Tight exponential evaluation for information theoretical secrecy based on universal composablity

We adopt L_1 distance as Eve's distinguishability in secret key generation from a common random number 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 universal_2 hash function. It is also shown that this bound gives the tight exponential rate of decrease in the independent and identical distribution. The result is applied to wire-tap channel model and secret key distillation (agreement) by public discussion.