# Verifiable Threshold Multi-Party Fully Homomorphic Encryption from Share Resharing
## 6 Conclusions  
This work proposes a verifiable threshold multi-party fully homomorphic encryption (VTMFHE) scheme based on the LWE problem, addressing the high complexity and expense issues of existing TMFHE schemes in asynchronous settings, as well as their insufficient security against malicious adversaries. Our scheme features a t-out-of-N threshold access structure, making it more compact than multi-key FHE schemes, and utilizes Shamir secret sharing and share resharing techniques for key generation. Compared to traditional TMFHE schemes, our scheme only introduces additional interactions during the key generation, and allows any user set above the threshold \(t\) to jointly perform decryption without causing the smudging noise to grow too quickly. This balances user overhead and makes it more flexible for practical applications. Additionally, we introduced commitments and NIZK to construct an effective verification mechanism. This mechanism proves the correctness and validity of the actions of each party during the execution of the scheme, addressing the issue of TMFHE schemes being insecure against malicious adversaries. Finally, we demonstrated, through hybrid experiments, that the VTMFHE scheme is secure against malicious adversaries. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Scheme** & **Threshold Setting** & **Adversary Model** & **Public Key Size** & \begin{tabular}{c} **Evaluation** \\ **Key Size** \\ \end{tabular} & **Compactness** \\ \hline Mukherjee and & N-out-of-N & \begin{tabular}{c} Passive \\ semi-honest \\ adversary \\ Passive \\ \end{tabular} & 2 Nn(n+1)\lceil\log q\rceil^{2}\) & 0 & Non-compact \\ [PERSON] et al. [19] & t-out-of-N & \begin{tabular}{c} semi-honest \\ adversary \\ \end{tabular} & 2n(n+1)\lceil\log q\rceil^{2}\) & 0 & Partially compact \\ [PERSON] et al. [21] & t-out-of-N & \begin{tabular}{c} passive \\ semi-honest \\ adversary \\ \end{tabular} & 2n\lceil\log q\rceil\) & 6 Ln\(\lceil\log q\rceil^{2}\) & Compact \\ [PERSON] et al. [35] & N-out-of-N & \
\begin{tabular}{c} malicious \\ adversary \\ \end{tabular} & \((n+2)N\lceil\log q\rceil\) & 0 & Non-compact \\ Ours & t-out-of-N &
\begin{tabular}{c} Active \\ malicious \\ adversary \\ \end{tabular} & 2n(n+1)\lceil\log q\rceil^{2}\) & 0 & Compact \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Performance comparison of multi-user FHE schemes. **Outlook. ** The proposed VTMFHE scheme offers strong structural generality and security, making it particularly suitable for low-latency, communication-efficient environments, where multiple parties can efficiently perform joint homomorphic computation over ciphertexts. However, since each party must generate and transmit verification information (e.g., commitments and NIZK proofs), the scheme's performance may degrade in high-communication-cost settings or large-scale systems, as the communication overhead grows linearly with the number of parties N. Furthermore, the scheme's homomorphic evaluation capability is constrained by the circuit depth bound \(l\), which depends on the underlying LWE parameter settings, especially when handling complex computations. To address these limitations, future work will focus on optimizing the LWE parameters using techniques such as modulus-dimension switching and binary secret reduction, aiming to achieve smaller moduli and larger circuit depth bounds. Additionally, it is worth exploring the application of this scheme in broader cryptographic settings, such as constructing threshold signature schemes with active security to support joint signing and decentralized systems. Conceptualization, Y.X. and R.H.; methodology, Y.X. ; validation, Y.X., R.H. and J.Q. ; formal analysis, Y.X. ; investigation, Y.X. and J.Q. ; writing--original draft preparation, Y.X. ; writing--review and editing, Y.X. and R.H.; supervision, R.H.; funding acquisition, R.H. All authors have read and agreed to the published version of the manuscript. This research was funded by the Guangxi Key Research and Development Program Project under grant No. AB24010340.