Exponential Decreasing Rate of Leaked Information in Universal Random Privacy Amplification

We derive a new upper bound for Eve´s information in secret key generation from a common random number without communication. This bound improves on Bennett ´s bound based on the Rényi entropy of order 2 because the bound obtained here uses the Rényi entropy of order 1+s for s ∈ [0,1]. This bound is applied to a wire-tap channel. Then, we derive an exponential upper bound for Eve´s information. Our exponent is compared with Hayashi ´s exponent. For the additive case, the bound obtained here is better. The result is applied to secret key agreement by public discussion.