Pseudorandom bits for non-commutative programs

Abstract

We obtain new explicit pseudorandom generators for several computational models involving groups. Our main results are as follows: 1. We consider read-once group-products over a finite group $G$, i.e., tests of the form $\prod_{i=1}^n g_i^{x_i}$ where $g_i\in G$, a special case of read-once permutation branching programs. We give generators with optimal seed length $c_G \log(n/\varepsilon)$ over any $p$-group. The proof uses the small-bias plus noise paradigm, but derandomizes the noise to avoid the recursion in previous work. Our generator works when the bits are read in any order. Previously for any non-commutative group the best seed length was $\ge\log n\log(1/\varepsilon)$, even for a fixed order. 2. We give a reduction that "lifts" suitable generators for group products over $G$ to a generator that fools width-$w$ block products, i.e., tests of the form $\prod g_i^{f_i}$ where the $f_i$ are arbitrary functions on disjoint blocks of $w$ bits. Block products generalize several previously studied classes. The reduction applies to groups that are mixing in a representation-theoretic sense that we identify. 3. Combining (2) with (1) and other works we obtain new generators for block products over the quaternions or over any commutative group, with nearly optimal seed length. In particular, we obtain generators for read-once polynomials modulo any fixed $m$ with nearly optimal seed length. Previously this was known only for $m=2$. 4. We give a new generator for products over "mixing groups." The construction departs from previous work and uses representation theory. For constant error, we obtain optimal seed length, improving on previous work (which applied to any group). This paper identifies a challenge in the area that is reminiscent of a roadblock in circuit complexity -- handling composite moduli -- and points to several classes of groups to be attacked next.