# Practical quantum somewhat-homomorphic encryption with coherent states. (arXiv:1710.03968v1 [quant-ph])

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.