We consider discrete distribution estimation over $k$ elements under $eps$-local differential privacy from $n$ samples. The samples are distributed across users who send privatized versions of their sample to the server. All  previously known sample optimal algorithms require linear (in $k$) communication complexity in the high privacy regime $(eps<1)$, and have a running time that grows as $ncdot k$, which can be prohibitive for large domain size $k$. 

We study the task simultaneously under four resource constraints, privacy, sample complexity, computational complexity, and communication complexity. We propose emph{Hadamard Response (HR)}, a local non-interactive privatization mechanism with order optimal sample complexity (for all privacy regimes), a communication complexity of $log k+2$ bits, and runs in time $tilde{O}(n)$. 

Our encoding and decoding mechanisms are based on Hadamard matrices, and are simple to implement. The gain in sample complexity comes from the large Hamming distance between rows of Hadamard matrices, and the gain in time complexity is achieved by using the Fast Walsh-Hadamard transform. 

We compare our approach with Randomized Response (RR), RAPPOR, and subset-selection mechanisms (SS), theoretically, and experimentally.