Practical quantum somewhat-homomorphic encryption with coherent states

TL;DR

This paper introduces a practical quantum homomorphic encryption scheme using coherent states and phase-shift encoding, enabling secure, linear optics-based computations like matrix multiplication with quantifiable security.

Contribution

It presents a novel encryption scheme compatible with passive linear optics and non-linear phase operations, expanding quantum homomorphic encryption to continuous-variable systems.

Findings

Encryption operations are simple phase rotations in phase space.

The scheme supports computations like matrix multiplication.

Security is quantified via indistinguishability of encrypted states.

Abstract

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.