On the universality of the next bit test

Abstract

The next bit test 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 of how to test other properties of sources whose output distribution is not necessarily uniform. We prove the surprising result 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.