Tight exponential evaluation for universal composablity with privacy amplification and its applications

We adopt the universal composablity as Eve's distinguishability in secret key generation from a common random number between two distinct players without communication. Under this secrecy criterion, using the Renyi entropy of order 1+s for s in [0,1], we derive a new upper bound of Eve's distinguishability under the application of the universal2 hash functions. It is also shown that this bound gives the tight exponential rate of decrease in the case of independent and identical distributions. The result is applied to the wire-tap channel model and to secret key generation (distillation) by public discussion.