Randomness in interactive proofs

The quantitative aspects of randomness in interactive proof systems are studied. The result is a randomness-efficient error-reduction technique: given an Arthur-Merlin proof system (error probability ⩽1/3) in which Arthur sends l=l(n ) random bits per rounds, a proof system that achieves error probability 2^-k at the cost of Arthur sending only 2l+O(k) random bits per round is constructed. The method maintains the number of rounds in the game. Underlying the transformation is a novel sampling method for approximating the average value of an arbitrary function f: {0,1}^l→[0,1]. The method evaluates the function on O(ε^-2 log δ^-1) sample points generated using only 2l+O(log δ^-1) coin tosses to get an estimate which with probability ⩾1-δ is within ε of the average value of the function