# Verifiable Threshold Multi-Party Fully Homomorphic Encryption from Share Resharing
## 5 Correctness, Security, and Performance Analysis
### Security
Instead of honestly computing \(p_{i}\leftarrow\mathrm{T.PartDec}(sk_{i},\hat{c}t)\), the simulator uniformly samples \(p^{\prime}_{i}\stackrel{{ R}}{{\leftarrow}}\mathbb{Z}_{q}\) under the condition that the final decryption result remains unchanged, i.e., \(\sum_{i\in\mathcal{P}\setminus\mathcal{C}}p^{\prime}_{i}=\mu-\sum_{j\in \mathcal{C}}p_{j}\). Since \(p^{\prime}_{i}\) is randomly sampled from a uniform distribution, according to the definition of the LWE problem, \(p^{\prime}_{i}\) is indistinguishable from \(p_{i}=\left\langle\hat{c}t,sk_{i}\right\rangle+\epsilon^{im}_{i}\), and the decryption result is not affected by the introduced randomness. Therefore, \(Game_{3}\) and \(Game_{4}\) are indistinguishable. _Game\({}_{5}\)_: Based on \(Game_{4}\), all partial decryption results of honest parties no longer depend on the secret key shares \(sk_{i}\). Therefore, we replace the zero-knowledge proofs \(\pi_{\mathbf{b}_{i}}\) generated by honest parties during the VDKeyGen process with proofs generated by the simulator \(Sim\). Similarly to \(Game_{1}\), the proofs generated by the simulator are computationally indistinguishable from those generated by an honest prover. Therefore, the experiments \(Game_{4}\) and \(Game_{5}\) are indistinguishable. _Game\({}_{6}\)_: Similarly to \(Game_{2}\), we replace the commitments in the VDKeyGen protocol that do not need to be opened, such as \(com_{s_{i}}\). Since we have already replaced the proofs in the key generation process in \(Game_{5}\), replacing these commitments will not affect the protocol. According to the commitment hiding property defined in Definition 14, the adversary \(\mathcal{A}\) cannot obtain any information about the committed values from the commitments themselves. Therefore, the experiments \(Game_{5}\) and \(Game_{6}\) are indistinguishable. We now prove that \(Game_{6}\) is indistinguishable from the ideal world simulated by \(Sim\). If the adversary \(\mathcal{A}\) wins \(Game_{6}\), then \(\mathcal{A}\) can break the LWE problem, i.e., it can effectively solve the hard lattice problems SVP and GapSVP. Ultimately, \(Game_{6}\) is clearly secure, because all components are generated by the simulator \(Sim\) and no private information is leaked. By proving step by step the indistinguishability between each pair of adjacent experiments, we can conclude that the initial experiment and the final experiment are statistically indistinguishable, thereby proving the security of the VTMFHE scheme against active malicious adversaries. This concludes the proof. We observe that most existing TMFHE schemes still rely on the semi-honest adversary model for their security guarantees. This assumption often lacks robustness against malicious adversaries or execution faults, where collusion, input forgery, or deliberately incorrect partial decryption by corrupted parties can threaten the data security of honest participants, particularly in the key sharing and threshold decryption processes. Moreover, traditional mechanisms generally lack systematic verification structures. Compared with representative TMFHE schemes, such as those by [PERSON] et al. [27], [PERSON] et al. [25], and [PERSON] et al. [21], our proposed VTMFHE scheme achieves multi-dimensional improvements in security. In particular, we construct a verifiable framework for key generation and threshold decryption that addresses the need to prove complex structural constraints and maintain secret consistency across protocol phases, all without relying on a trusted third party.