Joseph Naor and Moni Naor. Small-bias probability spaces: Efficient constructions and applications. In Proc. 22nd Annual ACM Symposium on Theory of Computing, pages 213--223, Baltimore, 1990.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....can be applied to the schemes from [20, 27] 26 GF(q) implying that at least n data strings must be tested. However, settling for a small probability of one sided error, a much more efficient solution to this problem can be obtained as a typical application of small bias probability spaces [22]. The following fact is proved in [22] Fact 2. For any n 2 N and ffl 0, there is (an efficiently constructible) meta test set T n;ffl (GF(q) where l = O(log ) such that: ffl jT n;ffl j is polynomial in n and 1=ffl. ffl For every y 2 GF(q) y 6= 0, at most an ffl fraction of ....

....27] 26 GF(q) implying that at least n data strings must be tested. However, settling for a small probability of one sided error, a much more efficient solution to this problem can be obtained as a typical application of small bias probability spaces [22] The following fact is proved in [22]. Fact 2. For any n 2 N and ffl 0, there is (an efficiently constructible) meta test set T n;ffl (GF(q) where l = O(log ) such that: ffl jT n;ffl j is polynomial in n and 1=ffl. ffl For every y 2 GF(q) y 6= 0, at most an ffl fraction of the test tuples (w 1 ; w l ) 2 ....

[Article contains additional citation context not shown here]

J. Naor and M. Naor. Small-bias probability spaces: Efficient constructions and applications. In Proc. of 22th STOC, pages 213--223, 1990.

....a non zero row vector fi of length L such that fiff = 0. Now, recall from Section 1.1 that U 1 (ffi) is exactly the theory capturing NC computations. Examples of fis needed in Facts 1, 2 are known to be NC computable from ff. For Fact 1 one could apply the standard derandomization procedure [16]; the NC algorithm for Fact 2 is based upon computing the matrix rank [15] Hence U 1 (ffi) can define relationals ffi witnessing Facts 1, 2 in the real world. It is not clear, however, to which extent U 1 (ffi) can prove the desired properties of these relationals. The point is that the ....

J. Naor and M. Naor. Small-bias probability spaces: Efficient constructions and applications. In Proceedings of the 22th ACM STOC, pages 213--223, 1990.

....codes of length O(log j Sigmaj) each so that the following property holds for any ordered triple of codes: the first two codes in the triple are identical and different from the third code on at least one bit. Amir et al. 3] use the ffl biased k wise independent sample space of Naor and Naor [5] to obtain these codes in O(j Sigmaj) time. They use the following parameter values: k = 3 and ffl 1=8. Our contribution is two fold. First, we give a randomized algorithm for the case of alphabet sets of size 2. This algorithm produces the right answer with probability at least 1 Gamma m ....

J. Naor, M. Naor. Small Bias Probability Spaces: Efficient Constructions and Applications, SIAM J. Computing, 22, pp. 838--856, 1993.

....independent sample space is a probability space on m bit sequences such that any k bits are almost independent. A ffl biased sample space is a space in which any (boolean) linear combination of the m bits has the value 1 with probability close to 1=2. These notions were introduced by Naor and Naor [17] and further studied in [1] due to their applications to algorithms and complexity theory. However, there are also cryptographic applications: Krawczyk applied ffl biased sample spaces to the construction of authentication codes [13] In this paper, we investigate several new relationships between ....

J. Naor and M. Naor. Small bias probability spaces: efficient constructions and applications. SIAM Journal on Computing 22 (1993), 838--856.

.... (i.e. every k bits are distributed almost uniformly) The reason being that the latter distributions can be generated using fewer random bits (i.e. O(k log(n=ffl) bits suffice, where ffl is the variation distance of these k projections to the uniform distribution) See the work of Naor and Naor [5] (as well as subsequent simplifications in [2] Note that, in both cases, replacing the algorithm s random tape by strings taken from a distribution of a smaller support requires verifying that the original analysis still holds for the replaced distribution. It would have been nicer, if instead ....

....unbiased over J . Then fi fi Pr[ Phi i2J Y i = 1] Gamma fi fi fi fi fi (1 Gamma p) Delta Pr[ Phi i2J X i = 1] p Delta Gamma fi fi = 1 Gamma p) Delta fi fi Pr[ Phi i2J X i = 1] Gamma fi fi The theorem follows. On one hand, we know (cf. 2] following [5]) that there exists ffl bias distributions of support size (n=ffl) On the other hand, we will show (in Lemma 3.1) that every k wise independent distribution, not only has large support (as proven, somewhat implicitly, in [6] and explicitly in [3] and [1] but also has a large min entropy ....

J. Naor and M. Naor. Small-bias Probability Spaces: Efficient Constructions and Applications. SIAM J. on Computing, Vol 22, 1993.

....class NC) Berger and Rompel [BR89] and Motwani, Naor, and Naor [MNN89] showed how to apply the method of conditional probabilities in a careful search of these sample spaces, and were able to obtain parallel derandomizations for problems like the discrepancy problem. A year later, Naor and Naor [NN90] introduced the notion of approximate independence and approximate k wise independence, and showed substantially easier ways to obtain the same derandomizations. The easier part is modulo the construction of random variables that are suitably approximately independent. After k wise ....

J. Naor and M. Naor. Small-bias probability spaces: efficient constructions and applications. In Proc. 22nd Annual ACM Symposium on the Theory of Computing, pages 213--223, 1990.

....success probability 3 16 with only (n) q(n) nondeterministic moves, but still using q 2 many random bits and quadratic time. Subsequent to our finding the application of error correcting codes to make the time quasilinear and r(n) 2n, we discovered that a trick of Naor and Naor [NN90, NN93] can also be applied to this reduction: Build a probabilistic NTM N that first 10 uses 2q 2 coin flips to determine, for each k q(n) 1, a hash function h k 2 H k . Next N flips q 1 more coins to form u 2 f0; 1g q 1 . Then N nondeterministically guesses y 2 f0; 1g q and k, 1 k q 1, ....

J. Naor and M. Naor. Small-bias probability spaces: efficient constructions and applications. SIAM J. Comput., 22:838--856, 1993.

....constant success probability 3 16 with only (n) q(n) nondeterministic moves, but still using q 2 many random bits and quadratic time. Subsequent to our finding the application of error correcting codes to make the time quasilinear and r(n) 2n, we discovered that a trick of Naor and Naor [NN90, NN93] can also be applied to this reduction: Build a probabilistic NTM N that first 10 uses 2q 2 coin flips to determine, for each k q(n) 1, a hash function h k 2 H k . Next N flips q 1 more coins to form u 2 f0; 1g q 1 . Then N nondeterministically guesses y 2 f0; 1g q and k, 1 k ....

J. Naor and M. Naor. Small-bias probability spaces: efficient constructions and applications. In Proc. 22nd Annual ACM Symposium on the Theory of Computing, pages 213--223, 1990.

....simulation is the solution of a system of linear equations over a finite field. We first prove a probabilistic lemma of general interest. Under the assumption of the existence of a sparse set hard for P, we obtain an RNC 2 simulation of P. Using a small bias sample space construction ( NN90, AGHP90] we derandomize this algorithm to obtain an NC 2 simulation. Finally, exploiting additional algebraic properties of a closely related construction, we arrive at a Vandermonde system. We then solve the system using closed formulae involving the elementary symmetric polynomials over a ....

J. Naor and M. Naor. Small-bias probability spaces: efficient constructions and applications. In Proc. 22nd Annual ACM Symposium on the Theory of Computing, pages 213--223, 1990.

....of an N bit witness z that satisfies R 2 (x; z) one can construct in polynomial time a witness y that satisfies RL (x; y) 1. 4 A construction using small sample spaces Partly inspired the result of the last section, in this section, we use an wise ffi biased sample space construction of [NN93, AGHP92] as the inner code, and construct a proof whose verifier works correctly, given any N 2 3 ffl bits of an N bit witness. Definition 1 ( NN93, AGHP92] A Q Theta k matrix M is said to be an wise ffi biased sample space of k variables if for any subset S of columns and any bit ....

....sample spaces Partly inspired the result of the last section, in this section, we use an wise ffi biased sample space construction of [NN93, AGHP92] as the inner code, and construct a proof whose verifier works correctly, given any N 2 3 ffl bits of an N bit witness. Definition 1 ( NN93, AGHP92] A Q Theta k matrix M is said to be an wise ffi biased sample space of k variables if for any subset S of columns and any bit vector t, the number of rows in which the pattern t occurs in the columns in S is within the interval [ Q=2 ) Gamma ffi Q; Q=2 ) ffi Q] The ....

[Article contains additional citation context not shown here]

J. Naor and M. Naor. Small-bias probability spaces: efficient constructions and applications. SIAM Journal on Computing, 22:838--856,

....iteration to the next, but carefully limit the amount of this fall. By employing the above seen inclusion exclusion expansion to evaluate the conditional expectation of the total weight of the final IS, we show that it suffices to choose each r i at random from a polynomial sized small bias space [22], instead of choosing from f0; 1g n . Thus, we will be able to choose a good value for each additional r i in NC, via parallel exhaustive search. Theorem 5. Consider the family of hypergraphs satisfying properties (P2) and (P3) of Section 1.2. Then, given any constant c 0 and any ffl n ....

J. Naor and M. Naor. Small--bias probability spaces: efficient constructions and applications. SIAM Journal on Computing, 22:838--856, 1993.

....we mention several suggestions by Noga Alon [1] which could provide interesting directions for further work. The problem of guessing secrets is closely related to the study of small sample spaces supporting k wise independent (or nearly independent) random variables, which has a rich literature [2, 21, 3, 4]. The problem of interest there is to find a sample space as small as possible, and n binary random variables defined on it, with the property, called k wise independence, that for any choice of k random variables X 1 ; X k , the probabilities satisfy: P rob(X 1 : X k = a 1 : a k ....

....0, respectively. Noga Alon [1] pointed out that the t columns of the generating matrix of any linear binary codes of dimension n and length t in which the weight of every nontrivial code word deviates from half the length by less than 1=14 the length provides such an F . The known constructions in [3, 21] gives an explicit, oblivious, inner product strategy with t = O(log N) queries. In fact, the construction described in Section 6 here can be obtained from one of the codes of [4] in the same manner. By using results from coding theory (or by applying some probabilistic arguments, together with an ....

J. Naor and M. Naor, Small-bias probability spaces: Efficient constructions and applications, 22nd STOC, (1990), 213-223.

.... 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 ....

J. Naor, and M. Naor, "Small-Bias Probability Spaces: Efficient Constructions and Applications", STOC 90, and SIAM J. on Computing, Vol 22, No. 4, 1993, pp. 838--856.

....universal hash function mapping s bits to t Gamma 2k bits, one needs at least s bits. One attempt around this is to use the extractor of [NZ] however that is only useful if t=s is large. Here we show how to use only O(t log s) bits and achieve a good result. We first recall Definition 3.1. [NN]) A d wise ae biased sample space S of n bit vectors has the property that if X = X 1 ; Xn ) is sampled uniformly at random from S, then 8I f1; 2; ng; jI j d; 8b 1 ; b 2 ; b jIj 2 f0; 1g, jP r X2S [ i2I X i = b i ] Gamma 2 GammajI j j ae: 3.1) ....

....sample space S of n bit vectors has the property that if X = X 1 ; Xn ) is sampled uniformly at random from S, then 8I f1; 2; ng; jI j d; 8b 1 ; b 2 ; b jIj 2 f0; 1g, jP r X2S [ i2I X i = b i ] Gamma 2 GammajI j j ae: 3. 1) Simplifying the construction in [NN], d wise ae biased spaces of cardinality O( d log n=ae) 2 ) were constructed explicitly in [AG ] In addition, given the random bits to sample from S, any bit of X can be computed in poly(d; log n; log(ae Gamma1 ) time. Lemma 3.2. Let A f0; 1g s ; jAj 2 t , k 0, and ffl 2 ....

J. Naor and M. Naor, "Small--Bias Probability Spaces: Efficient Constructions and Applications, " SIAM J. Comput., 22(4): 838-856, 1993.

....below. 2.1 e biased Sample Spaces Definition 5. A sample space S f0;1g n is called e biased if for all nonempty a [n] f1; ng, Exp s2S a2a ( 1) s a # e: 4 Small probability spaces with these properties were initially constructed by Naor and Naor [21] and Peralta [23] We will use a construction, due to Alon, Goldreich, Hastad and Peralta [2] which gives an e biased sample space in f0;1g n of size about ( n e ) 2 . The sample space is given as the image of a certain function s n;m : F 2 m F 2 m f0;1g n . Here F 2 n denotes ....

Joseph Naor and Moni Naor. Small-bias probability spaces: Efficient constructions and applications. SIAM Journal on Computing, 22(4):838--856, August 1993.

....the following code construction due to [GHSZ00] Fact 2.6. There is a polynomial time (in fact, Logspace) constructible [n; k] code with combinatorial list decoding property a, where n = O(k=a 4 ) Simpler and more efficient constructions can be achieved with somewhat worse parameters, e.g. [NN93, AGHP92]. 2.4 Reed Muller Codes Our construction uses a (h;D) Reed Muller code over F q . In such a code the message specifies a polynomial f in D variables over F q of total degree at most h, and the output is all the values of f over F D q . Every polynomial in D variables of total degree h can be ....

J. Naor and M. Naor. Small--bias probability spaces: Efficient constructions and applications. SIAM Journal on Computing, 22(4):838--856, 1993.

....information. 1 this setting for the basic xor function, and show direct sum type of results relating the rounds complexity and the 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 ....

J. Naor, and M. Naor, "Small-Bias Probability Spaces: Efficient Constructions and Applications", STOC 90, and SIAM J. on Computing, Vol 22(4), pp. 838--856, 1993.

....(and then possibly observed by the adversary) Our most important requirement is that the hash function in the strong extractor be describable by a very short random string. This requirement is met by the strong extractor of Srinivasan and Zuckerman [22] using the hash families of Naor and Naor [15]. Their results can be summarized as follows: Lemma 1 ( 22] For any and t =2, there exists a family H of hash functions mapping f0; 1g n to a range f0; 1g k , where k = Gamma 2t, such that the following holds: A random member of H can be described by and efficiently computed using ....

J. Naor, M. Naor. Small-Bias Probability Spaces: Efficient Constructions and Applications. In SIAM J. Computing, 22(4):838-856, 1993.

....variables 2 for random x. Recent work [2, 4, 12, 1] on the distribution of Jacobi symbols lends much theoretical support to this assumption. In particular, these sequences are ffl biased, a type of distribution which can be proven to closely approximate the distribution of fair independent coins [10, 12]. Thus, we again expect the probability of error to be proportional to 1 2 k . This motivates the following: Definition2. Given x 2 ZN and an integer k, the Jacobi signature ( of length k ) of x is the bit vector J (k) x) x 1 N ) x k N ) Algorithm 2 1. If ....

Naor, J., Naor, M.: Small-bias probability spaces: efficient constructions and applications. SIAM Journal on Computing (1993).

....: s n , j (S) j (s 1 ) j (s 2 ) j (s n ) Theorem 1 [2] Let P be a pattern, T a text, both over an arbitrary alphabet Sigma, and let S = f 1 ; k g be a ( Sigma; 3) universal set. P swap matches T at location i iff for all j, j (P ) swap matches j (T ) at location i. 11 In [23] it was shown how to construct ( Sigma; 3) universal set of cardinality k = O(log j Sigmaj) yielding the following. Corollary 1 A solution of the Swap Matching problem over alphabet fa; bg of time O(f(n; m) implies a solution of time O(log j Sigmajf (n; m) over a general alphabet Sigma. 8 ....

J. Naor and M. Naor. Small-bias probability spaces: Efficient constructions and applications. SIAM J. Comp., pages 838--856, 1993.

....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 approach, we search for a good point in a large sample space by improving certain conditional probabilities (or expectations) in an adaptive manner; in the latter method, we construct a sample space of polynomial size while guaranteeing the existence of a good point so that we ....

....of a good point so that we could accomplish finding such a point by exhaustive search. The most commonly used tools for constructing small sized sample spaces are: k wise independent hashing [CW79] sample spaces with limited independence [KW84, Lub85, ABI86] sample spaces with small bias [NN90], and expander graphs [AKS87] Remark: Here we have assumed that the polynomial time randomized computation has one side error, in which case, finding a good sample point is sufficient for derandomization. In fact, for almost all the problems that are known to have randomized algorithms, their ....

[Article contains additional citation context not shown here]

J. Naor and M. Naor. Small-bias probability spaces: efficient constructions and applications. SIAM J. Comput., 22(4):838-856, 1993.

....probabilities (see [44] Raman [56] showed how to deterministically find a good universal hash function bit by bit in total time O(n 2 w) which also becomes the total time for constructing the FKS dictionary. Alon and Naor [6] used another derandomization tool, small bias probability spaces [5, 46], to derandomize a variant of the FKS scheme, achieving construction time O(nw log 4 n) However, lookup requires evaluation of a linear function in time (w= log n) so the hash function is not efficient unless w = O(log n) Another variant of the FKS scheme reduces the number of random bits to ....

Joseph Naor and Moni Naor. Small-bias probability spaces: efficient constructions and applications. SIAM J. Comput., 22(4):838--856, 1993.

....still with quadratic runtime when q(n) n. Gupta [Gup93] gives a randomized reduction to parity which achieves constant success probability 3 16 with only (n) q(n) nondeterministic moves, but still using q 2 many random bits and quadratic time. The first construction of Naor and Naor [NN90, NN93] boils down to the following idea in this setting: N flips 2q 2 coins to determine functions h k 2 H k for all k, and then flips q 1 more coins to form u 2 f 0; 1 g q 1 . Then N nondeterministically guesses y 2 f 0; 1 g q and k, 1 k q 1, and accepts iff B(x; y) h k (y) 0 u k = 1. ....

...., which is O(q log q loglog q) Our construction achieves better constants than this, and is better by an order of magnitude in (randomness or number of nondeterministic moves) and running time than all of the previous ones. The idea of using error correcting codes is mentioned by Naor and Naor [NN93] and ascribed to Bruck, referring the reader to [ABN 92] for details. However, the construction in [ABN 92] uses a code of Justesen (see [Jus72, MS77] whose implementation in our setting seems to require exponentiation of field elements of length polynomial in n, which is not known to be ....

J. Naor and M. Naor. Small-bias probability spaces: efficient constructions and applications. SIAM J. Comp., 22:838--856, 1993.

....here is m = k log n=ffl O(1) We are thinking in terms of ffl 1 2 k , so we get m = O(2 k log n) Relative to our earlier upper and lower bounds on the size of a k wise independent sample space, this space is exponentially smaller in n. The preceding construction is due to Naor and Naor [NN90]. For further constructions, see also the paper by Alon et al. AGHP90] ....

J. Naor and M. Naor, "Small-bias Probability Spaces: Efficient Constructions and Applications," Proceedings of the 22nd STOC, 1990, pp. 213-223. 5

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J. Naor and M. Naor, Small-bias probability spaces: efficient constructions and applications, SIAM J. Comput., vol. 22(4), 1993, pp. 838-856.

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J. Naor and M. Naor, Small-bias probability spaces: efficient constructions and applications, SIAM J. on Computing 22, 1993, pp. 838--856.

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J. Naor and M. Naor. Small-bias probability spaces: efficient constructions and applications. SIAM Journal on Computing, 22 (1993), pp. 838--856.

....clique or independent set on t = 2 log n) 2 ( nodes, i.e. N t log log log log t . As we shall see in the next section, by taking smaller sample spaces we can do better. 3 A construction based on small bias probability spaces We now introduce small bias probability spaces, as defined in [11]. A probability space with n random variables in f0; 1g is called k wise ffl bias if for any k or fewer random variables the probabilities that their parity is 0 or 1 is differ by at most ffl. It is known that in a k wise ffl bias probability space, the probability that any k given random ....

....2 ) 2 ffl) Taking k to be 2 log n and ffl to be 2 Gammak we get that this probability is less than log n . Consider now a D that is defined by this probability space, i.e. every graph corresponds to a point in the probability space. The constructions of ffl bias probability spaces in [11] and [2] are of size (n=ffl) for some fixed c. Therefore, m = 2 for some fixed c . For the graph H in Lemma 1 we have that N = 2 k2 and t is 2 , i.e. N t 0 p . We now take a concrete example of an k wise ffl bias probability space given in [2] It is based on ....

J. Naor, M. Naor, Small-bias probability spaces: efficient constructions and applications, Proc. 22-nd ACM Symp. Theory of Comput. (1990), 213--223. 4

....main memory cannot be much larger than mt. 3 Hashing tools Several hashing techniques are used in the paper. Some of them rely on cryptographic assumptions while others do not. We review these hashing techniques in this section. 3. 1 ffl biased hash functions This hashing scheme is drawn from [13]. We briefly describe the result in a communication complexity setting. Suppose two players A and B have n bit strings x and y respectively and would like to decide if x = y. The scheme in [13] allows A to define a hash function h using O(log n k) random bits such that h(x) is small (O(k) bits) ....

....hashing techniques in this section. 3.1 ffl biased hash functions This hashing scheme is drawn from [13] We briefly describe the result in a communication complexity setting. Suppose two players A and B have n bit strings x and y respectively and would like to decide if x = y. The scheme in [13] allows A to define a hash function h using O(log n k) random bits such that h(x) is small (O(k) bits) and h(x) h(y) with probability 1=2 k if x 6= y. The hash function description is treated as a source of O(log n k) bits. These bits are expanded into l = O(k) distinguisher strings r 1 ....

J. Naor and M. Naor, Small-bias probability spaces: efficient constructions and applications, Proc. 22nd ACM Symposium on Theory of Computing, pages 213--223, 1990.

....with the property that summing the hash function over different sets yields a different value with high probability. These two applications are discussed in Section 9. In Section 10 we briefly survey recent papers that have used the constructions described in this paper since it appeared in [42]. Independent of our work, Peralta [44] has considered ffl bias probability spaces as well, and showed some applications to number theoretic algorithms. His construction is based on quadratic residues and Weil s Theorem. 2 Preliminaries and definitions Let x = x 1 ; x n be f Gamma1; 1g ....

....The family H is accessible with O(log n log m) bits and for any subsets A and B of f1 : ng, the probability that P a2A h(a) P b2B h(b) is smaller than 1 m fi for fi constant, where addition is bitwise XoR. 10 Further results Since the preliminary version of this paper appeared in [42], several new applications of small bias probability spaces have been discovered. Alon [4] used them to obtain an NC 1 algorithm for the parallel version of Beck s algorithm for the Lovasz local lemma. Feder, Kushilevitz and Naor [24] have applied them to amortize the communication complexity of ....

J. Naor and M. Naor, Small-bias probability spaces: efficient constructions and applications, Proceedings of the 22nd annual ACM Symposium on Theory of Computing (1990), pp. 213223.

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Joseph Naor and Moni Naor. Small-bias probability spaces: Efficient constructions and applications. In Proc. 22nd Annual ACM Symposium on Theory of Computing, pages 213--223, Baltimore, 1990.

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J. Naor, and M. Naor, "Small-Bias Probability Spaces: Efficient Constructions and Applications", STOC 90, and SIAM J. on Computing, Vol 22, No. 4, 1993, pp. 838--856.

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J. Naor, and M. Naor, "Small-Bias Probability Spaces: Efficient Constructions and Applications", STOC 90, and SIAM J. on Computing, Vol 22(4), pp. 838--856, 1993.

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J. Naor, and M. Naor, "Small-Bias Probability Spaces: Efficient Constructions and Applications", STOC 90, and SIAM J. on Computing, Vol 22, No. 4, 1993, pp. 838--856.

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J. Naor, and M. Naor, "Small-Bias Probability Spaces: Efficient Constructions and Applications ", STOC 90, and SIAM J. on Computing, Vol 22, No. 4, 1993, pp. 838--856.

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J. Naor and M. Naor. Small-bias probability spaces: efficient constructions and applications. SIAM J. Comput., 22(4):838--856, 1993.

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Joseph Naor, Moni Naor. Small-Bias Probability Spaces: Efficient Constructions and Applications. SIAM J. Comput. 22(4): 838-856 (1993).

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Naor and Naor. Small-bias probability spaces: Efficient constructions and applications. SIAM Journal on Computing, 22, 1993.

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J. Naor and M. Naor. Small-bias probability spaces: efficient constructions and applications. In 22 nd ACM Symposium on Theory of Computing, pages 213--223, 1990.

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Naor, J., Naor, M.: Small-bias probability spaces: Efficient constructions and applications. In: Proc. 22nd Annual ACM Symposium on Theory of Computing, Baltimore (1990) 213--223

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Naor and Naor, Small-bias probability spaces: Efficient constructions and applications, SIAM J. Comput. 22 (1993).

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J. Naor and M. Naor, Small-bias probability spaces: Efficient constructions and applications, SIAM J Comput 22 (1993), 838 -- 856.

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J. Naor and M. Naor. Small-bias probability spaces: Efficient constructions and applications. In Proc. of the 22nd ACM Symposium on Theory of Computing (STOC), pages 213--223, 1990.

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J. Naor and M. Naor. Small-bias Probability Spaces: Efficient Constructions and Applications. Proceedings of the 22nd Annual ACM Symposium on Theory of Computing, pages 213--223, 1990.

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J. Naor and M. Naor. Small--bias probability spaces: Efficient constructions and applications. SIAM Journal on Computing, 22(4):838--856, 1993.

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J. Naor, M. Naor. Small-Bias Probability Spaces: Efficient Constructions and Applications. In SIAM J. Computing, 22(4):838-856, 1993.

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J. Naor and J. Naor. Small-bias probability spaces: Efficient constructions and applications. SIAM Journal on Computing, 22(4):838--856, 1993.

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Joseph Naor and Moni Naor. Small-bias probability spaces: Efficient constructions and applications. SIAM Journal on Computing, 22(4):838--856, August 1993.

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J. Naor and M. Naor. Small-bias probability spaces: Efficient constructions and applications. In Proc. of 22th STOC, pages 213--223, 1990.

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J. Naor and M. Naor. Small-bias probability spaces: efficient constructions and applications. Siam J. Computing, 22:838--856, 1993.

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