  We present a scheme for implementing homomorphic encryption on coherent
states encoded using phase-shift keys. The encryption operations require only
rotations in phase space, which commute with computations in the codespace
performed via passive linear optics, and with generalized non-linear phase
operations that are polynomials of the photon-number operator in the codespace.
This encoding scheme can thus be applied to any computation with coherent state
inputs, and the computation proceeds via a combination of passive linear optics
and generalized non-linear phase operations. An example of such a computation
is matrix multiplication, whereby a vector representing coherent state
amplitudes is multiplied by a matrix representing a linear optics network,
yielding a new vector of coherent state amplitudes. By finding an orthogonal
partitioning of the support of our encoded states, we quantify the security of
our scheme via the indistinguishability of the encrypted codewords. Whilst we
focus on coherent state encodings, we expect that this phase-key encoding
technique could apply to any continuous-variable computation scheme where the
phase-shift operator commutes with the computation.
