Finding explicit extractors is an important derandomization goal that has received a lot of attention in the past decade. This research has focused on two approaches, one related to hashing and the other to pseudorandom generators. A third view, regarding extractors as good error correcting codes, was noticed before. Yet, ... more >>>

We consider a general approach to the hoary problem of (im)proving circuit lower bounds. We define notions of hardness condensing and hardness extraction, in analogy to the corresponding notions from the computational theory of randomness. A hardness condenser is a procedure that takes in a Boolean function as input, as ... more >>>

We give new constructions of randomness extractors and lossless condensers that are optimal to within constant factors in both the seed length and the output length. For extractors, this matches the parameters of the current best known construction [LRVW03]; for lossless condensers, the previous best constructions achieved optimality to within ... more >>>

We present new explicit constructions of *deterministic* randomness extractors, dispersers and related objects. We say that a
distribution $X$ on binary strings of length $n$ is a
$\delta$-source if $X$ assigns probability at most $2^{-\delta n}$
to any string of length $n$. For every $\delta>0$ we construct the
following poly($n$)-time ... more >>>

We explicitly construct extractors for two independent $n$-bit sources of $(\log n)^{1+o(1)}$ min-entropy. Previous constructions required either $\mathrm{polylog}(n)$ min-entropy \cite{CZ15,Meka15} or five sources \cite{Cohen16}.

Our result extends the breakthrough result of Chattopadhyay and Zuckerman \cite{CZ15} and uses the non-malleable extractor of Cohen \cite{Cohen16}. The main new ingredient in our construction ... more >>>

In their seminal work, Chattopadhyay and Zuckerman (STOC'16) constructed a two-source extractor with error $\varepsilon$ for $n$-bit sources having min-entropy $poly\log(n/\varepsilon)$. Unfortunately, the construction running-time is $poly(n/\varepsilon)$, which means that with polynomial-time constructions, only polynomially-large errors are possible. Our main result is a $poly(n,\log(1/\varepsilon))$-time computable two-source condenser. For any $k ... more >>>

We initiate a comprehensive study of the question of randomness extractions from two somewhat dependent sources of defective randomness.
Specifically, we present three natural models, which are based on different natural perspectives on the notion of bounded dependency between a pair of distributions.
Going from the more restricted model ... more >>>

In 2007 Guruswami, Umans and Vadhan gave an explicit construction of a lossless condenser based on Parvaresh-Vardy codes. This lossless condenser is a basic building block in many constructions, and, in particular, is behind the state of the art extractor constructions.

We give an alternative construction that is based on ... more >>>

A Chor--Goldreich (CG) source [CG88] is a sequence of random variables $X = X_1 \circ \ldots \circ X_t$, each $X_i \sim \{0,1 \{^d$, such that each $X_i$ has $\delta d$ min-entropy for some constant $\delta > 0$, even conditioned on any fixing of $X_1 \circ \ldots \circ X_{i-1}$. We typically ... more >>>

While the existence of randomness extractors, both seeded and seedless, has been thoroughly studied for many sources of randomness, currently, very little is known regarding the existence of seedless condensers in many settings. Here, we prove several new results for seedless condensers in the context of three related classes of ... more >>>

We give the first explicit constant rate, constant relative distance, linear codes with an encoder that runs in time $n^{1 + o(1)}$ and space $\mathop{polylog}(n)$ provided random access to the message. Prior to this work, the only such codes were non-explicit, for instance repeat accumulate codes [DJM98] and the codes ... more >>>

We present the first explicit construction of two-sided lossless expanders in the unbalanced setting (bipartite graphs that have many more nodes on the left than on the right). Prior to our work, all known explicit constructions in the unbalanced setting achieved only one-sided lossless expansion.

Specifically, we show ... more >>>

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 >>>

A natural model of a source of randomness consists of a long stream of symbols $X = X_1\circ\ldots\circ X_t$, with some guarantee on the entropy of $X_i$ conditioned on the outcome of the prefix $x_1,\dots,x_{i-1}$. We study unpredictable sources, a generalization of the almost Chor--Goldreich (CG) sources considered in [DMOZ23]. ... more >>>

We investigate the task of deterministically condensing randomness from Online Non-Oblivious Symbol Fixing (oNOSF) sources, a natural model of defective random sources for which it is known that extraction is impossible [AORSV, EUROCRYPT'20]. A $(g,\ell)$-oNOSF source is a sequence of $\ell$ blocks $\mathbf{X} = (\mathbf{X}_1, \dots, \mathbf{X}_{\ell})\sim (\{0, 1\}^{n})^{\ell}$, where ... more >>>