We show that in device independent quantum key distribution protocols theprivacy of randomness is of crucial importance. For sublinear test sample sizeseven the slightest guessing probability by an eavesdropper will completelycompromise security. We show that a combined attack exploiting test sample andmeasurement choices compromises the security even with a linear size testsample and otherwise device independent security considerations. We explicitlyderive the sample size needed to retrieve security as a function of therandomness quality. We demonstrate that exploiting features of genuinely higherdimensional systems one can reduce this weakness and provide device independentsecurity more robust against weak randomness sources.
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