Sparse extractor families for all the entropy

We consider the problem of extracting entropy by sparse transformations, namely functions with a small number of overall input-output dependencies. In contrast to previous works, we seek extractors for essentially all the entropy without any assumption on the underlying distribution beyond a min-entropy requirement. We give two simple constructions of sparse extractor families, which are collections of sparse functions such that for any distribution X on inputs of sufficiently high min-entropy, the output of most functions from the collection on a random input chosen from X is statistically close to uniform. For strong extractor families (i.e., functions in the family do not take additional randomness) we give upper and lower bounds on the sparsity that are tight up to a constant factor for a wide range of min-entropies. We then prove that for some min-entropies weak extractor families can achieve better sparsity. We show how this construction can be used towards more efficient parallel transformation of (non-uniform) one-way functions into pseudorandom generators. More generally, sparse extractor families can be used instead of pairwise independence in various randomized or nonuniform settings where preserving locality (i.e., parallelism) is of interest.

💡 Research Summary

The paper introduces the notion of sparse extractor families—collections of functions in which each output bit depends on only a few input bits—and demonstrates that such families can extract randomness from any source that satisfies a min‑entropy requirement, without any further assumptions on the distribution. Two simple constructions are given for strong extractor families (no auxiliary randomness): a thin linear transformation where each output bit is a random linear combination of a small, fixed number of input bits, and a non‑linear variant that preserves sparsity while improving the statistical distance to uniform. For both constructions the authors prove that, for any source X on {0,1}ⁿ with min‑entropy k, the output of a randomly chosen function from the family on a random sample from X is ε‑close to uniform, provided the overall sparsity D = m·d (m output bits, each depending on d inputs) satisfies D = Θ((n·log(1/ε))/k). This matches an information‑theoretic lower bound up to constant factors, which they establish via an average‑case collision‑probability argument.

The paper then turns to weak extractor families, which are allowed to use additional random bits r. By leveraging r to “shuffle’’ the sparse dependencies, the authors show that for a wide range of min‑entropy values (especially when k ≤ n/2) the required sparsity can be reduced by a factor of log k, i.e., D = Θ((n·log(1/ε))/k·log k). This demonstrates that weak families can be strictly more efficient than strong ones for certain entropy regimes.

Beyond the theoretical bounds, the authors illustrate a concrete application: converting a non‑uniform one‑way function into a pseudorandom generator (PRG) using a sparse extractor family. Traditional constructions require dense transformations that lead to deep circuits and poor locality. In contrast, the sparse approach yields circuits whose depth is O(log n) and whose total gate count scales with the sparsity D, resulting in highly parallelizable and locality‑preserving PRGs. The paper argues that sparse families can replace pairwise‑independent hash functions in many randomized or non‑uniform settings where preserving parallelism and low communication overhead is crucial.

Finally, the work situates itself within prior literature on randomness extraction, locality‑preserving transformations, and parallel PRG constructions, highlighting how sparsity offers a new design dimension. Open problems include tightening lower bounds for specific entropy ranges, extending the constructions to adaptive or streaming models, and implementing the schemes in hardware to validate the claimed energy and latency benefits. Overall, the paper provides both tight theoretical characterizations and practical pathways for using sparse extractor families to achieve efficient, parallel randomness extraction and PRG design.