Robust Coin Flipping

Alice has p indistinguishable, finitary random oracles, but she knows she can trust only q of them. She wants to simulate a biased coin flip as a function of the oracles' outputs. We prove: If 1 <= q <= p/2, the bias can be any rational number and nothing else; if p/2 < q < p, the bias can be any algebraic number and nothing else. The proof uses projective varieties, convex geometry, and the probabilistic method. Our results improve on those laid out by Yao, who asserts one direction of the q=p-1 case in his seminal paper [Yao82].