  We show that in device independent quantum key distribution protocols the
privacy of randomness is of crucial importance. For sublinear test sample sizes
even the slightest guessing probability by an eavesdropper will completely
compromise security. We show that a combined attack exploiting test sample and
measurement choices compromises the security even with a linear size test
sample and otherwise device independent security considerations. We explicitly
derive the sample size needed to retrieve security as a function of the
randomness quality. We demonstrate that exploiting features of genuinely higher
dimensional systems one can reduce this weakness and provide device independent
security more robust against weak randomness sources.
