Pseudorandomness, symmetry, smoothing: I

TL;DR

This paper establishes new limits on the effectiveness of small-bias distributions in fooling various computational tests, demonstrating their optimality, limitations, and implications for pseudorandom generator constructions.

Contribution

It proves that small-bias distributions cannot outperform bounded uniform distributions in several tests, closing longstanding research questions and clarifying their limitations.

Findings

Small-bias distributions achieve optimal statistical distance bounds.

They have heavier tail mass than uniform distributions.

They rule out certain pseudorandom generator paradigms.

Abstract

We prove several new results about bounded uniform and small-bias distributions. A main message is that, small-bias, even perturbed with noise, does not fool several classes of tests better than bounded uniformity. We prove this for threshold tests, small-space algorithms, and small-depth circuits. In particular, we obtain small-bias distributions that 1) achieve an optimal lower bound on their statistical distance to any bounded-uniform distribution. This closes a line of research initiated by Alon, Goldreich, and Mansour in 2003, and improves on a result by O'Donnell and Zhao. 2) have heavier tail mass than the uniform distribution. This answers a question posed by several researchers including Bun and Steinke. 3) rule out a popular paradigm for constructing pseudorandom generators, originating in a 1989 work by Ajtai and Wigderson. This again answers a question raised by several researchers. For branching programs, our result matches a bound by Forbes and Kelley. Our small-bias distributions above are symmetric. We show that the xor of any two symmetric small-bias distributions fools any bounded function. Hence our examples cannot be extended to the xor of two small-bias distributions, another popular paradigm whose power remains unknown. We also generalize and simplify the proof of a result of Bazzi.