Large deviation analysis for classical and quantum security via approximate smoothing

It is known that the security evaluation can be done by smoothing of R\'{e}nyi entropy of order 2 in the classical and quantum settings when we apply universal$_2$ hash functions. Using the smoothing of R\'{e}nyi entropy of order 2, we have derived security bounds for universal composability and modified mutual information criterion under the classical and quantum setting, and have derived these exponential decreasing rates. These results have been extended to the case when we apply $\epsilon$-almost dual universal$_2$ hash functions. Further, we have applied this analysis to the secret key generation with error correction.