  We study the problem of estimating $k$-ary distributions under
$\varepsilon$-local differential privacy. $n$ samples are distributed across
users who send privatized versions of their sample to a central server. All
previously known sample optimal algorithms require linear (in $k$)
communication from each user in the high privacy regime $(\varepsilon=O(1))$,
and run in time that grows as $n\cdot k$, which can be prohibitive for large
domain size $k$.
  We propose Hadamard Response (HR}, a local privatization scheme that requires
no shared randomness and is symmetric with respect to the users. Our scheme has
order optimal sample complexity for all $\varepsilon$, a communication of at
most $\log k+2$ bits per user, and nearly linear running time of $\tilde{O}(n +
k)$.
  Our encoding and decoding are based on Hadamard matrices, and are simple to
implement. The statistical performance relies on the coding theoretic aspects
of Hadamard matrices, ie, the large Hamming distance between the rows. An
efficient implementation of the algorithm using the Fast Walsh-Hadamard
transform gives the computational gains.
  We compare our approach with Randomized Response (RR), RAPPOR, and
subset-selection mechanisms (SS), both theoretically, and experimentally. For
$k=10000$, our algorithm runs about 100x faster than SS, and RAPPOR.
