Coin Flipping in Dynamic Programming Is Almost Useless

Collective Coin Flipping

Optimal Coin Flipping

This paper studies the problem of simulating a coin of arbitrary real bias q with a coin of arbitrary real bias p with minimum loss of entropy. We establish a lower bound that is strictly greater than the information-theoretic bound. We show that as a function of q, it is an everywhere-discontinuous selfsimilar fractal. We provide efficient protocols that achieve the lower bound to within any d...

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 prob...

Robust Coin Flipping

Alice seeks an information-theoretically secure source of private random data. Unfortunately, she lacks a personal source and must use remote sources controlled by other parties. Alice wants to simulate a coin flip of specified bias α, as a function of data she receives from p sources; she seeks privacy from any coalition of r of them. We show: If p/2 ≤ r < p, the bias can be any rational numbe...

Adaptively Secure Coin-Flipping, Revisited

The full-information model was introduced by Ben-Or and Linial in 1985 to study collective coin-flipping: the problem of generating a common bounded-bias bit in a network of n players with t = t(n) faults. They showed that the majority protocol, in which each player sends a random bit and the output is the majority of the players’ bits, can tolerate t(n) = O( √ n) even in the presence of adapti...