Pseudorandom generators for combinatorial shapes

  • Authors:

  • Parikshit Gopalan;Raghu Meka;Omer Reingold;David Zuckerman

  • Affiliations:

  • Microsoft Research, Silicon Valley, Mountain View, CA, USA;University of Texas at Austin, Austin, TX, USA;Microsoft Research, Silicon Valley, Mountain View, CA, USA;University of Texas at Austin, Austin, TX, USA

  • Venue:
  • Proceedings of the forty-third annual ACM symposium on Theory of computing
  • Year:
  • 2011

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Abstract

We construct pseudorandom generators for combinatorial shapes, which substantially generalize combinatorial rectangles, ε-biased spaces, 0/1 halfspaces, and 0/1 modular sums. A function f:[m]n - {0,1} is an (m,n)-combinatorial shape if there exist sets A1,...,An ⊆ [m] and a symmetric function h:{0,1}n - {0,1} such that f(x1,...,xn) = h(1A1(x1),...,1An(xn)). Our generator uses seed length O(log m + log n + log2(1/ε)) to get error ε. When m = 2, this gives the first generator of seed length O(log n) which fools all weight-based tests, meaning that the distribution of the weight of any subset is ε-close to the appropriate binomial distribution in statistical distance. For our proof we give a simple lemma which allows us to convert closeness in Kolmogorov (cdf) distance to closeness in statistical distance. As a corollary of our technique, we give an alternative proof of a powerful variant of the classical central limit theorem showing convergence in statistical distance, instead of the usual Kolmogorov distance.