## Simple Constructions of Almost k-wise Independent Random Variables (1992)

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Citations: | 270 - 40 self |

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@INPROCEEDINGS{Alon92simpleconstructions,

author = {Noga Alon and Oded Goldreich and Johan Håstad and René Peralta},

title = {Simple Constructions of Almost k-wise Independent Random Variables},

booktitle = {},

year = {1992},

pages = {289--304},

publisher = {}

}

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### Abstract

We present three alternative simple constructions of small probability spaces on n bits for which any k bits are almost independent. The number of bits used to specify a point in the sample space is (2 + o(1))(log log n + k/2 + log k + log 1 ɛ), where ɛ is the statistical difference between the distribution induced on any k bit locations and the uniform distribution. This is asymptotically comparable to the construction recently presented by Naor and Naor (our size bound is better as long as ɛ < 1/(k log n)). An additional advantage of our constructions is their simplicity.

### Citations

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Citation Context ...n the sample point, induces a 0-1 random variable with bias bounded by ɛ. Another construction with similar parameters can be given [3] by applying the known properties of the duals of BCH codes (see =-=[23]-=-, page 280). Our three constructions are so simple that they can be described in the three corresponding paragraphs below: 1. A point in the first sample space is specified by two bit strings of lengt... |

367 | A Simple Parallel Algorithm for the Maximal Independent Set Problem
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Citation Context ...n the development of efficient algorithms. Notable examples are the massive use of randomness in computational number theory (e.g., primality testing [26, 28, 18, 1]) and in parallel algorithms (e.g. =-=[21, 24]-=-). ∗ Sackler Faculty of Exact Sciences, Tel Aviv University, Israel, and IBM Almaden Research Center, San Jose, CA 95120. † Computer Science Dept., Technion, Haifa, Israel. Supported by grant No. 86-0... |

259 | Small-bias probability spaces: efficient constructions and applications - Naor, Naor - 1993 |

190 | Probabilistic algorithm for testing primality - Rabin - 1980 |

183 | How to recycle random bits
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Citation Context ...path of length O(m) starting at this point. With probability at least 1 − 2 −m , at least one of these points specifies a sequence of m polynomials containing at least one irreducible polynomial (cf. =-=[2, 13, 20, 17, 8]-=-). This sampling requires O(m) bits. Call the resulting sample space Em. • A sample point in Em specifies O(m 2 ) polynomials and with overwhelming probability at least one of them is irreducible. Say... |

165 | Matching is as easy as matrix inversion
- Mulmuley, Vazirani, et al.
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Citation Context ...ence, University of Wisconsin, Milwaukee, WI, 53201. Supported by NSFgrant No. CCR-8909657. 1in computational number theory (e.g., primality testing [26, 28, 18, 1]) and in parallel algorithms (e.g. =-=[21, 24]-=-). A randomized algorithm can be viewedasatwo-stage procedure in which rst a \sample point" is chosen at random and next a deterministic procedure is applied to the sample point. In the generic case t... |

153 | The bit extraction problem or t-resilient functions
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Citation Context ...ample spaces of size approximately n k/2 , where n is (as above) the length of the strings in the sample space. This result is very close to best possible, in view of the lower bound of Chor. et. al. =-=[11]-=-. Hence, kwise independent sample spaces of size polynomial in n are only possible for constant k. This fact led Naor and Naor to introduce the notion of almost k-wise independent sample spaces. Loose... |

137 |
A Fast Monte-Carlo Test for Primality
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Citation Context ...t years, randomization has played a central role in the development of efficient algorithms. Notable examples are the massive use of randomness in computational number theory (e.g., primality testing =-=[26, 28, 18, 1]-=-) and in parallel algorithms (e.g. [21, 24]). ∗ Sackler Faculty of Exact Sciences, Tel Aviv University, Israel, and IBM Almaden Research Center, San Jose, CA 95120. † Computer Science Dept., Technion,... |

115 | Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs
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Citation Context ...ear combination over GF (2) of the random variables, the probability that it is 0 is between 1/2 − ɛ and 1/2 + ɛ. Our objective is to study the function m(n, ɛ). A very similar function is studied in =-=[6]-=-, and most of the techniques applied there can be used in our case as well, as we briefly describe below. Besides these techniques, we need a new result, stated in proposition 4 below. As mentioned in... |

98 | Explicit construction of linear size superconcentrators - Gabber, Galil - 1981 |

98 |
Deterministic simulation in logspace
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Citation Context ... The minimal polynomial for x will give the feedback rule. Remark: It is possible to get slightly more bits without a ecting the bias of linear tests. Let v 1, v 2 :::vm be a basis of GF [2 m ]over GF=-=[2]-=-. Then we can extract nm bits by letting bij =(bin(vjx i )� bin(y)). The bias of any xor is still n2 ;m . The proof of this fact is almost identical to the present proof. The only di erence is that we... |

91 | On the Power of Two-Point Based Sampling
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Citation Context ...ependent sampling to specify m monic polynomials of degree m. With probability at least 1 2 , at least one of these polynomials is irreducible. The pairwise independent sampling requires 2m bits (cf. =-=[12]-=-). Call the resulting sample space Pm. • Use an expander-path of length O(m) to specify O(m) points in the sample space Pm. This is done by using O(m) bits to specify a starting point and then using O... |

72 |
Equations over finite fields: an elementary approach.LectureNotesinMath.536
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Citation Context ...reducible monic polynomials of degree m (i.e. 2m m ) the proposition follows. 8s4 The Quadratic Character Construction Our second construction is based on Weil’s Theorem regarding character sums (cf. =-=[27]-=-, p. 43, Thm. 2C). A special case of this theorem is stated below. Definition 5 (Quadratic Character): Let p be an odd prime and x be an integer relatively prime to p. The Quadratic Character of x mod... |

72 |
Explicit expanders and the Ramanujan conjectures
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Citation Context ... exponent depends, 2 among other things, on the constants involved in an explicit construction of an expander (namely the degree and second eigenvalue of the expander). Using the best known expanders =-=[22]-=- this constant is slightly larger than 4. We present three alternative constructions of sample spaces of size roughly ( n ) 2 for which the exclusive-or of any xed bit locations, in the sample point, ... |

69 | Almost all primes can be quickly certified
- Goldwasser, Kilian
- 1986
(Show Context)
Citation Context ...t years, randomization has played a central role in the development of efficient algorithms. Notable examples are the massive use of randomness in computational number theory (e.g., primality testing =-=[26, 28, 18, 1]-=-) and in parallel algorithms (e.g. [21, 24]). ∗ Sackler Faculty of Exact Sciences, Tel Aviv University, Israel, and IBM Almaden Research Center, San Jose, CA 95120. † Computer Science Dept., Technion,... |

58 | A parallel algorithmic version of the Local Lemma
- Alon
- 1991
(Show Context)
Citation Context ...nt sample space. In view of this property it is not surprising that these spaces can be used in many applications. Some applications are presented in [25], while more recent applications are given in =-=[4]-=- and [10]. Another, more detailed application, is that one can get an alternative (slightly nicer) proof of Lemma 8 (on page 8) of [15]. Naor and Naor presented an efficient construction of an almost ... |

56 |
E cient and Secure Pseudo-Random Number Generation
- Vazirani, Vazirani
- 1984
(Show Context)
Citation Context ...can be interpreted as trying to refute the randomness of a probability space by taking a fixed linear combination of the bits in the sample. The following lemma, attributed to Vazirani [29] (see also =-=[30]-=-, [11]), links the ability to pass linear tests with almost independence. 5sLemma 1 (Vazirani): Let Sn ⊂ {0, 1} n be a sample space that is ɛ-biased with respect to linear tests of size at most k. The... |

42 |
Randomness, Adversaries and Computation
- Vazirani
- 1986
(Show Context)
Citation Context ...s. A linear test can be interpreted as trying to refute the randomness of a probability space by taking a xed linear combination of the bits in the sample. The following lemma, attributed to Vazirani =-=[29]-=- (see also [30], [11]), links the ability to pass linear tests with almost independence. 5Lemma 1 (Vazirani): Let Sn f0� 1g n be a sample space that is -biased with respect to linear tests of size at... |

36 | Deterministic simulation - Ajtai, Komlos, et al. - 1987 |

34 |
Shift Register Sequences, Aegean Park Press
- Golomb
- 1982
(Show Context)
Citation Context ...(n − 1)2 −m (1 + O(2 −m/2 ))-biased when r is selected uniformly in A 2m n . Proof: For the rest of this section we consider only polynomials over GF (2). The number of irreducible monic polynomials (=-=[19]-=-, p. 39) of degree m is 1 m � d|m µ( m d )2d , where µ is the ordinary Möbius function (i.e. µ(x) = (−1) s where s is the number of primes that divide x if x is squarefree and µ(x) = 0 otherwise). Sin... |

31 |
A fast and simple randomized algorithm for the maximal independent set problem
- Alon, Babai, et al.
- 1986
(Show Context)
Citation Context ...probability distribution, induced on every k bit locations in a string randomly selected in the sample space, should be uniform. Such a sample space is called k-wise independent. Alon, Babai and Itai =-=[5]-=- presented an efficient construction of k-wise independent sample spaces of size approximately n k/2 , where n is (as above) the length of the strings in the sample space. This result is very close to... |

8 |
Matching is as Easy as
- Mulmuley, Vazirani, et al.
- 1987
(Show Context)
Citation Context ...n the development of efficient algorithms. Notable examples are the massive use of randomness in computational number theory (e.g., primality testing [26, 28, 18, 1]) and in parallel algorithms (e.g. =-=[21, 24]-=-). ∗ Sackler Faculty of Exact Sciences, Tel Aviv University, Israel, and IBM Almaden Research Center, San Jose, CA 95120. † Computer Science Dept., Technion, Haifa, Israel. Supported by grant No. 86-0... |

7 | Security Preserving Amplification of Hardness", 31st FOCS - Goldreich, Impagliazzo, et al. - 1990 |

7 |
Almost All Primes Can be Quickly Certi ed
- Goldwasser, Kilian
- 1986
(Show Context)
Citation Context ...x Dept. of Electrical Engineering and Computer Science, University of Wisconsin, Milwaukee, WI, 53201. Supported by NSFgrant No. CCR-8909657. 1in computational number theory (e.g., primality testing =-=[26, 28, 18, 1]-=-) and in parallel algorithms (e.g. [21, 24]). A randomized algorithm can be viewedasatwo-stage procedure in which rst a \sample point" is chosen at random and next a deterministic procedure is applied... |

4 |
Private communication
- Even
- 2013
(Show Context)
Citation Context ...ary values considered as elements of Zd). The first such generalization is due to Azar, Motwani and Naor [7] (extending the characters construction). The second such generalization is due to Guy Even =-=[14]-=- (extending the LFSR construction). Our third construction can be easily generalized as well. However, if one is interested in distributions over d-ary sequences which are statistically close to k-wis... |

4 | Dispensers, Deterministic Amplification - Cohen, Wigderson - 1989 |

3 |
Explicit Constructions of Linear Sized
- Gabber, Galil
- 1981
(Show Context)
Citation Context ...onstructions of expander graphs which do not use a large prime or anything else that might be hard to find deterministically. An example of such a construction is the construction by Gabber and Galil =-=[16]-=-. 15sConstruction 4 suffices as a randomized preprocessing for Constructions 2 and 3, and for a modification of Construction 1 (sketched below). However, for Construction 1 (as appearing in Section 3)... |

2 |
An e cient construction of a multiple value small bias probability space
- Azar, Motwani, et al.
(Show Context)
Citation Context ...ave small bias with respect to linear tests (which compute a linear combination mod d of the d-ary values considered as elements of Zd). The first such generalization is due to Azar, Motwani and Naor =-=[7]-=- (extending the characters construction). The second such generalization is due to Guy Even [14] (extending the LFSR construction). Our third construction can be easily generalized as well. However, i... |

2 |
Approximating Clique is Almost NPComplete”, 32nd FOCS
- Feige, Goldwasser, et al.
- 1991
(Show Context)
Citation Context ...re presented in [25], while more recent applications are given in [4] and [10]. Another, more detailed application, is that one can get an alternative (slightly nicer) proof of Lemma 8 (on page 8) of =-=[15]-=-. Naor and Naor presented an efficient construction of an almost k-wise independent sample space [25]. Points in their sample space are specified by O(log log n + k + log 1 ɛ ) bits, where ɛ is a boun... |

2 |
Security Preserving Amplification of Hardness”, 31st FOCS
- Goldreich, Impagliazzo, et al.
- 1990
(Show Context)
Citation Context ...path of length O(m) starting at this point. With probability at least 1 − 2 −m , at least one of these points specifies a sequence of m polynomials containing at least one irreducible polynomial (cf. =-=[2, 13, 20, 17, 8]-=-). This sampling requires O(m) bits. Call the resulting sample space Em. • A sample point in Em specifies O(m 2 ) polynomials and with overwhelming probability at least one of them is irreducible. Say... |

1 |
On Dependent Random Variables Over Small Sample Spaces
- Ben-Natan
- 1990
(Show Context)
Citation Context ...= 2−k . In many applications it suffices that a bit sequence is “almost” k-wise independent. There are several standard ways of quantifying this condition (i.e. interpreting the phrase “almost”): cf. =-=[9]-=-. We use two very natural ways corresponding to the L∞ and L1 norms: Definition 2 (almost k-wise independence): Let Sn be sample space and X = x1 · · · xn be chosen uniformly from Sn. • (max-norm): Sn... |

1 | Recognizing - Adleman, Huang - 1987 |

1 | Subquadratic ZeroKnowledge ", 32nd FOCS - Boyar, Brassard, et al. - 1991 |

1 | Approximating Clique is almost NP-complete", 32nd FOCS - Feige, Goldwasser, et al. - 1991 |

1 |
Subquadratic ZeroKnowledge", 32nd FOCS
- Boyar, Brassard, et al.
- 1991
(Show Context)
Citation Context ...e space. In view of this property it is not surprising that these spaces can be used in many applications. Some applications are presented in [25], while more recent applications are given in [4] and =-=[10]-=-. Another, more detailed application, is that one can get an alternative (slightly nicer) proof of Lemma 8 (on page 8) of [15]. Naor and Naor presented an e cient construction of an almost k-wise inde... |

1 |
Dispensers, Deterministic Ampli cation
- Cohen, Wigderson
- 1989
(Show Context)
Citation Context ... path of length O(m) starting at this point. With probability at least 1 ; 2 ;m , at least one of these points speci es a sequence of m polynomials containing at least one irreducible polynomial (cf. =-=[2, 13, 20, 17, 8]-=-). This sampling requires O(m) bits. Call the resulting sample space Em. A sample point in Em speci es O(m2 ) polynomials and with overwhelming probability at least one of them is irreducible. Say we ... |

1 |
Security Preserving Ampli cation of Hardness", 31st FOCS
- Goldreich, Impagliazzo, et al.
- 1990
(Show Context)
Citation Context ... path of length O(m) starting at this point. With probability at least 1 ; 2 ;m , at least one of these points speci es a sequence of m polynomials containing at least one irreducible polynomial (cf. =-=[2, 13, 20, 17, 8]-=-). This sampling requires O(m) bits. Call the resulting sample space Em. A sample point in Em speci es O(m2 ) polynomials and with overwhelming probability at least one of them is irreducible. Say we ... |

1 |
Equations over nite elds, an elementary approach
- Schmidt
- 1976
(Show Context)
Citation Context ...reducible monic polynomials of degree m (i.e. 2m m ) the proposition follows. 84 The Quadratic Character Construction Our second construction is based on Weil's Theorem regarding character sums (cf. =-=[27]-=-, p. 43, Thm. 2C). A special case of this theorem is stated below. De nition 5 (Quadratic Character): Let p be an odd prime and x be an integer relatively prime to p. The Quadratic Character of x mod ... |