Sparse extractor families for all the entropy
This work addresses entropy extraction for cryptography and randomized algorithms, offering incremental improvements in sparsity and parallelism.
The paper tackles the problem of extracting entropy using sparse transformations without distributional assumptions beyond min-entropy, achieving tight bounds on sparsity for strong extractor families and showing that weak families can be sparser for some min-entropies.
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.