The problem of preserving the privacy of individual entries of a database with constrained additive noise is considered. An adversary can submit linear queries to an agent possessing the entire database. The agent returns a response to the query that is corrupted by an additive random noise whose support is a subset or equal to a constraint set. The Cramer-Rao bound is used to bound the variance of the estimation error of the database, which may be used by the adversary, to the trace of the inverse of the Fisher information matrix. A measure of privacy using the Fisher information matrix is developed. The probability density that minimizes the Fisher information (as a proxy for maximizing the measure of privacy) is computed.