Alon, N., Goldreich, O., Hastad, J., Peralta, R.: Simple construction of almost kwise independent random variables. Random Struct. Algorithms 3 (1992) 289--304  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... is called biased (or an biased sample space) if for all nonempty [n] f1; ng, Exp s2S Q sa : Small sets with these properties were initially constructed by Naor and Naor [16] and Peralta [18] We will use a construction, due to Alon, Goldreich, H astad and Peralta [1], which gives an biased sample space in f0; 1g of size about ( The sample space is given as the image of a certain function n;m : F 2 m F 2 m . Here F 2 n denotes the nite eld with 2 elements. To de ne , let bin : F 2 m f0; 1g be a bijection satisfying bin(0) ....

....the componentwise exclusive or of and . Then (x; y) r = r 0 ; r n 1 ) where r i = hbin(x ) bin(y)i 2 , the inner product, modulo two, of x and y. The size of the sample space is 2 . Let S n;m f0; 1g be the collection of points so de ned. They show that Theorem 1 ([1]) S n;m = im m;n is 2 m biased. Observe that when m = log n 1 2 m . As elements of S m;n are constructed during the encryption (and decryption) phase of the 0 entropic encryption system, we analyze the complexity of computing the function above. First, we need to nd an ....

Noga Alon, Oded Goldreich, Johan Hastad, and Rene Peralta. Simple constructions of almost k-wise independent random variables. In 31st Annual Symposium on Foundations of Computer Science, volume II, pages 544-553, St. Louis, Missouri, 22-24 October 1990. IEEE.

....columns are indexed by at most k element subsets and the corresponding entry is 1 if the sets are disjoint and 0 otherwise. This matrix has full rank, see, e.g. 7] or a more general Lemma 4 below. Thus we get a lower bound n=2 There are several constructions that achieve k = 346 n) see [2], the most popular is the Paley graph. Hence we obtain a lower bound of the form n 452 . An upper bound on the rank of the associated matrices We shall show that the last mentioned construction gives asymptotically the largest rank that is possible for the matrices associated with structures ....

N. Alon, O. Goldreich, J. Hastad, R. Peralta, Simple constructions of almost k-wise independent random variables. Random Structures and Algorithms 3, (1992), 289-304.

....satisfying the following property: if at most 2 sets in all are chosen from the various partitions with no two sets coming from the same partition, then the union of these sets does not cover N . Partitions with the above properties can be constructed deterministically in polynomial time [AGHP92,NSS95]. Let P i ; P i respectively denote the first and second sets in the ith partition. We describe the construction of SC next. Using P j i s to construct SC. The base set B for SC is defined to be f(e; i)je 2 E; 1 i n g. The collection C of subsets of B contains a set C(v; a) for each ....

N. Alon, O. Goldreich, J. Hastad, R. Perralta. Simple Constructions of Almost k-Wise Independent Random Variables. Random Structures and Algorithms, 3, 1992.

....in order to reduce the influence on the error probability. The large size of the hash function that has to be communicated for privacy amplification can be reduced by using almost universal hash functions based on almost k wise independent random variables that can be constructed efficiently [AGHP92] Such functions g : X Y can be described with about 5 log jYj instead of log jX j bits and can replace universal hash functions in privacy amplification [GW96, SZ94] 5.4.6 Discussion Our results show that unconditional security can be based on assumptions about the adversary s available ....

Noga Alon, Oded Goldreich, Johan Hastad, and Ren'e Peralta, Simple constructions of almost k-wise independent random variables, Random Structures and Algorithms 3 (1992), no. 3, 289--304, Preliminary version presented at 31st FOCS (1990).

....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 almost k wise independent ....

....universal hash families which allowed the construction of multiple A codes. Here, we prove that almost k wise independent sample spaces are equivalent to multiple A codes. This allows us to obtain a more efficient construction of multiple A codes from the almost k wise independent sample spaces of [1]. Next, we present a lower bound on the size of the keyspace in a multiple A code. Numerical examples show that the multiple A codes we construct are quite close to this bound. Further, from the above equivalence, a lower bound on the size of almost k wise independent sample spaces is obtained ....

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N. Alon, O. Goldreich, J. Hastad, and R. Peralta. Simple constructions of almost k-wise independent random variables. Random Structures and Algorithms 3 (1992), 289--304.

....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 quadratic ....

....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 quadratic characters. Let p be a prime which is 1 mod 4 such that p (k=ffl) and also greater than ) A point in the probability space is defined by i 2 Z p . The random variable x j at point i is the quadratic character of i j mod p, i.e if i j is a quadratic residue ....

N. Alon, O. Goldreich, J. Hastad and R. Peralta, Simple constructions for almost k-wise independent random variables, Proc. 31st IEEE Symp. on Foundations of Computer Science, 1990, pp. 544-553.

....will divide the elements of X so that no more than k 2 fall in any hash bucket. We then continue recursively. Our construction of these hash functions is based on the explicit construction of almost k wise independent distributions on N bit binary strings. We use the following result from [2]: Proposition 1 We can construct a family of N bit strings which are # away (in the L 1 norm) from k wise independence, such that log is at most k 2 log( k log N 2# ) 2. We use this proposition to construct an almost k wise independent family of hash functions from [n] to [r] where we ....

N. Alon, O. Goldreich, J. Hastad, and R. Peralta. Simple constructions of almost k-wise independent random variables. Random Structures and Algorithms, 3(3):289--304, 1992.

....is satis ed for our choice of C out . Hence the number of codewords consistent with the received word y is at most 1= O(1= thus proving Part (iii) of the lemma as well. 2 (Lemma 11) Remark: The previous results from [19] obtained by appealing to Lemma 2 for code constructions from [2, 3]) along the lines of the above theorem achieved a block length of N = minfO( 2 ) O( 3 )g. Thus, our result achieves the best features of both these bounds and gets N = O(K= hiding the lg(1= factor) 3.2 Some improvements 3.2.1 Obtaining near linear encoding and ....

N. Alon, O. Goldreich, J. Hastad and R. Peralta. Simple constructions of almost k-wise independent random variables. Random Structures and Algorithms, 3 (1992), pp. 289-304.

....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 certain ....

....and we will find the output of C(x) in particular. We have proved: Theorem 2 If there is a sparse set that is hard for P under logspace or NC 2 many one reductions, then P RNC 2 . 7 4 Deterministic construction In this section, we use a small sample space construction due to Alon et al. AGHP90] and generalize their result concerning the construction. We apply the generalization to derandomize the probabilistic simulation of Section 3. Under the hypothesis about sparse hard sets, this yields a collapse of P to NC 2 . As before we have B = f0; 1g n = Z n 2 considered as an ....

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N. Alon, O. Goldreich, J. Hastad, and R. Peralta. Simple constructions of almost k-wise independent random variables. In Proc. 31st Annual IEEE Symposium on Foundations of Computer Science, pages 544--553, 1990.

....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 vector ....

....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 goal in ....

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N. Alon, O. Goldreich, J. Hastad, and R. Peralta. Simple constructions of almost k-wise independent random variables. Random Structures and Algorithms, 3(3):289--303,

....independent. However they satisfy a weaker form of independence which will suce. We rst introduce this notion and show how the second moment analy sis can still be applied here. Our notion seems to be slightly weaker than some weak forms of independence that have been used in the literature [21, 1] and incomparable to some of the others [18, 23] The de nition that follows is given only for the special case of boolean random variables, but is easily generalized. We let [N ] f1; Ng. Refer to bottom of Section 2 for notation and conventions. De nition 3.7 Let X 1 ; Xn be ....

N. Alon, O. Goldreich, J. H astad and R. Peralta. Simple constructions of almost k-wise independent random variables. FOCS 90.

.... AG codes from a fraction of errors approaching (1 Gamma 1=q) Code families with minimum distance approaching (1 Gamma 1=q Gamma ) can in principle be list decoded to a radius of (about) 1 Gamma 1=q Gamma O( p ) and constructions of such codes are known without relying on concatenation [4], but no efficient list decoding algorithm is known for these codes. 1.1 Previous Work Let us first consider the case of erasures. It is shown in [19] that a Reed Solomon code concatenated with a Hadamard code together with the outer (list) decoder for Reed Solomon codes of [27, 14] 2 implies ....

....1=q) c) Algebraic geometric code concatenated with Hadamard code These codes are by no means new to our paper and have been often considered in the past. The Reed Solomon code concatenated with a Hadamard code, in particular, has been a popular code and has been considered for instance in [1, 4, 19, 28]. The novel aspect of our work is in the (list) decoding algorithms we give to decode these codes in the presence of a very large number of errors and erasures. Our decoding algorithms for these concatenated codes begin by a decoding of the inner code which can be accomplished by brute force since ....

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N. Alon, O. Goldreich, J. H astad and R. Peralta. Simple constructions of almost k-wise independent random variables. Random Structures and Algorithms, 3 (1992), pp. 289-304.

....B satisfy ( Take z 1 = 0; z 2 = 1. Then the matrix R A;B; z has full rank. This is because R A;B; z is the well known disjointness matrix D(n=2; k) 1 e.g. 5] or a more general Lemma 1 below. Thus we get a lower bound n=2 k : There are several constructions that achieve k = 299 n) see [1], the most popular being the Paley graph, hence we obtain a lower bound of the form n 405 n) Relation to self avoiding families A family A of subsets of V is called self avoiding (cf. 2] if, no two elements of A are comparable by inclusion and with each a 2 A, one can associate a subset T ....

N. Alon, O. Goldreich, J. Hastad, R. Peralta, Simple constructions of almost k-wise independent random variables. Random Structures and Algorithms 3, (1992), 289-304.

....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 ....

....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 argument similar to the one used in the study of perfect hash families) the following lower bound for f (k) 0 (N) can be derived: f (k) 0 (N) c Delta 2 2k log 2 N; where c ....

N. Alon, O. Goldreich, J. Hastad and P. Peralta, Simple constructions of almost k-wise independent random variables, Random Structures and Algorithms,3 (1992), 289-304.

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Alon, N., Goldreich, O., H astad, J., and Peralta, R. Simple constructions of almost k wise independent random variables. Journal of Random Structures and Algorithms 3, 3 (Fall 1992), 289-304.

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N. Alon, O. Goldreich, J. Hastad, and R. Peralta, Simple constructions of almost k-wise independent random variables, J Random Struct Alg 3(3) (1992), 289 --304.

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Alon, N., Goldreich, O., H astad, J., and Peralta, R. Simple constructions of almost k wise independent random variables. Journal of Random Structures and Algorithms 3, 3 (Fall 1992), 289-304.

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N. Alon, O. Goldreich, J. Hastad, R. Peralta, "Simple Constructions of Almost k-wise Independent Random Variables", Journal of Random structures and Algorithms, Vol. 3, No. 3, (1992), pp. 289--304.

.... 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 of re analyzing the algorithm for the case of ....

....for any (ffl; k) approximation X, the bias of the distribution X on every non empty subset of size at most k is bounded above by ffl. On the other hand, if X has bias at most ffl on every nonempty subset of size at most k then X is an (2 Delta ffl; k) approximation (see [7] and the Appendix in [2]) Since we are willing to give up on exp(k) factors, we state our results in terms of distributions of bounded bias. Theorem 2.1 (Upper Bound) Let X = X 1 : X n ) be a distribution over f0; 1g such that the bias of X on any non empty subset of size upto k is at most ffl. Then X is ffi(n; ....

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N. Alon, O. Goldreich, J. Hastad, R. Peralta. Simple Constructions of Almost k-wise Independent Random Variables. Journal of Random structures and Algorithms, Vol. 3, No. 3, (1992), pages 289--304.

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Alon, N., Goldreich, O., Hastad, J., Peralta, R.: Simple construction of almost kwise independent random variables. Random Struct. Algorithms 3 (1992) 289--304

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N. Alon, O. Goldreich, J. Hastad, and R. Peralta, "Simple Constructions of Almost k-wise Independent Random Variables", FOCS 90 and Random Structures & Algorithms, Vol 3, pp. 289-304, 1992. (Addendum: Vol 4. pp. 119-120, 1993). 36

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N. Alon, O. Goldreich, J. Hastad and R. Peralta, Simple constructions for almost k-wise independent random variables, Random Structures and Algorithms, vol. 3, 1992, pp. 289-304.

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Noga Alon, Oded Goldreich, Johan Hastad, and Rene Peralta. Simple constructions of almost k-wise independent random variables. In Proc. of the 31st FOCS, pages 544--553, 1990.

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N. Alon, O. Goldreich, J. Hastad, and R. Peralta, "Simple constructions of almost k-wise independent random variables", FOCS 90 and Random Structures & Algorithms, Vol 3, pp. 289-304,

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N. Alon, O. Goldreich, J. Hastad, and R. Peralta. Simple constructions of almost k wise independent random variables. Random Structures and Algorithms, 3(3):289-304, 1992. (preliminary version in FOCS'90).

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N. Alon, O. Goldreich, J. Hastad, and R. Peralta, "Simple constructions of almost k-wise independent random variables", Proc. of 31st FOCS, 1990, pp. 544--553.

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N. Alon, O. Goldreich, J. Hastad, and R. Peralta, "Simple constructions of almost k-wise independent random variables", FOCS 90 and Random Structures & Algorithms, Vol 4, 1993.

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N. Alon, O. Goldreich, J. Hastad, and R. Peralta. Simple constructions of almost k-wise independent random variables. In Proceedings of Annual IEEE Symposium on Foundations of Computer Science, pages 544-553, 1990.

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N. Alon, O. Goldreich, J. Hastad, and R. Peralta. Simple constructions of almost k-wise independent random variables. In Proc. 31st IEEE Symp. on Foundations of Computer Science, volume 2, pages 544--553, 1990.

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N. Alon, O. Goldreich, J. Hastad, R. Peralta, Simple construction of almost k-wise independent random variables, Random Structures and Algorithms, vol. 3, no. 3, pp. 289-304, 1992.

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Noga Alon, Oded Goldreich, Johan Hastad, Rene Peralta. Simple Construction of Almost k-wise Independent Random Variables. Random Structures and Algorithms 3(3): 289-304.

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N. Alon, O. Goldreich, J. Hastad and R. Peralta, Simple constructions of almost k- wise independent random variables. Random Structures and Algorithms, 3(3), 1992, pp. 289-303.

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N. Alon, O. Goldreich, J. Hastad, and R. Peralta. Simple constructions of almost k-wise independent random variables. Random Structures and Algorithms, 3(3):289-304, 1992. Addendum in Random Structures and Algorithms, 4(1):119-120, 1993.

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N. Alon, O. Goldreich, J. Hastad and R. Peralta, Simple constructions of almost k-wise independent random variables, Random Structures and Algorithms 3 (1992), 289-304. (Addendum: Random Structures and Algorithms 4 (1993), 119120. )

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N. Alon and O. Goldreich and J. Hastad and R. Peralta. Simple Constructions of Almost k-wise Independent Random Variables, Journal of Random Structures and Algorithms,3:3 (1992), pp 289--304

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N. Alon and O. Goldreich and J. Hastad and R. Peralta. Simple Constructions of Almost k-wise Independent Random Variables, Journal of Random Structures and Algorithms,3:3 (1992), pp 289--304

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N. Alon, O. Goldreich, J. Hastad, R. Peralta, Simple constructions of almost k--wise independent random variables, Random Structures and Algorithms 3 (3) (1992) 289--303.

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Alon, Goldreich, Hastad, and Peralta. Simple constructions of almost kwise independent random variables. Random Structures & Algorithms, 3, 1992.

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N. Alon, O. Goldreich, J. Hastad, and Peralta, Simple constructions of almost k-wise independent random variables, Random Struct. and Algorithms 3 (1992).

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N. Alon, O. Goldreich, J. Hastad, and R. Peralta. Simple constructions of almost k-wise independent random variables. Random Structures and Algorithms, 3(3):289--304, 1992.

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N. Alon, O. Goldreich, J. Hastad, and R. Peralta, "Simple constructions of almost k-wise independentrandomv ariables," in: Proceedings of the 3-e Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Angeles, 1990, pp. 544--553.

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Noga Alon, Oded Goldreich, Johan Hastad, and Rene Peralta. Simple constructions of almost k-wise independent random variables. Random Structures & Algorithms, 3(3):289--304, 1992.

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N. Alon, O. Goldreich, J. H astad, and R. Peralta. Simple constructions of almost k--wise independent random variables. Random Structures and Algorithms, 3(3):289--303, 1992.

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N. Alon, O. Goldreich, J. Hastad, and R. Peralta. Simple Constructions of Almost k-wise Independent Random Variables. Random Structures and Algorithms, 3 (1992), pp. 289--304.

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N. Alon, O. Goldreich, J. Hastad, and R. Peralta. Simple constructions of almost k--wise independent random variables. Random Structures and Algorithms, 3(3):289--303, 1992.

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N. Alon, O. Goldreich, J. Hastad, and R. Peralta. Simple constructions of almost k-wise independent random variables. Random Structures and Algorithms, 3(3):289--304, 1992.

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Alon, N., Goldreich, O., Hastad, J., and Peralta, R. (1992). Simple constructions of almost k-wise independent random variables. Random Structures and Algorithms 3, 289--304.

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N. Alon, O. Goldreich, J. Hastad, and R. Peralta. Simple constructions of almost k-wise independent random variables. Random Structures and Algorithms 3 (1992), 289--304.

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N. Alon, O. Goldreich, J. H&stad and R. Peralta, Simple constructions of almost k-wise independent random variables, 31st Annual Symposium on Foundations of Computer Science, IEEE Computer Society,

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N. Alon, O. Goldreich, J. Hastad, R. Peralta, Simple constructions of almost k-wise independent random variables, Random Structures and Algorithms, 3(3) (1992), pp. 289-304.

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