Universal tests for nonuniform distributions

The next bit test as introduced by Blum and Micali was shown by Yao to be a universal test for sources of unbiased independent bits. The aim of this paper is to provide a rigorous methodology for testing sources whose output distributions are not necessarily uniform. We first show that the natural extension of the next bit test, even in the simplest case of biased independent bits, is no longer universal: we construct a source of biased bits, whose bits are obviously dependent and yet none of these bits can be predicted with probability of success greater than the bias. To overcome this difficulty, we develop new universal tests for arbitrary models of (potentially imperfect) sources of randomness. These new tools contribute to the theoretical as well as practical study of sources of randomness.