# P System Design for Integer Factorization
## 2 Foundation
### Extended P System
However, it is still difficult

Figure 10: Simulation results using UPSimulator, where (**a**) experimental results with \(N=15\), which factored into 5 and 3 marked with the red square; (**b**) experimental results with \(N=39\), which factored into 13 and 3 marked with the red square. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & **Time Complexity** & **Space Complexity** \\ \hline General Number Field Sieve (GNFS) & \(O\left(e^{c(\log\ n)^{1/3}}\times(\log\ \log\ n)^{2/3}\right)\) & -1 \\ [PERSON]’s algorithm & \(O((\log\ n)^{3}\text{loglog}\ n)\) & \(O(\log\ n)\) \\ [PERSON]’s rho algorithm [23] & \(O(\sqrt{n})\) & \(O(1)\) \\ ours(PFLN) & \(O(\text{lnlog}\ n)\) & \(O(n)\)2  \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of space–time complexity of Factorization algorithm. to implement such systems in a biological sense, so it does not affect the security of existing Cryptography. We built related codes and experiments in UPSimulator and successfully calculated multiple prime factorization problems including \(N=15\). Due to how the numerical value of the power calculation increases too fast, UPSimulator overflows when calculating a slightly larger number, making the sample calculation unable to be very large. Future work can be divided into three parts:

1. The membrane structure can continue to be optimized, and a dedicated simulator can be established to test on larger data samples;
2. The P system has variants of various biological mechanisms. How to introduce these variants into the current model to improve its performance is worth considering. 3. Whether the hit rate of the periodic function has a mathematical law is still a question that can be discussed, which is related to the size of the space when the P system is executed. In this way, it can be determined whether a stable space complexity can be found when performing the integer factorization problem. Conceptualization, Z.X. and H.N. ; methodology, C.L. ; software, Z.X. ; validation, Z.X. and C.L. ; formal analysis, H.N. ; resources, Z.X. ; writing--original draft preparation, C.L. ; writing--review and editing, M.Z. and X.L. ; supervision, H.N. ; project administration, C.L. All authors have read and agreed to the published version of the manuscript. This research was funded by the Funding Results of Action Plan for High-quality Development of Postgraduate Education of Chongqing University of Technology (0103204167), and Humanities and Social Sciences Research Key Project of Chongqing Municipal Education Commission (23 SKGH247). Not applicable. The authors declare no conflict of interest. ## References

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