N. Alon, J. Bruck, J. Naor, M. Naor, and R. Roth, Construction of Asymptotically Good, Low-Rate Error-Correcting Codes through Pseudo-Random Graphs, IEEE Transactions on Information Theory 38 (192), pp. 509-516.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....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 ....

....secrets can be viewed as finding a small sample space satisfying a still weaker condition that the probability of any 4 random variables assuming the values 0011 or 1100 is nonzero. Therefore, the constructions of small sample spaces for almost k wise independent random variables in, for example, [3] can be used for constructing efficient oblivious algorithms. By using these sample spaces, we can get upper bounds for the minimum number f (k) 0 (N) of questions required for an oblivious algorithm giving a separating strategy of guessing k secrets in a space of size N of the form f (k) 0 ....

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N. Alon, J. Bruck, J. Naor, M. Naor and R. Roth, Construction of asymptotically good, low-rate error-correcting codes through pseudo-random graphs, IEEE Transactions on Information Theory, 38 (

....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: Prob(X 1 . X k = a 1 . a k ) 1 2 ....

....secrets can be viewed as finding a small sample space satisfying a still weaker condition that the probability of any 4 random variables assuming the values 0011 or 1100 is nonzero. Therefore, the constructions of small sample spaces for almost k wise independent random variables in, for example, [3] can be used for constructing e#cient oblivious algorithms. By using these sample spaces, we can get upper bounds for the minimum number f (k) 0 (N) of questions required for an oblivious algorithm giving a separating strategy of guessing k secrets in a space of size N of the form f (k) 0 ....

[Article contains additional citation context not shown here]

N. Alon, J. Bruck, J. Naor, M. Naor and R. Roth, Construction of asymptotically good, low-rate error-correcting codes through pseudo-random graphs, IEEE Transactions on Information Theory, 38 (

....0) # n 2 m # #. This, combined with the above observations, helps show that bias D (T ) # #. Thus we have an e#cient construction of a multiset S # 0, 1 n with S # O(n 2 # 2 ) such that U(S) is # biased; a di#erent such construction with S # O(n # 3 ) is presented in [2]. For k wise # bias, the reduction of [21] mentioned at the end of 2.1 yields constructions of size O(min ( k log n) #) 2 , k log n) # 3 ) 6) We now sketch a di#erent approach to small bias spaces due to [21] which has connections to coding theory. An e#cient construction of a ....

....of codewords x, y, of the Hamming distance between x and y) equals the minimum weight (the minimum, over all nonzero codewords x, of the number of nonzero symbols in x) of the code. The minimum distance is a fundamental parameter of a code. Let us specialize the discussion to the case where F = GF [2]. Suppose we have an e#cient construction of a matrix G # (GF [2] nm , with m # poly(n) and with minimum distance at least #m, for some constant # # (0, 1) If we consider the set A of m column vectors of G, a moment s reflection shows that (7) holds. Indeed, matrices G with these ....

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N. Alon, J. Bruck, J. Naor, M. Naor, and R. Roth. Construction of asymptotically good, low-rate error-correcting codes through pseudo-random graphs. IEEE Trans. Info. Theory, 38:509--516, 1992.

....second method yields smaller sample spaces for the efficient derandomization of randomized algorithms. Three major known approaches to derandomization are the techniques of limited independence ( 18, 19, 1] the method of conditional probabilities ( 26, 29] and small bias probability spaces ([22, 3, 2]) We now discuss these methods briefly. Let the set f1; 2; ng be denoted by [n] Definition 1 Random variables X 1 ; X 2 ; Xn are k wise independent if for any set I [n] of at most k indices and for any choice of v 1 ; v 2 ; we have P r( V i2I (X i = v i ) Q i2I ....

....for x 2 f0; 1g n . From this it can be shown that if all the Fourier coefficients of D S are bounded in magnitude by ffl, then the sample space is ffl approximate. NC 1 constructions of sample spaces of size O(n=ffl 3 ) with Fourier coefficients bounded in magnitude by ffl were developed in [2], improving on those of [22] NC 1 constructions of three different sample spaces of size O(n 2 =ffl 2 ) were presented by [3] The basic idea behind our construction of approximate sample spaces is the following simple and well known fact: Fact 1 Let X 1 ; X k be independent ....

[Article contains additional citation context not shown here]

N. Alon, J. Bruck, J. Naor, M. Naor, and R. Roth, Construction of asymptotically good, low-rate error-correcting codes through pseudo-random graphs, IEEE Trans. Info. Theory, 38:509--516, 1992.

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N. Alon, J. Bruck, J. Naor, M. Naor and R. Roth, Construction of asymptotically good, low-rate error-correcting codes through pseudorandom graphs, IEEE Transactions on Information Theory, 38 (1992), 509--516.

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N. Alon, J. Bruck, J. Naor, M. Naor and R. Roth, \Construction of asymptotically good, low-rate error-correcting codes through pseudo-random graphs", iIEEE Transactions on Information Theory, 38 1992, pp. 509-516.

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N. Alon, J. Bruck, J. Naor, M. Naor and R. Roth, Construction of asymptotically good, low-rate error-correcting codes through pseudo-random graphs, IEEE Transactions on Information Theory, in press.

....d wise ffl bias probability spaces are described as almost d wise independent. The size of the constructions of small bias probability spaces is P oly(d; 1=ffl; logm) More precisely, there are constructions of size: ffl O( d log m ffl 3 ) Naor Naor [18] optimized in Alon et al. [2]) ffl O( d 2 log 2 m ffl 2 ) Alon et al. 3] If 1 ffl = 2 d 1 then the size of the first probability space is 2 O(d) log m. 8.3 Intersection independence For a ground set B and subset S ae B let S 0 denote the complement of S, i.e. B Gamma S, and let S 1 denote S itself. ....

N. Alon, J. Bruck, J. Naor, M. Naor, and R. Roth, Construction of asymptotically good, low-rate error-correcting codes through pseudo-random graphs, IEEE Trans. Information Theory 38 (1992), pp. 509-516.

....Assume that k is a power of 2. Let R be a k log k wise ffi bias probability space on n log k random variables which takes values in f0; 1g. They are indexed as Y ij for 1 i n and 1 j log k. There are explicit constructions of such probability spaces of size 2 O(k log k) log n (see [8] [1]) Each function h corresponds to a point in the probability space. h(x) is the value of Y x1 ; Y x2 ; Y x log k treated as a number between 0 and 2 k Gamma 1. It can be shown that for all x 1 ; x 2 ; x k 2 f1; ng and for all y 1 ; y 2 ; y k 2 f0; 2 k Gamma 1g we have ....

N. Alon, J. Bruck, J. Naor, M. Naor and R. Roth, Construction of asymptotically good, low-rate error-correcting codes through pseudo-random graphs, IEEE Transactions on Information Theory, 38 (1992), 509-516.

....d wise ffl bias probability spaces are described as almost d wise independent. The size of the constructions of small bias probability spaces is P oly(d; 1=ffl; logm) More precisely, there are constructions of size: ffl O( d log m ffl 3 ) Naor Naor [25] optimized in Alon et al. [4]) ffl O( d 2 log 2 m ffl 2 ) Alon et al. 5] If 1 ffl = 2 d 1 then the size of the first probability space is 2 O(d) log m. In order to obtain the d intersection independent collection, set ffl 2 Gammad (say 1 ffl = 2 d 1) From a d wise ffl bias probability space ....

N. Alon, J. Bruck, J. Naor, M. Naor, and R. Roth, Construction of asymptotically good, low-rate error-correcting codes through pseudo-random graphs, IEEE Trans. Information Theory 38 (1992), pp. 509-516.

....that the number of random bits required to sample from them is O(log 1=ffl log c log log n) Therefore if 1=ffl is logarithmic in n and c is at most logarithmic in n the size of the probability space is still polylogarithmic in n. To be more precise, the construction of [23] as optimized in [2], yields a probability space of size O( c log n ffl 3 ) and the ones in [5] yield probability spaces of size O( c 2 log 2 n ffl 2 ) 2 Boolean matrix multiplication with witnesses All the matrices in this section are n by n matrices, unless otherwise specified. If M is such a matrix, we ....

....Our scheme consists of two or three levels. We keep the same form as the one in [12] for the bottom (i.e. third, and in case there is no third then second) level hash functions, but for the first level our function h will have the form h(x) Ax where A is a log n Theta log m matrix over GF [2] and x is treated as a vector of length log m over GF [2] This will also be the form of the second level hash function, whenever there is a third level. Given the first level hash function h we distinguish between those i s for which s i (h) is small and those for which s i (h) is large. The ....

[Article contains additional citation context not shown here]

N. Alon, J. Bruck, J. Naor, M. Naor and R. Roth, Construction of asymptotically good, low-rate error-correcting codes through pseudo-random graphs, IEEE Transactions on Information Theory, 38 (1992), 509-516.

....Assume that k is a power of 2. Let R be a k log k wise ffi bias probability space on n log k random variables which takes values in f0; 1g. They are indexed as Y ij for 1 i n and 1 j log k. There are explicit constructions of such probability spaces of size 2 O(k log k) log n (see [8] [1]) Each function h corresponds to a point in the probability space. h(x) is the value of Y x1 ; Y x2 ; Y x log k treated as a number between 0 and 2 k Gamma 1. It can be shown that for all x 1 ; x 2 ; x k 2 f1; ng and for all y 1 ; y 2 ; y k 2 f0; 2 k Gamma 1g we ....

N. Alon, J. Bruck, J. Naor, M. Naor and R. Roth, Construction of asymptotically good, low-rate error-correcting codes through pseudo-random graphs, IEEE Transactions on Information Theory, 38 (1992), 509-516.

No context found.

N. Alon, J. Bruck, J. Naor, M. Naor, and R. Roth, Construction of Asymptotically Good, Low-Rate Error-Correcting Codes through Pseudo-Random Graphs, IEEE Transactions on Information Theory 38 (192), pp. 509-516.

No context found.

N. Alon, J. Bruck, J. Naor, M. Naor, and R. Roth. Construction of asymptotically good, low-rate error-correcting codes through pseudo-random graphs. IEEE Transactions on Information Theory, 38:509--516, 1992.

No context found.

N. Alon, J. Bruck, J. Naor, M. Naor, and R. Roth. Construction of asymptotically good, low-rate error-correcting codes through pseudo-random graphs. IEEE Transactions on Information Theory, 28:509--516, 1992.

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