  The Differential Privacy (DP) literature often centers on meeting privacy
constraints by introducing noise to the query, typically using a pre-specified
parametric distribution model with one or two degrees of freedom. However, this
emphasis tends to neglect the crucial considerations of response accuracy and
utility, especially in the context of categorical or discrete numerical
database queries, where the parameters defining the noise distribution are
finite and could be chosen optimally. This paper addresses this gap by
introducing a novel framework for designing an optimal noise Probability Mass
Function (PMF) tailored to discrete and finite query sets. Our approach
considers the modulo summation of random noise as the DP mechanism, aiming to
present a tractable solution that not only satisfies privacy constraints but
also minimizes query distortion. Unlike existing approaches focused solely on
meeting privacy constraints, our framework seeks to optimize the noise
distribution under an arbitrary $(\epsilon, \delta)$ constraint, thereby
enhancing the accuracy and utility of the response. We demonstrate that the
optimal PMF can be obtained through solving a Mixed-Integer Linear Program
(MILP). Additionally, closed-form solutions for the optimal PMF are provided,
minimizing the probability of error for two specific cases. Numerical
experiments highlight the superior performance of our proposed optimal
mechanisms compared to state-of-the-art methods. This paper contributes to the
DP literature by presenting a clear and systematic approach to designing noise
mechanisms that not only satisfy privacy requirements but also optimize query
distortion. The framework introduced here opens avenues for improved
privacy-preserving database queries, offering significant enhancements in
response accuracy and utility.
