Pseudorandom Generators for Combinatorial Shapes

Parikshit Gopalan, Raghu Meka, Omer Reingold, David Zuckerman

Abstract: We construct pseudorandom generators for combinatorial shapes, which substantially generalize combinatorial rectangles, $\epsilon$-biased spaces, 0/1 halfspaces, and 0/1 modular sums. A function $f:[m]^n\rightarrow\{0,1\}$ is an $(m,n)$-combinatorial shape if there exist sets $A_1,\ldots,A_n\subseteq[m]$ and a symmetric function $h:\{0,1\}^n\rightarrow\{0,1\}$ such that $f(x_1,\ldots,x_n)=h(1_{A_1}(x_1),\ldots,1_{A_n}(x_n))$. Our generator uses seed-length $O(\log m+\log n+\log^2(1/\varepsilon))$ to get error $\varepsilon$. When $m=2$, this gives the first generator of seed-length $O(\log n)$ that fools all weight-based tests, meaning that the distribution of the weight of any subset is $\varepsilon$-close to the appropriate binomial distribution in statistical distance. Along the way, we give a generator for combinatorial rectangles with seed-length $O(\log^{3/2}n)$ and error $1/\mathrm{poly}(n)$, matching Lu's bound from ICALP 1998. 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.