# A Compact Multi-Identity Fully Homomorphic Encryption Scheme Without Fresh Ciphertexts
## 6 Conclusions  
In this study, we optimized the ABB10 scheme using the pre-image sampling algorithm based on YJW23, resulting in a foundational IBE scheme that satisfies IND-sID-CPA security. We modified the relationship and generation method of the identity vector and identity public key in our IBE scheme to meet the requirements of the decomposition method. Utilizing the decomposition method and the modified IBE scheme, we proposed a compact MIBFHE scheme. Comparisons with other MIBFHE schemes show that our proposal optimizes parameters such as lattice dimension, public and private key sizes, and noise growth rates, making it more concise and efficient. Overall, WZ-MIBFHE offers a practical solution in the field of multi-identity fully homomorphic encryption, demonstrating good security and scalability. In the future, we aim to extend the functionality of WZ-MIBFHE to include multi-hop attributes. Additionally, we are keenly aware of the practical application prospects of MIBFHE in areas like privacy-preserving cloud computing [29; 30; 31; 32]. We also plan to extend WZ-MIBFHE scheme to rings, leveraging the unique properties of rings to further enhance the efficiency of the MIBFHE scheme. To the best of our knowledge, no existing scheme has addressed the IBFHE and MIBFHE problems from the perspective of ring-based LWE.  
Conceptualization, Z.W. and R.H.; methodology, X.W.; software, Z.W.; validation, R.H., Z.W. and X.W.; formal analysis, Z.W.; investigation, Z.W.; resources, X.W.; data curation, Z.W.; writing--original draft preparation, Z.W.; writing--review and editing, Z.W.; visualization, Z.W.; supervision, Z.W.; project administration, X.W.; funding acquisition, R.H. All authors have read and agreed to the published version of the manuscript.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Scheme**} & \multirow{2}{*}{**Dimension**} & \multirow{2}{*}{**Secret Key Size**} & \multicolumn{2}{c}{**Expanded**} & \multicolumn{2}{c}{**Noise Expansion**} \\  & & & & **Ciphertext Size** & **Rate** \\ \hline
[11] & \(6n\text{log}\,q\) & \({Dm^{\prime}}^{2}\) & \({D^{2}{m^{\prime}}^{2}\omega^{2}}\) & \(1+n\omega\) \\
[20] & \(2n\text{log}\,q\) & \({Dm^{\prime}}^{2}\) & \({D^{2}{m^{\prime}}^{2}\omega^{2}}\) & \(1+7n\omega\) \\
[22] & \(2n\text{log}\,q\) & \({Dm^{\prime}}\) & \({D^{2}{m^{\prime}}^{2}\omega}\) & \((1+2n)+3n(1+2n\omega)^{3}\) \\ WZ-MIBFHE & \(n\text{log}\,q+\text{log}_{b}\,q\) & \({Dm^{\prime}}\) & \({D^{2}{m^{\prime}}^{2}\omega}\) & \(D\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Comparison of Dimension, Secret Key Size, Ciphertext Size, and Noise Expansion Rate in Similar Schemes.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Scheme**} & \multirow{2}{*}{**Trapdoor**} & \multirow{2}{*}{**IBE Architecture**} & \multicolumn{2}{c}{**Ciphertext Extension**} & \multicolumn{1}{c}{**Must Fresh**} & \multicolumn{1}{c}{**Multi-Hop**} \\  & & & **Method** & **Ciphertext** & **Support** \\ \hline
[11] & GPV08 & Dual Regev & Mask system & Yes & No \\
[20] & MP12 & Dual Regev & Mask system & Yes & Yes \\
[22] & MP12 & ABB10 & Mask system & Yes & No \\ WZ-MIBFHE & YJW23 & Our modified IBE & The decomposition method & No & No \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of Trapdoor, IBE Architecture, Ciphertext Extension Method, Fresh Ciphertext Requirement, and Multi-hop Support in Similar Schemes.  
**Funding:** This work was supported in part by the National Natural Science Foundation Project of China under Grant No. 62062009 and The Guangxi Key Research and Development Program Project No. AB24010340.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The original contributions presented in this study are included in the article.  
**Conflicts of Interest:** The authors declare no conflicts of interest.