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The work concerns the problem of finding a protocol for randomness amplification secure against non-signaling adversary with polynomial number of devices, which allows for correlations between
the device and the source of weak randomness. We focus on the epsilon-Santha-Vazirani sources, and provide two results in this direction.
First we revisit the seminal protocol of R. Colbeck and R. Renner (CR protocol) of randomness amplification using Santha-Vazirani (SV) sources, and prove its security relaxing partially assumptions of independence between the devices and the source at a price of narrowed range of epsilon. The relaxation allows that the SV source can indicate as a final device from which randomness is taken choosen with uniform probability from the insecure devices. The proof of relaxation bases on the assumption which is a generalization of Santha-Vazirani condition - the SV condition for boxes: there does not exist a device such that given its inputs and outputs one can get to know the value of SV source by more than epsilon.
Second, we prove security of the CR protocol allowing arbitrary correlations between SV source and device, up to the mentioned SV-box condition, and the assumption that the devices are not correlated with each other. We prove that
if the final device chosen in the protocol was not secure, an independent tester could guess the value of SV source bits more than the SV-box condition allows. The strategy of a tester is to choose a random device out of the ones which do not satisfy condition of the Chain Bell inequality. The idea of the proof of the second result indicates that the CR protocol may be secure under attack which arbitrarily correlates SV with devices of arbitrary type, and is promising in studying this problem.
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