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Description

We give an explicit construction of a pseudorandom generator for read-once formulas whose inputs can be read in arbitrary order. For formulas in $n$ inputs and arbitrary gates of fan-in at most $d = O(n/\log n)$, the pseudorandom generator uses $(1 - \Omega(1))n$ bits of randomness and produces an output that looks $2^{-\Omega(n)}$-pseudorandom to all such formulas. Our analysis is based on the following lemma. Let $pr = Mz + e$, where $M$ is the parity-check matrix of a sufficiently good binary error-correcting code of constant rate, $z$ is a random string, $e$ is a small-bias distribution, and all operations are modulo 2. Then for every pair of functions $f, g\colon \B^{n/2} \to \B$ and every equipartition $(I,J)$ of $[n]$, the distribution $pr$ is pseudorandom for the pair $(f(x|_I), g(x|_J))$, where $x|_I$ and $x|_J$ denote the restriction of $x$ to the coordinates in $I$ and $J$, respectively.

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