## 4 Scheme Construction
; validation, D.P. and Y.T. ; formal analysis, D.P. and L.L. ; writing--original draft preparation, D.P. ; writing--review and editing, Y.T. and P.Q. ; supervision, Y.T. ; funding acquisition, Y.T. and P.Q. All authors have read and agreed to the published version of the manuscript. Figure 5: The GS-MR scheme verification parameter size. [PERSON] [24], [PERSON] [27], [PERSON] [19]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Scheme**} & \multicolumn{4}{c}{**Size of Verification Parameter (KB)**} \\  & \(N\) **= 128** & \(N\) **= 256** & \(N\) **= 512** & \(N\) **= 1024** & \(N\) **= 2048** \\ \hline Katsumata [24] & 741.80 & 1477.75 & 2949.75 & 5893.75 & 11781.70 \\ Canard [27] & 3033.75 & 3033.75 & 3033.75 & 3033.75 & 3033.75 \\ [PERSON] [19] & 693.10 & 1033.10 & 1513.70 & 2673.70 & 3373.70 \\ Ours & 293.75 & 521.75 & 977.75 & 1889.75 & 3649.75 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of the size of verification parameter. **Funding:** This research was partially supported by the Support Plan of Scientific and Technological Innovation Team in Universities of Henan Province (20 IRTSTHN013) and the Henan Province Key R&D and Promotion Special Project (No.212102210166). **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** Not applicable. **Conflicts of Interest:** The authors declare no conflict of interest. **Glossary: Symbol Definitions**

**Notations**: **Explanation**

\(\mathcal{R}_{q}\): Polynomial ring \(\mathcal{R}_{q}=\mathbb{Z}_{q}[x]/(x^{n}+1)\)

\(\mathcal{A}\): Gram-Schmidt orthogonalization of matrix \(\mathbf{A}\)

\(|x|\): Bit length when identifying \(x\) with binary

\(|x|^{n}\): Takes \(n\) bits from the high binary bit \(x\) to the low bit

\(|x|_{n}\): Takes \(n\) bits from the low binary bit \(x\) to the high bit

\(a\otimes b\): The convolutional computing of two polynomials: \((a\cdot b)/<x^{n}+1>\)

\(rot(a)\): Circular matrix of \(a\in\mathcal{R}_{q}\)

\(\tau(a)\): Vector of coefficients of the polynomial \(a\in\mathcal{R}_{q}\)

\(\tau^{-1}(a)\): Transformation of the vector \(a\in\mathbb{Z}^{n}\) into the corresponding polynomial

## References

* (1) [PERSON]; [PERSON]; [PERSON]; [PERSON] Towards personalized federated learning. _IEEE Trans. Neural Netw. Learn. Syst._**2022**, 1-17. [CrossRef]
* (2) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Privacy-Preserving and Traceable Federated Learning for data sharing in industrial IoT applications. _Expert Syst. Appl._**2023**, _213_, 119036. [CrossRef]
* (3) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Reviewing Federated Machine Learning and Its Use in Diseases Prediction. _Sensors_**2023**, _23_, 2112. [CrossRef] [PubMed]
* (4) [PERSON]; [PERSON]; [PERSON]; [PERSON] Optimizing federated learning with deep reinforcement learning for digital twin empowered industrial IoT. _IEEE Trans. Ind. Inform._**2022**, _19_, 1884-1893. [CrossRef]
* (5) [PERSON]; [PERSON] signatures. In Proceedings of the Workshop on the Theory and Application of Cryptographic Techniques, Brighton, UK, 8-11 April 1991; pp. 257-265. Available online: [[https://dl.acm.org/doi/abs/10.5555/1754868.1754897](https://dl.acm.org/doi/abs/10.5555/1754868.1754897)]([https://dl.acm.org/doi/abs/10.5555/1754868.1754897](https://dl.acm.org/doi/abs/10.5555/1754868.1754897)) (accessed on 1 December 2022). * (6) [PERSON]; [PERSON] A new signature scheme based on the DSA giving message recovery. In Proceedings of the 1 st ACM Conference on Computer and Communications Security, Fairfax, VA, USA, 3-5 November 1993; pp. 58-61. [CrossRef]
* (7) [PERSON]; [PERSON] A novel chipless RFID tag using spiral resonator to achieve the pentamerous data encoding form. _J. Electromagn. Waves Appl._**2014**, _28_, 13-27. [CrossRef]
* (8) [PERSON]; [PERSON]; [PERSON] Using a hybrid encoding method based on the hexagonal resonators to increase the coding capacity of chipless RFID tags. _Int. J. RF Microw. Comput. Aided Eng._**2022**, _32_, e23474.