Certifiable quantum dice: Or, true random number generation secure against quantum adversaries

We introduce a protocol through which a pair of quantum mechanical devices may be used to generate n bits that are ε-close in statistical distance from n uniformly distributed bits, starting from a seed of O(log n log 1/ε) uniform bits. The bits generated are certifiably random based only on a simple statistical test that can be performed by the user, and on the assumption that the devices do not communicate in the middle of each phase of the protocol. No other assumptions are placed on the devices' inner workings. A modified protocol uses a seed of O(log ³ n) uniformly random bits to generate n bits that are poly ^-1(n)-indistinguishable from uniform even from the point of view of a quantum adversary who may have had prior access to the devices, and may be entangled with them.