# LPCP: An efficient Privacy-Preserving Protocol for Polynomial Calculation Based on CRT
## 8 Performance and Evaluation
### Practical Performance
In the multiplicative arithmetic operations, the running time of both 64-bit and 1024-bit Helib multiplication is slower than LPCP. Next, we analyze the two-party secure computation experimental results comparison between LPCP and Emp-sh2 pc. Figure 3 shows the running time comparison of different arithmetic operations, and Figure 4 shows the communication overhead comparison. We can see that LPCP comes in the lead in both the running time and communication overhead. The running time of our protocol is more than 100 times faster than Emp-sh2 pc in 128-bit and 256-bit additive and multiplicative tests. In particular, in the 1024-bit modular exponential operation test, the running time of LPCP is more than 100,000 times faster than Emp-sh2 pc. Although the running time of LPCP's additive computation becomes almost as costly as Emp-sh2 pc when the bit length of the security parameter comes to 1024 bits. According to the analytic computational and space complexity, the running time of our protocol grows linearly with the largest exponent of the polynomial, which means the multiplicative computation of LPCP takes almost the same amount of time as the additive computation. For this reason, the running time of LPCP's multiplicative computation is exponentially faster than Emp-sh2 pc. As the communication latency increases, the additional one communication round of LPCP results in more communication time than the other 2 PC protocols. In the communication space comparison, the total communication space is more than 10,000 times smaller than the result of Emp-sh2 pc in 128-bit and 256-bit additive and multiplicative tests. Figure 2: The running time comparison of additive, multiplicative and exponential arithmetic operations of LPCP on ARM and X86 architecture. When it comes to three-party and four-party secure computation, we compare our LPCP experimental result with MP-SPDZ. To ensure fairness, we choose two kinds of MPC protocols, Semi and Hemi, which are both secure against the semi-honest adversary from the MP-SPDZ library. The difference of the chosen MPC protocol lies in that Semi is built based on oblivious transfer (OT) [31] and Hemi is built based on semi-homomorphic encryption [27]. Figure 5 shows the running time of Semi, Hemi and LPCP in the two-party, three-party and four-party experiments. Figure 6 shows the total communication overhead of Semi, Hemi and LPCP in the two-party, three-party and four-party experiments. The advantage of the fast execution and low communication cost of LPCP becomes much more clear as the number of parties grows because the time complexity and communication complexity grow approximately linearly to the party numbers. From the figures above, we can see that LPCP has better scalability than other traditional MPC protocols in the same condition. It is especially suitable for those mobile devices with low computing power and small storage space to execute LPCP for data privacy. \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Arithmetic** & **Encryption** & **Arithmetic** & **Decryption** & **Total Running** \\
**Operation Type** & **Time** & **Operation Time** & **Time** & **Time** \\ \hline
64-bit add & 0.643 ms & 0.000000011 ms & 2.010 ms & 2.653 ms \\
64-bit mul & 0.643 ms & 6.972 ms & 2.010 ms & 9.625 ms \\
1024-bit add & 935.44 ms & 0.000126427 ms & 589.312 ms & 1524.752 ms \\
1024-bit mul & 935.44 ms & 3843.84 ms & 589.312 ms & 5368.592 ms \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Performance of BGV implemented in HElib library on additive and multiplicative operations. Figure 3: The running time comparison of additive operation between LPCP and Emp-sh2 pc.