  We propose the first method that realizes the Laplace mechanism exactly
(i.e., a Laplace noise is added to the data) that requires only a finite amount
of communication (whereas the original Laplace mechanism requires the
transmission of a real number) while guaranteeing privacy against the server
and database. Our mechanism can serve as a drop-in replacement for local or
centralized differential privacy applications where the Laplace mechanism is
used. Our mechanism is constructed using a random quantization technique.
Unlike the simple and prevalent Laplace-mechanism-then-quantize approach, the
quantization in our mechanism does not result in any distortion or degradation
of utility. Unlike existing dithered quantization and channel simulation
schemes for simulating additive Laplacian noise, our mechanism guarantees
privacy not only against the database and downstream, but also against the
honest but curious server which attempts to decode the data using the dither
signals.
