  We study the mean estimation problem under communication and local
differential privacy constraints. While previous work has proposed
\emph{order}-optimal algorithms for the same problem (i.e., asymptotically
optimal as we spend more bits), \emph{exact} optimality (in the non-asymptotic
setting) still has not been achieved. In this work, we take a step towards
characterizing the \emph{exact}-optimal approach in the presence of shared
randomness (a random variable shared between the server and the user) and
identify several conditions for \emph{exact} optimality. We prove that one of
the conditions is to utilize a rotationally symmetric shared random codebook.
Based on this, we propose a randomization mechanism where the codebook is a
randomly rotated simplex -- satisfying the properties of the
\emph{exact}-optimal codebook. The proposed mechanism is based on a $k$-closest
encoding which we prove to be \emph{exact}-optimal for the randomly rotated
simplex codebook.
