# Smooth Entropy and {R}{é}nyi Entropy

## Christian Cachin

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```The notion of smooth entropy allows a unifying, generalized formulation of
privacy amplification and entropy smoothing. Smooth entropy is a measure for
the number of almost uniform random bits that can be extracted from a random
source by probabilistic algorithms. It is known that Rényi entropy of order
at least 2 of a random variable is a lower bound for its smooth entropy. On
the other hand, an assumption about Shannon entropy (which is Rényi entropy
of order 1) is too weak to guarantee any non-trivial amount of smooth entropy.
In this work, we close the gap between Rényi entropy of order 1 and 2. In
particular, we show that Rényi entropy of order $\alpha$ for any
$1<\alpha<2$ is a lower bound for smooth entropy, up to a small parameter
depending on $\alpha$, the alphabet size and the failure probability. The
results have applications in cryptography for unconditionally secure
protocols such as quantum key agreement, key agreement from correlated
information, oblivious transfer, and bit commitment.