  Locally Differentially Private (LDP) Reports are commonly used for collection
of statistics and machine learning in the federated setting. In many cases the
best known LDP algorithms require sending prohibitively large messages from the
client device to the server (such as when constructing histograms over large
domain or learning a high-dimensional model). This has led to significant
efforts on reducing the communication cost of LDP algorithms.
  At the same time LDP reports are known to have relatively little information
about the user's data due to randomization. Several schemes are known that
exploit this fact to design low-communication versions of LDP algorithm but all
of them do so at the expense of a significant loss in utility. Here we
demonstrate a general approach that, under standard cryptographic assumptions,
compresses every efficient LDP algorithm with negligible loss in privacy and
utility guarantees. The practical implication of our result is that in typical
applications the message can be compressed to the size of the server's
pseudo-random generator seed. More generally, we relate the properties of an
LDP randomizer to the power of a pseudo-random generator that suffices for
compressing the LDP randomizer. From this general approach we derive
low-communication algorithms for the problems of frequency estimation and
high-dimensional mean estimation. Our algorithms are simpler and more accurate
than existing low-communication LDP algorithms for these well-studied problems.
