Randomness condensers for efficiently samplable, seed-dependent sources

We initiate a study of randomness condensers for sources that are efficiently samplable but may depend on the seed of the condenser. That is, we seek functions Cond : {0,1} ⁿ × {0,1} ^d → {0,1} ^m such that if we choose a random seed S ← {0,1} ^d, and a source is generated by a randomized circuit of size t such that X has min-entropy at least k given S, then Cond(X;S) should have min-entropy at least some k′ given S. The distinction from the standard notion of randomness condensers is that the source X may be correlated with the seed S (but is restricted to be efficiently samplable). Randomness extractors of this type (corresponding to the special case where k′ = m) have been implicitly studied in the past (by Trevisan and Vadhan, FOCS '00). We show that: Unlike extractors, we can have randomness condensers for samplable, seed-dependent sources whose computational complexity is smaller than the size t of the adversarial sampling algorithm . Indeed, we show that sufficiently strong collision-resistant hash functions are seed-dependent condensers that produce outputs with min-entropy , i.e. logarithmic entropy deficiency. Randomness condensers suffice for key derivation in many cryptographic applications: when an adversary has negligible success probability (or negligible "squared advantage" [3]) for a uniformly random key, we can use instead a key generated by a condenser whose output has logarithmic entropy deficiency. Randomness condensers for seed-dependent samplable sources that are robust to side information generated by the sampling algorithm imply soundness of the Fiat-Shamir Heuristic when applied to any constant-round, public-coin interactive proof system.