  We consider a problem of analyzing a global property of private data through
randomized responses subject to a certain rule, where private data are used for
another cryptographic protocol, e.g., authentication. For this problem, the
security of private data was evaluated by a universally composable security
measure, which can be regarded as $(0,\delta)$-differential privacy. Here we
focus on the trade-off between the global accuracy and a universally composable
security measure, and derive an optimal solution to the trade-off problem. More
precisely, we adopt the Fisher information of a certain distribution family as
the estimation accuracy of a global property and impose
$(0,\delta)$-differential privacy on a randomization mechanism protecting
private data. Finally, we maximize the Fisher information under the
$(0,\delta)$-differential privacy constraint and obtain an optimal mechanism
explicitly.
