Quantum Fully Homomorphic Encryption by Integrating Pauli One-time Pad with Quaternions.

Quantum fully homomorphic encryption (QFHE) allows to evaluate quantum circuits on encrypted data. We present a novel QFHE scheme, which extends Pauli one-time pad encryption by relying on the quaternion representation of SU(2) to make more efficient evaluation of 1-qubit gates. By a new technique called encrypted-CROT, we establish a conversion between this encryption and previous Pauli one-time pad encryption, allowing to evaluate 1-qubit gates on our encryption, and evaluate multi-bit Clifford gates on Pauli one-time pad encryption. This makes our scheme compatible and complementary to previous QFHE schemes, thus leading to better evaluation of quantum circuits by taking advantage of both schemes. Our QFHE scheme achieves a high level of private quantum circuits evaluation. The security of the scheme relies on the hardness of the underlying quantum capable FHE scheme, and the latter sets its security on the learning with errors problem and the circular security assumption.