One of the earliest models of weak randomness is the Chor-Goldreich (CG) source. A $(t,n,k)$-CG source is a sequence of random variables $\mathbf{X}=(\mathbf{X}_1,\dots,\mathbf{X}_t) \sim (\{0,1\}^n)^t$, where each $\mathbf{X}_i$ has min-entropy $k$ conditioned on any fixing of $\mathbf{X}_1,\dots,\mathbf{X}_{i-1}$. Chor and Goldreich proved that there is no deterministic way to extract randomness ... more >>>

We prove that random low-degree polynomials (over $\mathbb{F}_2$) are unbiased, in an extremely general sense. That is, we show that random low-degree polynomials are good randomness extractors for a wide class of distributions. Prior to our work, such results were only known for the small families of (1) uniform sources, ... more >>>

We explicitly construct the first nontrivial extractors for degree $d \ge 2$ polynomial sources over $\mathbb{F}_2^n$. Our extractor requires min-entropy $k\geq n - \frac{\sqrt{\log n}}{(d\log \log n)^{d/2}}$. Previously, no constructions were known, even for min-entropy $k\geq n-1$. A key ingredient in our construction is an input reduction lemma, which allows ... more >>>

We continue a line of work on extracting random bits from weak sources that are generated by simple processes. We focus on the model of locally samplable sources, where each bit in the source depends on a small number of (hidden) uniformly random input bits. Also known as local sources, ... more >>>

Recently, there has been exciting progress in understanding the complexity of distributions. Here, the goal is to quantify the resources required to generate (or sample) a distribution. Proving lower bounds in this new setting is more challenging than in the classical setting, and has yielded interesting new techniques and surprising ... more >>>

We give an explicit construction of an affine extractor (over $\mathbb{F}_2$) that works for affine sources on $n$ bits with min-entropy $k \ge~ \log n \cdot (\log \log n)^{1 + o(1)}$. This improves prior work of Li (FOCS'16) that requires min-entropy at least $\mathrm{poly}(\log n)$.

Our construction is ... more >>>

An $(n,r,s)$-design, or $(n,r,s)$-partial Steiner system, is an $r$-uniform hypergraph over $n$ vertices with pairwise hyperedge intersections of size $0$, we extract from $(N,K,n,k)$-adversarial sources of locality $0$, where $K\geq N^\delta$ and $k\geq\text{polylog }n$. The previous best result (Chattopadhyay et al., STOC 2020) required $K\geq N^{1/2+o(1)}$. As a result, we ... more >>>

In a recent work, Kumar, Meka, and Sahai (FOCS 2019) introduced the notion of bounded collusion protocols (BCPs), in which $N$ parties wish to compute some joint function $f:(\{0,1\}^n)^N\to\{0,1\}$ using a public blackboard, but such that only $p$ parties may collude at a time. This generalizes well studied models in ... more >>>

Randomness extraction is a fundamental problem that has been studied for over three decades. A well-studied setting assumes that one has access to multiple independent weak random sources, each with some entropy. However, this assumption is often unrealistic in practice. In real life, natural sources of randomness can produce samples ... more >>>