Bounded Independence Plus Noise Fools Products

Elad Haramaty, Chin Ho Lee, Emanuele Viola

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 $f \colon \{0,1\}^m \to [-1,1]$ are $\varepsilon$-fooled by $D+E$, i.e., $|{\rm E}[f(D+E)] - {\rm E}[f(U)]| \le \varepsilon$, where $U$ is the uniform distribution. We show that $D+E$ $\varepsilon$-fools product tests $f : (\{0,1\}^n)^k \to [-1,1]$ given by the product of $k$ bounded functions on disjoint $n$-bit inputs with error $\varepsilon = k(1-\eta)^{\Omega(b^2/m)}$, where $m = nk$ and $b \ge n$. This bound is tight when $b = \Omega(m)$ and $\eta \ge (\log k)/m$. For $b \ge m^{2/3} \log m$ 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 $\varepsilon$-fool product tests $f\colon (\{0,1\}^n)^k \to [-1,1]$ under any permutation of the bits with seed lengths $2n + \tilde O(k^2 \log (1/\varepsilon))$ and $O(n) + \tilde O(\sqrt{nk \log 1/\varepsilon})$. Previous generators have seed lengths $\ge nk/2$ or $\ge n \sqrt{n k}$. For the special case where the $k$ bounded functions have range $\{0,1\}$ the previous generators have seed length $\ge (n+\log k)\log (1/\varepsilon)$.