Quantum homomorphic encryption (QHE) is a key technology in the field of cryptography, allowing remote servers to perform quantum computations on encrypted data without decrypting it. This paper proposes a two-party computing module based on quantum remote control, denoted as \(RC[U_{\varphi }]\) . This module supports clients in implementing non-interactive rotation gate operations on server-side quantum states. By constructing a quantum operation framework composed of Clifford and T gates, the universality of homomorphic computation is achieved. To address the S error problem generated during the T gate homomorphic evaluation process in quantum homomorphic encryption, this study designs a novel module \(RC[S^x]\) based on the new module, establishing a complete T gate quantum homomorphic evaluation mechanism. Compared with existing T gate homomorphic evaluation schemes, this scheme has two major innovations during the evaluation phase: (1) there is no need for the server to transmit quantum states to the client, and (2) the dependence on classical homomorphic encryption technology is eliminated. This significantly reduces communication costs and ensures information-theoretic security. To counter malicious server attacks, this paper innovatively proposes a publicly verifiable quantum homomorphic encryption (vQHE) scheme based on permutation technology. By implementing position permutation of computing circuits and verification circuits on the client side, this scheme achieves the public verifiability of quantum homomorphic encryption for the first time. Theoretical analysis shows that the proposed vQHE scheme also has the characteristics of F-homomorphism, quasi-compactness, information-theoretic security, and non-interactivity. Finally, this paper provides a comprehensive analysis and demonstration of the correctness, security, compactness, and public verifiability of the scheme.
