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Randomness extractors, widely used in classical and quantum cryptography andother fields of computer science, e.g., derandomization, are functions whichgenerate almost uniform randomness from weak sources of randomness. In thequantum setting one must take into account the quantum side information held byan adversary which might be used to break the security of the extractor. In thecase of seeded extractors the presence of quantum side information has beenextensively studied. For multi-source extractors one can easily see that highconditional min-entropy is not sufficient to guarantee security againstarbitrary side information, even in the classical case. Hence, the interestingquestion is under which models of (both quantum and classical) side informationmulti-source extractors remain secure. In this work we suggest a natural modelof side information, which we call the Markov model, and prove that anymulti-source extractor remains secure in the presence of quantum sideinformation of this type (albeit with weaker parameters). This improves onprevious results in which more restricted models were considered and thesecurity of only some types of extractors was shown.

Quantum-proof multi-source randomness extractors in the Markov model