We study the problem of extracting random bits from weak sources that are sampled by algorithms with limited memory. This model of small-space sources was introduced by Kamp, Rao, Vadhan and Zuckerman (STOC'06), and falls into a line of research initiated by Trevisan and Vadhan (FOCS'00) on extracting randomness from weak sources that are sampled by computationally bounded algorithms. Our main results are the following.

1. We obtain near-optimal extractors for small-space sources in the polynomial error regime. For space $s$ sources over $n$ bits, our extractors require just $k\geq s\cdot$polylog$(n)$ entropy. This is an exponential improvement over the previous best result, which required $k\geq s^{1.1}\cdot2^{\log^{0.51} n}$ (Chattopadhyay and Li, STOC'16).

2. We obtain improved extractors for small-space sources in the negligible error regime. For space $s$ sources over $n$ bits, our extractors require entropy $k\geq n^{1/2+\delta}\cdot s^{1/2-\delta}$, whereas the previous best result required $k\geq n^{2/3+\delta}\cdot s^{1/3-\delta}$ (Chattopadhyay, Goodman, Goyal and Li, STOC'20).

To obtain our first result, the key ingredient is a new reduction from small-space sources to affine sources, allowing us to simply apply a good affine extractor.

To obtain our second result, we must develop some new machinery, since we do not have low-error affine extractors that work for low entropy. Our main tool is a significantly improved extractor for adversarial sources, which is built via a simple framework that makes novel use of a certain kind of leakage-resilient extractors (known as cylinder intersection extractors), by combining them with a general type of extremal designs. Our key ingredient is the first derandomization of these designs, which we obtain using new connections to coding theory and additive combinatorics.

Improved presentation

We give significantly improved explicit constructions of three related pseudorandom objects.

1. Extremal designs: 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 $\lt s$. For all constants $r\geq s\in\mathbb{N}$ with $r$ even, we explicitly construct $(n,r,s)$-designs $(G_n)_{n\in\mathbb{N}}$ with independence number $\alpha(G_n)\leq O(n^{\frac{2(r-s)}{r}})$. This gives the first derandomization of a result by Rödl and Šinajová (Random Structures & Algorithms, 1994).

2. Extractors for adversarial sources: By combining our designs with leakage-resilient extractors (Chattopadhyay et al., FOCS 2020), we establish a new, simple framework for extracting from adversarial sources of locality $0$. As a result, we obtain significantly improved low-error extractors for these sources. For any constant $\delta>0$, we extract from $(N,K,n,$ polylog$(n))$-adversarial sources of locality $0$, given just $K\geq N^\delta$ good sources. The previous best result (Chattopadhyay et al., STOC 2020) required $K\geq N^{1/2+o(1)}$.

3. Extractors for small-space sources: Using a known reduction to adversarial sources, we immediately obtain improved low-error extractors for space $s$ sources over $n$ bits that require entropy $k\geq n^{1/2+\delta}\cdot s^{1/2-\delta}$, whereas the previous best result (Chattopadhyay et al., STOC 2020) required $k\geq n^{2/3+\delta}\cdot s^{1/3-\delta}$. On the other hand, using a new reduction from small-space sources to affine sources, we obtain near-optimal extractors for small-space sources in the polynomial error regime. Our extractors require just $k\geq s\cdot\log^Cn$ entropy for some constant $C$, which is an exponential improvement over the previous best result, which required $k\geq s^{1.1}\cdot2^{\log^{0.51}n}$ (Chattopadhyay and Li, STOC 2016).

Additional main theorem (small-space extractors for polylogarithmic entropy); other additional smaller results; improved presentation

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 less than $s$. An independent set in a hypergraph $G$ is a subset of vertices covering no hyperedge, and its independence number $\alpha(G)$ is the size of its largest independent set. For all constants $r\geq s\in\mathbb{N}$ with $r$ even, we explicitly construct $(n,r,s)$-designs $(G_n)_{n\in\mathbb{N}}$ with independence number $\alpha(G_n)\leq O(n^{\frac{2(r-s)}{r}})$. This gives the first derandomization of a result by Rödl and Šinajová (Random Structures & Algorithms, 1994).

By combining our designs with a recent explicit construction of a leakage-resilient extractor that works for low-entropy (Chattopadhyay et al., FOCS 2020), we obtain simple and significantly improved low-error explicit extractors for adversarial and small-space sources. In particular, for any constant $\delta>0$, 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 get extractors for small-space sources over $n$ bits with entropy requirement $k\geq n^{1/2+\delta}$, whereas the previous best result (Chattopadhyay et al., STOC 2020) required $k\geq n^{2/3+\delta}$.

Fixed abstract

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 get extractors for small-space sources over $n$ bits with entropy requirement $k\geq n^{1/2+\delta}$, whereas the previous best result (Chattopadhyay et al., STOC 2020) required $k\geq n^{2/3+\delta}$.

fixed abstract

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 get extractors for small-space sources over $n$ bits with entropy requirement $k\geq n^{1/2+\delta}$, whereas the previous best result (Chattopadhyay et al., STOC 2020) required $k\geq n^{2/3+\delta}$.

fixed abstract

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 get extractors for small-space sources over $n$ bits with entropy requirement $k\geq n^{1/2+\delta}$, whereas the previous best result (Chattopadhyay et al., STOC 2020) required $k\geq n^{2/3+\delta}$.