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Generalized Strong Extractors and Deterministic Privacy Amplification

Robert Koenig and Ueli Maurer

Extracting essentially uniform randomness from a somewhat random source $X$ is a crucial operation in various applications, in particular in cryptography where an adversary usually possesses some partial information about $X$. In this paper we formalize and study the most general form of extracting randomness in such a cryptographic setting. Our notion of strong extractors captures in particular the case where the catalyst randomness is neither uniform nor independent of the actual extractor input. This is for example important for privacy amplification, where a uniform cryptographic key is generated by Alice and Bob sharing some partially secret information $X$ by exchanging a catalyst $R$ over an insecure channel accessible to an adversary Eve. Here the authentication information for $R$ creates, from Eve's viewpoint, a dependence between $X$ and $R$. We provide explicit constructions for this setting based on strong blenders. In addition, we give strong deterministic randomness extractors for lists of random variables, where only an unknown subset of the variables is required to have some amount of min-entropy.