Bounded Independence Plus Noise Fools Products

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 η/2. We study which tests f : {0, 1}m → [−1, 1] are ε-fooled by D + E, i.e., |E[f(D+E)]−E[f(U)]| ≤ ε where U is the uniform distribution. We show that D+E ε-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 ε = k(1− η)Ω(b2/m), where m = nk and b ≥ n. This bound is tight when b = Ω(m) and η ≥ (log k)/m. For b ≥ m2/3 logm and any constant η 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 Ω(ηm) − O(logm) is required to decode one message symbol from a codeword with ηm errors, and communication O(ηm logm) suffices. Second, we obtain pseudorandom generators. We can ε-fool product tests f : ({0, 1}n)k → [−1, 1] under any permutation of the bits with seed lengths 2n+ Õ(k2 log(1/ε)) and O(n)+ Õ( √ nk log 1/ε). Previous generators had seed lengths ≥ nk/2 or ≥ n √ nk. ∗Supported by NSF grant CCF-1319206. Work done in part while at the College of Computer and Information Science, Northeastern University, in part while visiting Harvard University, with support from Salil Vadhan’s Simons Investigator grant, and in part while at the Simons Institute for the Theory of Computing. ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 169 (2016)