In this work, we continue the line of research on the complexity of distributions (Viola, Journal of Computing 2012), and study samplers defined by low degree polynomials. An $n$-tuple $\mathcal{P} = (P_1,\dots, P_n)$ of functions $P_i \colon \mathbb{F}_2^m \to \mathbb{F}_2$ defines a distribution over $\{0,1\}^n$ in the natural way: ... more >>>

The partial derivative method is a central tool in algebraic complexity, underlying lower bounds for multilinear formulas, bounded depth circuits, and algebraic branching programs. A key feature of this measure is its subadditivity and submultiplicativity, which are usually used to upper bound the measure. However, proving lower bounds requires bounding ... more >>>

Breaking the log-squared barrier in pseudorandom generator constructions for read-once branching programs, namely, achieving seed length $o(\log^2 n)$ for length-$n$ programs, has remained a longstanding open problem since Nisan's seminal construction.

We show that breaking this barrier, even achieving seed length $O(\log^{3/2} n)$ (for, say, constant width), would follow from ... more >>>