Large Deviation Analysis for Quantum Security via Smoothing of Rényi Entropy of Order 2

It is known that the security evaluation can be done by smoothing of Rényi entropy of order 2 in the classical and quantum settings when we apply universal
\(_{2}\)
hash functions. Using the smoothing of Rényi entropy of order 2, we derive security bounds for
\(L_{1}\)
distinguishability and modified mutual information criterion under the classical and quantum setting, and have derived these exponential decreasing rates. These results are extended to the case when we apply
\(\varepsilon \)
-almost dual universal
\(_{2}\)
hash functions. Furthermore, we apply this analysis to the secret key generation with error correction.