M. Bellare, O. Goldreich, and S. Goldwasser, "Randomness in Interactive Proofs", FOCS, 1990, pp. 563--571.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....t local extractor with t = poly(log(1 #) 1 #) It is an interesting open problem, posed in [Gol97] to construct averaging samplers whose sample and randomness complexities are both within a constant factor of optimal. Without the averaging constraint, there are samplers which achieve this [BGG93, GW97, Gol97]. 6.4 Previous Constructions. Some previous constructions of cryptosystems in the bounded storage model can be understood using our approach, namely Theorem 10 together with Theorem 5 (of [Lu02] For example, the cryptosystem of Cachin and Maurer [CM97] amounts to using pairwise independence ....

Mihir Bellare, Oded Goldreich, and Shafi Goldwasser. Randomness in interactive proofs. Computational Complexity, 3(4):319--354, 1993.

.... a consensus in an asynchronous distributed system with faults is impossible with deterministic protocols [25] but is possible with the use of randomized protocols (see [17] Various techniques to minimize the amount of randomness needed were extensively studied in computer science (e.g. [36, 52, 10, 47, 54, 18, 32, 4, 46, 1, 50, 35, 37, 40, 33, 34]) and tradeoffs between randomness and other resources were found (e.g. 14, 48, 38, 15, 21, 9, 8, 44, 7, 42, 40] Security vs. Randomness. It is not hard to show that some randomness is essential to maintain security (if all parties are deterministic then the adversary can infer information on ....

M. Bellare, O. Goldreich, and S. Goldwasser, "Randomness in Interactive Proofs", FOCS, 1990, pp. 563--571.

....[IZ, CoW] For the value of t we obtain, however, the running time of this simulation will not be polynomial but only quasi polynomial. On the other hand, our simulation satisfies a stronger requirement: it truly approximates the acceptance probability of a BPP machine. The result of [BGG] is similar in this regard, but does not yield an extractor. 2 Definitions and Notation Throughout this paper, we use the convention that capital letters denote random variables, sets, distributions, and probability spaces; other variables will be in small letters. Exceptions are R and R 0 , ....

....ck ffi 2 l. Then we can use O(k=ffi log n) random bits to pick l random variables X 1 ; X l in f1; 2; ng such that P r[ ffi 2 l=16 of the X i s lie in T ] 1 Gamma 2 Gammak : Proof: We combine 10 wise independence and random walks on expanders in a manner similar to [BGG]. We divide f1; 2; ng into m disjoint sets A 1 ; Am of size p = n=m, where m = 14k=ffi. Within each set A i , we use 10 log p bits to pick a set S i of size l 0 = l=m using 10 wise independence, as in Lemma 5 (in fact, pairwise independence would suffice, but we wish to quote ....

M. Bellare, O. Goldreich, and S. Goldwasser, Randomness in Interactive Proofs, Computational Complexity 5 (1993): 319-354.

....randomness complexity of such computations. Randomness is an important resource in computation. As a result, various methods for saving in randomness were studied [1, 6, 14, 19, 26, 28, 29, 30, 38, 39, 42, 44, 45, 47] In addition, the role of randomness in specific contexts was studied in, e.g. [41, 36, 4, 11, 8, 9]. One such case is the study of randomness in private multiparty computations [7, 12, 31, 34, 35, 22] In particular, in [7, 35, 31] the amount of randomness required for private computations of xor (the exclusive or function) was considered (this function was the subject of previous research in ....

M. Bellare, O. Goldreich, and S. Goldwasser, "Randomness in Interactive Proofs", Proc. of 31st FOCS, 1990, pp. 563--571.

....to the design of general derandomization techniques, it is important to identify the properties of the probabilistic systems that are susceptible to certain derandomization techniques. Following this direction of research, quantitative aspects of randomness in Arthur Merlin games are studied in [BGG93]. Also, FK95] proved that there exist natural interactive proof systems for which repetition using truly random bits can reduce the error rate to be polynomially small, while repetition using bits produced by any pseudo random generator cannot reduce the error below a constant. In this work, we ....

....algorithm. 1. 1 Previous Work The techniques for analyzing the hitting properties of random walks on expander graphs were first introduced by [AKS87] Then, random walks on expanders used for decreasing the error rate of randomized algorithms [CW89, IZ89, MR95] and interactive proof systems [BGG93], while not substantially increasing the random bit complexity. Up to our knowledge, the first time that recursive random walks on expanders were used for decreasing the random bit complexity of certain hypothetical one side error probabilistic systems was in [FS96] Hitting sets are one of the ....

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M. Bellare, O. Goldreich, and S. Goldwasser. Randomness in Interactive Proofs. Computational Complexity 3, pp. 319--354, 1993.

....Furthermore, randomization is used to construct both more e#cient and, not less significant, much simpler algorithms. Randomness as a resource was extensively studied in the last decade. One line of research was devoted to a quantitative study of the role of randomness in specific contexts, e.g. [35, 28, 5, 12, 8, 9, 32]; another direction was developing general purpose methods for saving random bits. These methods range over pseudo random generators [38, 7, 34] techniques for recycling random bits [22, 19] sources of weak randomness [18, 37, 40] and construction of various kinds of small probability spaces ....

M. Bellare, O. Goldreich, and S. Goldwasser, Randomness in interactive proofs, in Proc. of 31st Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1990, pp. 563--571.

....to the design of general derandomization techniques, it is important to identify the properties of the probabilistic systems that are susceptible to certain derandomization techniques. Following this direction of research, quantitative aspects of randomness in Arthur Merlin games are studied in [BGG93]. Also, FK95] proved that there exist natural interactive proof systems for which repetition using truly random bits can reduce the error rate to be polynomially small, while repetition using bits produced by any pseudo random generator cannot reduce the error below a constant. In this work, we ....

....algorithm. 1. 1 Previous Work The techniques for analyzing the hitting properties of random walks on expander graphs were first introduced by [AKS87] Then, random walks on expanders used for decreasing the error rate of randomized algorithms [CW89, IZ89, MR95] and interactive proof systems [BGG93], while not substantially increasing the random bit complexity. Up to our knowledge, the first time that recursive random walks on expanders were used for decreasing the random bit complexity of certain hypothetical one side error probabilistic systems was in [FS96] Hitting sets are one of the ....

[Article contains additional citation context not shown here]

M. Bellare, O. Goldreich, and S. Goldwasser. Randomness in Interactive Proofs. Computational Complexity 3, pp. 319--354, 1993.

....amplification [11, 8] For the value of t we obtain, however, the running time of this simulation will not be polynomial but only quasi polynomial. On the other hand, our simulation satisfies a stronger requirement: it truly approximates the acceptance probability of a BPP machine. The result of [3] is similar in this regard, but does not yield an extractor. 2 Definitions and Notation Throughout this paper, we use the convention that capital letters denote random variables, sets, distributions, and probability spaces; other variables will be in small letters. Exceptions are R and R 0 , ....

....ck ffi 2 l. Then we can use O(k=ffi log n) random bits to pick l random variables X 1 ; X l in f1; 2; ng such that P r[ ffi 2 l=16 of the X i s lie in T ] 1 Gamma 2 Gammak : Proof: We combine 10 wise independence and random walks on expanders in a manner similar to [3]. We divide f1; 2; ng into m disjoint sets A 1 ; Am of size p = n=m, where m = 14k=ffi. Within each set A i , we use 10 log p bits to pick a set S i of size l 0 = l=m using 10 wise independence, as in Lemma 5 (in fact, pairwise independence would suffice, but we wish to quote ....

M. Bellare, O. Goldreich, and S. Goldwasser, Randomness in Interactive Proofs, Computational Complexity 5 (1993): 319-354.

....in [SZ95] style. To remove dependencies we approximate the average behavior of this set of functions (an object independent of any particular one function) with sufficient accuracy and high probability. This is achieved without space and random bit penalty via deterministic sampling of either [BGG93] or the recent extractor constructions of [Zuc96] This technique for removing dependencies has potential for generalization, and may become a useful derandomization tool. The rest of the paper is organized as follows. In section 2 we give basic notation, definitions and preliminary tools. In ....

....a function f : f0; 1g l [0; 1] we want to efficiently the expectation over a randomly chosen x 2 f0; 1g l of f(x) i.e. to estimate E[f ] E x2f0;1g l[f(x) 1 2 l Sigma x2f0;1g lf(x) to a given accuracy, minimizing the number of random bits used. Two optimal methods are known, BGG93] and [Zuc96] Definition 2.1 [BR94, Zuc96] An (ffl; fl) oblivious sampler (or sampler) A : f0; 1g r f0; 1g p Deltal is a deterministic algorithm that on an input r bit string y, outputs a sequence of p sample points A(y) 1] A(y) p] 2 f0; 1g l subject to the property that for ....

[Article contains additional citation context not shown here]

Mihir Bellare, Oded Goldreich, and Shafi Goldwasser. Randomness in interactive proofs. Computational Complexity, 3(4):319--354, 1993.

....randomization is used to construct both more efficient and, not less significant, much simpler algorithms. Randomness as a resource was extensively studied in the last decade. One line of research was devoted to a quantitative study of the role of randomness in specific contexts, e.g. [RS89, KPU88, BGG90, CG90, BGS94, BSV94, KR94]; another direction was developing general purpose methods for saving random bits. These methods range over pseudo random Dept. of Computer Science, Technion. This research was supported in part by MANLAM Fund. e mail: eyalk cs.technion.ac.il) y Dept. of Computer Science, Tel Aviv ....

M. Bellare, O. Goldreich, and S. Goldwasser, "Randomness in Interactive Proofs", Proc. of 31st FOCS, 1990, pp. 563--571.

.... a consensus in an asynchronous distributed system with faults is impossible with deterministic protocols [FLP85] but is possible with the use of randomized protocols (see [CD89] Various techniques to minimize the amount of randomness needed were extensively studied in computer science (e.g. [AGHP90, BGG90, BM82, CG85, IZ89, KK94, KM93, KM94a, KM94b, KM96, KY76, N90, NN90, S92, Y82a, Z91]) and tradeoffs between randomness and other resources were found (e.g. BDPV95, BGS94, BSV94, CG90, CK93, CRS93, KM96, KOR96, KPU88, KR94, RS89] Security vs. Randomness. It is not hard to show that, except for degenerate cases, some randomness is essential to maintain security (if all parties ....

M. Bellare, O. Goldreich, and S. Goldwasser, "Randomness in Interactive Proofs", FOCS, 1990, pp. 563--571.

....is the case t = 1. Randomness is an important resource in computation. As a result, various methods for saving in randomness were studied [1, 6, 14, 19, 25, 27, 28, 29, 36, 37, 40, 42, 43, 44] In addition, there was a quantitative study of the role of randomness in specific contexts, e.g. [39, 34, 4, 11, 8, 9]. One such case is the study of randomness in private multiparty computations [7, 12, 30, 32, 33, 22] In particular, in [7, 33, 30] the amount of randomness required for private computations of xor (the exclusive or function) was considered (this function was the subject of previous research in ....

M. Bellare, O. Goldreich, and S. Goldwasser, "Randomness in Interactive Proofs", Proc. of 31st FOCS, 1990, pp. 563--571.

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M. Bellare, O. Goldreich and S. Goldwasser. Randomness in interactive proofs. Proceedings of the Thirty First Annual Symposium on the Foundations of Computer Science, IEEE, 1990.

No context found.

M. Bellare, O. Goldreich and S. Goldwasser. Randomness in interactive proofs. Proceedings of the Thirty First Annual Symposium on the Foundations of Computer Science, IEEE, 1990.

No context found.

M. Bellare, O. Goldreich and S. Goldwasser. Randomness in interactive proofs. Proceedings of the Thirty First Annual Symposium on the Foundations of Computer Science, IEEE, 1990.

No context found.

M. Bellare, O. Goldreich, and S. Goldwasser "Randomness in Interactive Proofs", Computational Complexity, Vol. 4, No. 4 (1993), pp. 319--354.

No context found.

M. Bellare, O. Goldreich and S. Goldwasser. Randomness in interactive proofs. Proceedings of the Thirty First Annual Symposium on the Foundations of Computer Science, IEEE, 1990.

....a pseudorandom sequence of possible random pads by itself. V will then query the oracle as to which random pad to use, in the simulation of V , and complete its computation by invoking V with the specified random pad. To generate the pseudorandom sequence we use the sampling procedure of [BGG]. Specifically, for m 1=c this merely amounts to generating a pairwise independent sequence of uniformly distributed strings in f0; 1g , which can be done using randomness maxf2r; 2 log 2 mg. Otherwise (i.e. for m 1=c) the construction of [BGG] amounts to generating Theta(cm) such related ....

....sequence we use the sampling procedure of [BGG] Specifically, for m 1=c this merely amounts to generating a pairwise independent sequence of uniformly distributed strings in f0; 1g , which can be done using randomness maxf2r; 2 log 2 mg. Otherwise (i.e. for m 1=c) the construction of [BGG] amounts to generating Theta(cm) such related sequences, where the sequences are related via a random walk on a constant degree expander. Part (2) follows. The following corollary exemplifies the usage of the above proposition. In case c(n) n Gammaff r(n) O(log n) the gap is preserved ....

M. Bellare, O. Goldreich and S. Goldwasser. Randomness in interactive proofs. Computational Complexity, Vol. 3, No. 4, 1993, pp. 319--354.

....sequence of queries that, with high probability, needs to be representative of the average value of any function. We focus on the randomness and query complexities of samplers, and mention that any sampler yields a hitter with identical complexities. Theorem 16 (The Median of Averages Sampler [3]) There exists a polynomial time (oblivious) sampler of randomness complexity O(n log(1=ffi) and query complexity O(ffl Gamma2 log(1=ffi) Specifically, the sampler outputs the median value among O(log(1=ffi) values, where each of these values is the average of O(ffl Gamma2 ) distinct ....

M. Bellare, O. Goldreich, and S. Goldwasser. Randomness in Interactive Proofs. Computational Complexity, Vol. 4, No. 4, pages 319--354, 1993.

....extends to promise problems and preserves exponentially vanishing soundness error. 7 For a proof of this folklore theorem see [21, Apdx. C. 1] We mention that a somewhat more involved argument applies also to interactive proof systems with non perfect completeness (which we did not define) [10]. 7 Zero knowledge Our main results are the existence of certain constant round interactive proof systems. It turns out that these have some zero knowledge [28] property (defined below) A reader who does not care about this extra property may skip the following definition as well as any ....

M. Bellare, O. Goldreich, and S. Goldwasser. Randomness in Interactive Proofs. Computational Complexity, Vol. 4, No. 4, pages 319--354, 1993.

No context found.

M. Bellare, O. Goldreich, and S. Goldwasser, "Randomness in Interactive Proofs", FOCS, 1990, pp. 563--571.

No context found.

M. Bellare, O. Goldreich, and S. Goldwasser, "Randomness in Interactive Proofs", Proc. of 31st FOCS, 1990, pp. 563--571.

No context found.

M. Bellare, O. Goldreich, and S. Goldwasser, "Randomness in Interactive Proofs", Proc. of 31st FOCS, 1990, pp. 563--571.

No context found.

M. Bellare, O. Goldreich, and S. Goldwasser, "Randomness in Interactive Proofs", Proc. of 31st FOCS, 1990, pp. 563--571.

No context found.

M. Bellare, O. Goldreich, S. Goldwasser. "Randomness in interactive proofs", 31st Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, (1990), 563-572.

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