## 7 Conclusions  
Data authenticated aggregation is always a significant issue for marine WSNs. Most data authenticated aggregation is based on the multi-signature, which relies on the technique of bilinear pairing involving heavy computational overhead or the management of certificates beyond marine wireless sensors. We have constructed two efficient IBMS schemes (\(\mathbf{IBMSCR-1}\) and \(\mathbf{IBMSCR-2}\)) based on cubic residues, which are much more suitable for data authenticated aggregation in marine WSNs. Without employing the heavy overload of a bilinear pairing technique, our schemes have been  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Schemes** & **The Underlying Mathematical Assumptions** \\ \hline
[15] & Computational Diffie-Hellman (CDH) \\
[19] & Discrete Logarithm (DL) \\
[20] & RSA \\
[6] & Quadratic Residues \\ \(\mathbf{IBMSCR-1}\) & Cubic Residues \\ \(\mathbf{IBMSCR-2}\) & Cubic Residues \\ \hline \hline \end{tabular}
\end{table}
Table 1: The comparison of related work on the security assumptions.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Schemes** & **Extract** & **Sign** & **Verify** & **Total Time** & **Length** \\ \hline
[15] & \(2H_{m}+2M_{p}\) & \(1H_{m}+4M_{p}\) & \(3O_{p}\) & \(107.52\) & \(2\left|g\right|\) \\
[19] & \(1E_{n}\) & \(2E_{n}\) & \(2E_{n}\) & \(26.55\) & \(\ell+\left|N\right|\) \\
[20] & \(1E_{n}\) & \(2E_{n}\) & \(2E_{n}\) & \(26.55\) & \(\ell+2\left|N\right|\) \\
[6] & \(1E_{n}\) & \(2E_{n}\) & \(2E_{n}\) & \(26.55\) & \(\ell+\left|N\right|\) \\ \(\mathbf{IBMSCR-1}\) & \(1E_{n}\) & \(2E_{n}\) & \(2E_{n}\) & \(26.55\) & \(\ell+\left|N\right|\) \\ \(\mathbf{IBMSCR-2}\) & \(2E_{n}\) & \(1E_{n}\) & \(1E_{n}\) & \(21.24\) & \(\ell+\left|N\right|\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: The comparison of related work of IBMS on the computational performance.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Schemes** & **Underlying Cryptographic Primitive** & **Sign** & **Verify** & **Total Time** \\ \hline
[28] & IBMPS & \(3E_{n}\) & \(3E_{n}\) & \(6E_{n}\) \\
[26] & IBPMS & \(1E_{n}\) & \(3E_{n}\) & \(4E_{n}\) \\
[29] & IBMPMS & \(3E_{n}\) & \(3E_{n}\) & \(6E_{n}\) \\ \(\mathbf{IBMSCR-1}\) & IBMS & \(2E_{n}\) & \(2E_{n}\) & \(4E_{n}\) \\ \(\mathbf{IBMSCR-2}\) & IBMS & \(1E_{n}\) & \(1E_{n}\) & \(2E_{n}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: The comparison of related work on computational performance based on the cubic residues.  
designed efficiently. Our schemes have been proven to be secure under chosen identity attacks and chosen message attacks, relying only on the hardness of the integer factorization assumptions.  
This work was supported by the Natural Science Foundation of China (NSFC grant nos. 61402282, 61672339 and 41671431), the Shanghai Youth Talent Development Program (grant no. 14 YF1410400), and the Shanghai Local University Capacity Enhancement Program (grant no. 15590501900).  
The work was conducted under the cooperation of authors. [PERSON] conceived the scheme 1, and wrote the partial paper; [PERSON] conceived the scheme 2 and wrote the partial paper, [PERSON] guided the study and reviewed the manuscript; [PERSON] conceived the security proof and wrote the partial paper; [PERSON] gave the figures and verified the results; [PERSON] introduced the background of marine sensor networks.  
The authors declare no conflicts of interest.