On randomness extraction in AC0

We consider randomness extraction by AC⁰ circuits. The main parameter, n, is the length of the source, and all other parameters are functions of it. The additional extraction parameters are the min-entropy bound k = k(n), the seed length r = r(n), the output length m = m(n), and the (output) deviation bound ε = ε(n). For k ≤ n/log^ω(1) n, we show that AC⁰-extraction is possible if and only if m/r ≤ 1+poly(log n)·k/n; that is, the extraction rate m/r exceeds the trivial rate (of one) by an additive amount that is proportional to the min-entropy rate k/n. In particular, non-trivial AC⁰-extraction (i.e., m ≥ r + 1) is possible if and only if k · r > n/poly(log n). For k ≥ n/log^O(1) n, we show that AC⁰-extraction of r + Ω(r) bits is possible when r = O(log n), but leave open the question of whether more bits can be extracted in this case. The impossibility result is for constant ε, and the possibility result supports ε = 1/poly(n). The impossibility result is for (possibly) non-uniform AC⁰, whereas the possibility result hold for uniform AC⁰. All our impossibility results hold even for the model of bit-fixing sources, where k coincides with the number of non-fixed (i.e., random) bits. We also consider deterministic AC⁰ extraction from various classes of restricted sources. In particular, for any constant δ > 0, we give explicit AC⁰ extractors for poly(1/δ) independent sources that are each of min-entropy rate δ; and four sources suffice for δ = 0.99. Also, we give non-explicit AC⁰ extractors for bit-fixing sources of entropy rate 1/poly(log n) (i.e., having n/poly(log n) unfixed bits). This shows that the known analysis of the "restriction method" (for making a circuit constant by fixing as few variables as possible) is tight for AC⁰ even if the restriction is picked deterministically depending on the circuit.