# Public-Key Cryptography Based on Tropical Circular Matrices
## 5 Conclusions and Further Research  
In this paper, we present a new key exchange protocol and a new public-key encryption scheme based on tropical matrices. We use a class of tropical commuting matrix, that is, the tropical \(t\)-circular matrix, other than matrix powers or matrix polynomials. The security of new public-key cryptosystems relies on a two-side tropical circular matrix action problem (TCMAP). The use of \(t\)-circular matrices allows us to share less information with the attacker. Since tropical circular matrices cannot be represented by a known matrix, our public-key cryptosystems can resist KU attacks. There is no addition of tropical matrices in our schemes. So, the attack method proposed by [PERSON] and [PERSON] does not work for our public-key cryptosystems. Our public-key cryptosystem can resist all known attacks. As we know, the best way to solve TCMAP is to solve a tropical nonlinear system of equations, which is NP-hard. So, the new cryptosystems can be considered as a potential post-quantum cryptosystem.  
Future works worth studying include the following:  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Schemes** & **Mathematical Problems** & **KU Attack** & **RM Attack** \\ \hline [PERSON] et al. [24] & Two-side matrix action problem & \(\times\) & \(\surd\) \\ [PERSON] et al. [26] & Semidirect product problem & \(\surd\) & \(\times\) \\ Our scheme & Two-side tropical circular matrix action problem & \(\surd\) & \(\surd\) \\ \hline \hline \end{tabular} Note that \(\surd\) means that the scheme can resist the corresponding attack, while \(\times\) means it does not.  
\end{table}
Table 1: Comparison among relevant tropical schemes.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(k\) & **Size of sk (kB)** & **Size of pk (kB)** & **Complexity of Solving TCMAP** \\ \hline
10 & 0.0781 & 0.7813 & \(O(2^{81})\) \\
20 & 0.1563 & 3.1250 & \(O(2^{199})\) \\
30 & 0.2344 & 7.0313 & \(O(2^{331})\) \\
40 & 0.3125 & 12.5000 & \(O(2^{472})\) \\
50 & 0.3906 & 19.5313 & \(O(2^{620})\) \\
60 & 0.4688 & 28.1250 & \(O(2^{775})\) \\ \hline \hline \end{tabular} Note that “sk” means secret key and “pk” means public key.  
\end{table}
Table 2: Performance comparison under some different parameters.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Experimental Platform** & **Key Generation** & **Encryption** & **Decryption** \\ \hline Intel (R) i7-8550 1.80 GHz & 0.984 s & 1.018 s & 0.513 s \\ Intel (R) i5-5200 2.20 GHz & 0.624 s & 0.594 s & 0.297 s \\ Intel (R) i7-4700 2.40 GHz & 0.363 s & 0.346 s & 0.187 s \\ \hline \hline \end{tabular}
\end{table}
Table 3: Timings for cryptographic operations in our cryptosystem.  
1. A possible algorithm for solving TCMAP. If we can find some algorithms for solving the systems of min-plus polynomial equations, then they can be used to attack our schemes.
2. Other cryptographic applications of TCMAP. For example, we can try to design digital signature schemes and identity authentication schemes based on TCMAP.  
Conceptualization, H.H. and C.L.; methodology, H.H. and L.D.; software, H.H. and L.D.; validation, H.H. and L.D.; formal analysis, H.H., C.L. and L.D.; writing--original draft preparation, H.H. and L.D.; writing--review and editing, H.H., C.L. and L.D. All authors have read and agreed to the published version of the manuscript.  
This work is supported by the National Natural Science Foundation of China (No. 61962011, 61462016) and the Science and Technology Foundation of Guizhou Province (QIANKEFIEJICH-ZK [2021] 313).  
Not applicable.  
The authors declare no conflict of interest.