32, 3] and have come to be known as Nisan Wigderson designs; we refer to them as NW designs. The properties of NW designs are, unfortunately, not strong enough for our proof. No previously constructed set system we know has these properties, so we construct ....
....u U so that the resulting bipartite graph satisfies (1) In particular, if we have a set system # u#U , such that for all u 1 , u 2 U(u 1 u 2 ) 1 ) #(u 2 ) n d. 6 then (1) is automatically satisfied. A NW design has precisely this property. If we use the bounds from [12, 29, 30], then we can manage with a V of size at O( log m) This is how the upper bound in Theorem 3 is shown. For the lower bound, consider (1) again. It follows easily that no #(u) is included in the union of the neighborhoods of other vertices in U . That is, the family u#U is an ....
N. Nisan and A. Wigderson. Hardness vs randomness. Journal of Computer and System Sciences, 49:149-167, 1994.
Extractors and Pseudorandom Generators - Trevisan (1999) (7 citations) (Correct)
....in our work) Our second contribution is a construction that is simpler to describe and analyse (the generator of Impagliazzo and Wigderson is quite complicated) and that has somewhat better parameters. Our idea is to use a pseudorandom generator construction due to Nisan and Wigderson [NW94] that is considerably simpler than the one of Impagliazzo and Wigderson (indeed the construction of Impagliazzo and Wigderson contains the one of Nisan and Wigderson as one of its several components) The Nisan Wigderson generator has weaker properties than the Impagliazzo Wigderson generator, ....
....present in Section 2 our connection between pseudorandom generator constructions and extractors. The main result of Section 2 is that the Impagliazzo Wigderson generator [IW97] is a good extractor. In Section 3 we describe and analyse a simpler construction based on the NisanWigderson generator [NW94] and on error correcting codes. Some concluding remarks are contained in Section 4. Section 3 might be read independently of Section 2. 2 The Connection Between Pseudorandom Generators and Extractors This section describes our main idea of how to view a certain kind of pseudorandom generator ....
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N. Nisan and A. Wigderson. Hardness vs randomness. Journal of Computer and System Sciences, 49:149--167, 1994. Preliminary version in Proc. of FOCS'88.
Pseudorandom Generators and Extractors - Garg, Mazumdar (2002) (Correct)
....8i; jS i j = l 8i; j jS i S j j Lemma 2.1 8l; n, there exist (l,log n,d,n) designs where d = O( log n and which can be constructed deterministically in O(2 n) time. Let x jS denote the restriction of x 2 f0; 1g to the locations in S f1; 2 ; ng De nition NW Generator [5] For a function f : f0; 1g f0; 1g and a (l; d; n) design S = S 1 ; S 2 ; S n ) the Nisan Wigderson generator NW S : f0; 1g is de ned as S (x) f(x jS1 ) f(x jSn ) De nition Pairwise Independent sets generator: A function P I l;n : f0; 1g 3l (f0; 1g such ....
A. W. N. Nisan. Hardness vs randomness. Journal of computer and System Science, October 1994.
Derandomizing Arthur-Merlin Games using Hitting Sets - Miltersen, Variyam (1999) (16 citations) (Correct)
....Theorem 2 (Impagliazzo Wigderson) Let # 0 be any constant. If a language L in E exists so that L has circuit complexity at least 2 for all su#ciently large n, then P=BPP. The proof of this theorem is technical and is built on the results of many earlier papers, including [BM84, Yao82, NW94, GL89, BFNW93, Imp95] Although much research has gone into derandomizing BPP and RP, derandomization of classes like AM has received attention only recently. The class AM was defined, by Babai and Babai and Moran in [Bab85, BM88] as a natural randomized (and interactive) version of the class ....
....give polynomial size membership proofs for positive instances of Graph Nonisomorphism. In contrast, the lengths of the shortest proofs known, without any assumptions, are exponential in the sizes of the graphs [BL83, BKL83] In [AK97] Arvind and Kobler showed that the construction of [NW94] can be extended to the nondeterministic setting to get pseudorandom generators which can be used to completely derandomize AM. As in the case of [NW94] they needed an average case hardness assumption in order to construct the generator. To be precise, Arvind and Kobler show Theorem 3 ....
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Noam Nisan and Avi Wigderson. Hardness vs randomness. Journal of Computer and System Sciences, 49(2):149--167, October 1994.
Pseudorandom generators without the XOR Lemma - Sudan, Trevisan, Vadhan (2000) (38 citations) (Correct)
....existence of good pseudorandom generators. The construction of Impagliazzo and Wigderson goes through three phases of hardness amplification (a multivariate polynomial encoding, a first derandomized XOR Lemma, and a second derandomized XOR Lemma) that are composed with the Nisan Wigderson [NW94] generator. In this paper we present two different approaches to proving the main result of Impagliazzo and Wigderson. In developing each approach, we introduce new techniques and prove new results that could be useful in future improvements and or applications of hardness randomness ....
....2 Omega Gamma n) then P = BPP. Impagliazzo and Wigderson prove their result by presenting a randomness efficient amplification of hardness based on a derandomized version of Yao s XOR Lemma. The hardness amplification procedure is then composed with the Nisan Wigderson (NW) generator [NW94] to yield the result. The hardness amplification goes through three steps: an encoding using multivariate polynomials (from [BFNW93] a first derandomized XOR Lemma (from [Imp95] and a second derandomized XOR Lemma (which is the technical contribution of [IW97] In our first result, we show ....
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Noam Nisan and Avi Wigderson. Hardness vs randomness. Journal of Computer and System Sciences, 49(2):149--167, October 1994.
Derandomization That is Rarely Wrong From Short Advice.. - Goldreich, Wigderson (2002) Self-citation (Wigderson) (Correct)
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N. Nisan and A. Wigderson. Hardness vs Randomness. JCSS, Vol. 49, No. 2, pages 149--167, 1994.
Research Reports on - Mathematical And Computing (Correct)
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N. Nisan and A. Wigderson, Hardness vs randomness, J. Comput. Syst. Sci. 49, 149--167, 1994.
Pseudorandomness and Average-case Complexity via Uniform.. - Trevisan, Vadhan (2002) (5 citations) (Correct)
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N. Nisan and A. Wigderson. Hardness vs randomness. J. Comput. Syst. Sci., 49(2):149--167, Oct. 1994.
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N. Nisan and A. Wigderson. Hardness vs Randomness. J. Comput. Syst. Sci. 49(2), pages 149-167, 1994.
Lecturer: Madhu Sudan Scribe: Ryan O'Donnell - Error Correcting Codes (Correct)
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Noam Nisan and Avi Wigderson. Hardness vs randomness. Journal of Computer and System Sciences, 49(2):149-167, October 1994.
Uniform hardness vs. randomness tradeoffs for Arthur-Merlin .. - Gutfreund, Shaltiel, al. (Correct)
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N. Nisan and A. Wigderson. Hardness vs randomness. Journal of Computer and System Sciences, 49(2):149-- 167, Oct. 1994.
Some Applications of Coding Theory in Computational Complexity - Trevisan (2004) (Correct)
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N. Nisan and A. Wigderson. Hardness vs randomness. Journal of Computer and System Sciences, 49:149--167, 1994. Preliminary version in Proc. of FOCS'88.
The Computational Complexity Column - Lance Fortnow Nec (Correct)
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N. Nisan and A. Wigderson. Hardness vs randomness. Journal of Computer and System Sciences, 49(2):149-167, October 1994.
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N. Nisan and A. Wigderson. Hardness vs randomness. Journal of Computer and Systems Sciences, 49, 1994.
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N. Nisan and A. Wigderson, Hardness vs randomness, in Journal of Computer and System Sciences, 49:149-167, 1994.
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N. Nisan and A. Wigderson. Hardness vs randomness. Journal of Computer and System Sciences, 49:149--167, 1994. Preliminary version in Proc. of FOCS'88.
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N. Nisan and A. Wigderson. Hardness vs randomness. Journal of Computer and Systems Sciences, 49, 1994.
On the Distribution of the Number of Roots of Polynomials and.. - Hartman, Raz (2003) (1 citation) (Correct)
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N. Nisan and A. Wigderson, Hardness vs randomness, J Comput System Sci 49(2) (October 1994), 149 --167.
On Pseudorandomness and Resource-Bounded Measure - Arvind Institute Of (1997) (9 citations) (Correct)
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N. Nisan and A. Wigderson. Hardness vs randomness. Journal of Computer and System Sciences, 49:149-167, 1994.
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N. Nisan and A. Wigderson, Hardness vs randomness, Journal of Computer and System Sciences, 49 (1994), 149-167.
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