Quantum homomorphic encryption scheme with flexible number of evaluator based on (k, n)-threshold quantum state sharing

Quantum homomorphic encryption provides the ability to perform calculations on encrypted data without decrypting them. The number of evaluators in most previous schemes is 1, and in other schemes it is n. In this paper, we propose a novel quantum homomorphic encryption scheme with flexible number of evaluator. One of our main contributions is that the number of evaluators is d (k ⩽ d ⩽ n), here n ≥ 2. These evaluators are allowed to alternately complete the evaluation of single-qubit unitary operations on the shared encrypted sequence. Other work is that for n = 1, we also give the quantum homomorphic encryption scheme that one evaluator can complete the evaluation. Then, for (3, 5)-case and (2, 2)-case, we give two examples (i.e. example-I and example-II) to illustrate the flexibility of evaluators. Finally, it is shown that the scheme is secure from analyzing the private key, plaintext sequence and encrypted sequence.