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Bounded Independence Plus Noise Fools Products

Authors Elad Haramaty, Chin Ho Lee, Emanuele Viola

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  • ounded independence
  • Noise
  • Product tests
  • Error-correcting codes
  • Pseudorandomness

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Abstract

Let D be a b-wise independent distribution over {0,1}^m. Let E be the "noise" distribution over {0,1}^m where the bits are independent and each bit is 1 with probability eta/2.  We study which tests f: {0,1}^m -> [-1,1] are epsilon-fooled by D+E, i.e., |E[f(D+E)] - E[f(U)]| <= epsilon where U is the uniform distribution.

We show that D+E epsilon-fools product tests f: ({0,1}^n)^k -> [-1,1] given by the product of k bounded functions on disjoint n-bit inputs with error epsilon = k(1-eta)^{Omega(b^2/m)}, where m = nk and b >= n.  This bound is tight when b = Omega(m) and eta >= (log k)/m. For b >= m^{2/3} log m and any constant eta the distribution D+E also 0.1-fools log-space algorithms.

We develop two applications of this type of results. First, we prove communication lower bounds for decoding noisy codewords of length m split among k parties. For Reed-Solomon codes of dimension m/k where k = O(1), communication Omega(eta m) - O(log m) is required to decode one message symbol from a codeword with eta m errors, and communication O(eta m log m) suffices. Second, we obtain pseudorandom generators. We can epsilon-fool product tests f: ({0,1}^n)^k -> [-1,1] under any permutation of the bits with seed lengths 2n + O~(k^2 log(1/epsilon)) and O(n) + O~(sqrt{nk log 1/epsilon}). Previous generators have seed lengths >= nk/2 or >= n sqrt{n k}. For the special case where the k bounded functions have range {0,1} the previous generators have seed length >= (n+log k)log(1/epsilon).

Cite As Get BibTex

Elad Haramaty, Chin Ho Lee, and Emanuele Viola. Bounded Independence Plus Noise Fools Products. In 32nd Computational Complexity Conference (CCC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 79, pp. 14:1-14:30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017) https://doi.org/10.4230/LIPIcs.CCC.2017.14

Author Details

Elad Haramaty
Chin Ho Lee
Emanuele Viola

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