Coin flipping from a cosmic source: On error correction of truly random bits

We study a problem related to coin flipping, coding theory, and noise sensitivity. Consider a source of truly random bits x ∈ {0, 1}n, and k parties, who have noisy version of the source bits yi ∈ {0, 1}n, when for all i and j, it holds that P[ymath image = xj] = 1 − ϵ, independently for all i and j. That is, each party sees each bit correctly with probability 1 − ϵ, and incorrectly (flipped) with probability ϵ, independently for all bits and all parties. The parties, who cannot communicate, wish to agree beforehand on balanced functions fi: {0, 1}n → {0, 1} such that P[f1(y1) = … = fk(yk)] is maximized. In other words, each party wants to toss a fair coin so that the probability that all parties have the same coin is maximized. The function fi may be thought of as an error correcting procedure for the source x. When k = 2,3, no error correction is possible, as the optimal protocol is given by fi(yi) = ymath image. On the other hand, for large values of k, better protocols exist. We study general properties of the optimal protocols and the asymptotic behavior of the problem with respect to k, n, and ϵ. Our analysis uses tools from probability, discrete Fourier analysis, convexity, and discrete symmetrization.