# An Identity-Based Anti-Quantum Privacy-Preserving Blind Authentication in Wireless Sensor Networks
## 4 Analyzing the Security and Performances
### Performances  
Here, we will compare our IDBS-NTRU's performances with other IDBS schemes. First of all, we will compare NTRU-IDBS scheme with traditional IDBS schemes in terms of performance, which were constructed based on number theory. Secondly, we will compare our IDBS-NTRU scheme with lattice-based BS schemes in terms of performance. 1. Comparing with traditional IDBS schemes As shown in Table 1, we compare IDBS-NTRU'performance with ZK scheme [35], HCZ scheme [10], and CZYW scheme [36]. The ZK scheme is constructed based on computational diffie-hellman problem of bilinear pairings. The HCZ scheme is constructed based on discrete logarithm problem of ellipse curve. The CZYW scheme is constructed based on big integer factoring problem. The IDBS-NTRU scheme's signing speed and verification speed are O(n), which outperform ZK scheme, HCZ scheme, and CZYW schemes. Its moves are 2, it is shorter than ZK scheme and HCZ scheme. Its signing secret key is \(2 nlog(s\sqrt{n})\), it is larger than ZK scheme and HCZ scheme. However, the rsa has to use \(O(n^{3})\) to achieve n bits security, the signing secret key of IDBS-NTRU scheme is shorter than CZYW scheme. The signature size of IDBS-NTRU scheme is \(2 nlog(12\sigma)+n(log\lambda+1)\), it is larger than ZK, HCZ, and CZYW schemes. For the same reason, it is also shorter than CZYW scheme. The most important of all, the BS schemes based on number theory are considered to be insecure to resist quantum computers attack [4], our IDBS-NTRU scheme is more secure than other three traditional schemes. 2. Comparing with lattice-based BS schemes We compare IDBS-NTRU's performance with GHWX [37], ZTZ [4], Ruckert [32], and ZM schemes [38] in Table 2, \(n\) denotes safety parameter. GHWX scheme and ZM scheme are constructed based on small integer solution problem of lattice. ZTZ scheme is constructed based on closest vector problem of lattice. Ruckert is constructed based on ideal-lattice shortest vector problem. As presented in Table 2, IDBS-NTRU's signing speed is \(O(n)\), which outperforms all the other schemes. IDBS-NTRU's verification speed is \(O(n)\), which outperforms GHWX and ZM schemes. Our IDBS-NTRU scheme has two moves, it is shorter than Ruckert scheme, and ZM scheme. In Ruckert scheme, the parameters satisfy \(m>\lfloor c_{m}log(1)\rfloor+1\), \(c_{m}>1/log(2d_{s})\). In ZM schemes, the parameters satisfy \(m>2 nlogq,q>2\). The signing secret key of our IDBS-NTRU scheme is \(2 nlog(s\sqrt{n})\), it is shorter than all the other schemes. The signature size of our IDBS-NTRU scheme is \(2 nlog(12\sigma)+n(log\lambda+1)\) \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**IDBS Scheme** & **ZK**[35] & **HCZ**[10] & **CZYW**[36] & **Ours** \\ \hline Hard Problem & CDHP & DLP & Factoring & R-SIS \\ Signing Speed & \(O(n^{3})\) & \(O(n^{3})\) & \(O(n^{3})\) & \(O(n)\) \\ Verifying Speed & \(O(n^{3})\) & \(O(n^{3})\) & \(O(n^{3})\) & \(O(n)\) \\ Moves & 3 & 3 & 2 & 2 \\ Signing Secret key & \(2n\) & logk+\(2n\) & \(n\) & \(2 nlog(s\sqrt{n})\) \\ Signature size & \(3n\) & logk+\(4n\) & \(n\) & \(2 nlog(12\sigma)+n(log\lambda+1)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Performance comparison with traditional IDBS schemes. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Lattice-Based** & **GHWX**[37] & **ZTZ**[4] & **Rückert**[32] & **ZM**[38] & **Ours** \\ \hline Hard Problem & SIS & CVP & ISVP & SIS & R-SIS \\ \hline Signing Speed & \(O(n^{2})\) & \(O(n^{2}logn)\) & \(O(n(logn)^{c})\) & \(O(n^{2})\) & \(O(n)\) \\ \hline Verifying Speed & \(O(n^{3})\) & O(\(n\)) & O(\(n\)) & \(O(n^{3})\) & O(\(n\)) \\ \hline Moves & 2 & 2 & 4 & 3 & 2 \\ \hline Signing secret key & \(nm\log(q+1)\) & \(dn^{2}(\log n+1)\) & \(mn\log\left(2d_{s}+1\right)\) & \(m^{2}\log(q+1)\) & \(2n\log(s\sqrt{n})\) \\ \hline Signature size & \(m\log(q+1)\) & \(dn(\log n+1)\) & \(n^{2}+mn\) & \(2m\log(q+1)\) & \(\frac{2 nlog(12\sigma)+n(\log\lambda+1)}{n(\log\lambda+1)}\) \\ \hline Identity Based & yes & no & no & yes & yes \\ \hline \hline \end{tabular}
\end{table}