  We study the problem of estimating non-linear functionals of discrete
distributions in the context of local differential privacy. The initial data
$x_1,\ldots,x_n \in [K]$ are supposed i.i.d. and distributed according to an
unknown discrete distribution $p = (p_1,\ldots,p_K)$. Only $\alpha$-locally
differentially private (LDP) samples $z_1,...,z_n$ are publicly available,
where the term 'local' means that each $z_i$ is produced using one individual
attribute $x_i$. We exhibit privacy mechanisms (PM) that are interactive (i.e.
they are allowed to use already published confidential data) or
non-interactive. We describe the behavior of the quadratic risk for estimating
the power sum functional $F_{\gamma} = \sum_{k=1}^K p_k^{\gamma}$, $\gamma >0$
as a function of $K, \, n$ and $\alpha$. In the non-interactive case, we study
two plug-in type estimators of $F_{\gamma}$, for all $\gamma >0$, that are
similar to the MLE analyzed by Jiao et al. (2017) in the multinomial model.
However, due to the privacy constraint the rates we attain are slower and
similar to those obtained in the Gaussian model by Collier et al. (2020). In
the interactive case, we introduce for all $\gamma >1$ a two-step procedure
which attains the faster parametric rate $(n \alpha^2)^{-1/2}$ when $\gamma
\geq 2$. We give lower bounds results over all $\alpha$-LDP mechanisms and all
estimators using the private samples.
