Randomness-efficient oblivious sampling

We introduce a natural notion of obliviousness of a sampling procedure, and construct a randomness-efficient oblivious sampler. Our sampler uses O(l+log δ^-1·log l) coins to output m=poly(ε^-1, log δ^-1, log l) sample points x₁, …, x_m, ∈ {0, 1}¹ such that Pr[|1/mΣ_i=1^mf(x_i )-E[f]|<ε]⩾1-δ for any function f: {0, 1}¹ →[0, 1]. We apply this sampling procedure to reduce the randomness required to halve the number of rounds of interaction in an Arthur Merlin proof system. Given a 2g(n) round AM proof for L in which Arthur sends l(n) coins per round and Merlin responds with a q(n) bit string, we construct a g(n) round AM proof for L in which Arthur sends O(l+(q+log g)·log l) coins per round and Merlin responds with a poly(n) bit string