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 e = e(n). For k ≤ n/logω(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/logO(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 e, and the possibility result supports e = 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.