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 et al(1995)'s bound based on the Rényi entropy of order 2 because the bound obtained here uses the Rényi entropy of order $1+s$ for $s \in [0,1]$. 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(2006)'s exponent. For the additive case, the bound obtained here is better. The result is applied to secret key agreement by public discussion.