  We find separation rates for testing multinomial or more general discrete
distributions under the constraint of local differential privacy. We construct
efficient randomized algorithms and test procedures, in both the case where
only non-interactive privacy mechanisms are allowed and also in the case where
all sequentially interactive privacy mechanisms are allowed. The separation
rates are faster in the latter case. We prove general information theoretical
bounds that allow us to establish the optimality of our algorithms among all
pairs of privacy mechanisms and test procedures, in most usual cases.
Considered examples include testing uniform, polynomially and exponentially
decreasing distributions.
