Sampling of Min-Entropy Relative to Quantum Knowledge
Robert König, Renato Renner · IEEE Transactions on Information Theory · 2011
Let$X_1, \ldots, X_n$be a sequence of$n$classical random variables and consider a sample$X_{s_1}, \ldots, X_{s_r}$of$r \leq n$positions selected at random. Then, except with (exponentially in$r$) small probability, the min-entropy$H_{\min}(X_{s_1} \cdots X_{s_r})$of the sample is not smaller than, roughly, a fraction${r\over n}$of the overall entropy$H_{\min}(X_1 \cdots X_n)$, which is optimal. Here, we show that this statement, originally proved in [S. Vadhan, LNCS 2729, Springer, 2003] for the purely classical case, is still true if the min-entropy$H_{\min}$is measured relative to a quantum system. Because min-entropy quantifies the amount of randomness that can be extracted from a given random variable, our result can be used to prove the soundness of locally computable extractors in a context where side information might be quantum-mechanical. In particular, it implies that key agreement in the bounded-storage model—using a standard sample-and-hash protocol—is fully secure against quantum adversaries, thus solving a long-standing open problem.