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 ans quantum settings when we apply universal$_2$ hash functions. This fact can be extended to the case when we apply $\epsilon$-almost dual universal$_2$ hash functions. We approximately demonstrate the smoothing of R\'{e}nyi entropy of order 2, and derived security bounds for universal composability and mutual information criterion under the condition in the classical and quantum setting. Further, we apply this analysis to the secret key generation with error correction.