Tight exponential evaluation for information theoretical secrecy based on L1 distance

We adopt L1 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 universal2 hash function. It is also shown that this bound gives the tight exponential decreasing rate. The result is applied to wire-tap channel model and secret key distillation (agreement) by public discussion.