# An Improved Multi-Chaotic Public Key Algorithm Based on Chebyshev Polynomials
## 4 Software Implementation
### Feasibility Analysis
#### 4.1.3 Applications of the Selection of Coefficient \(k_{i}\)  
The calculation process for selecting coefficient \(k_{i}\) follows the scheme described in Section 3.1. An example is provided below for illustration.  
Selection process for \(k_{i}\):  
1. [PERSON] chooses two large prime numbers \(P=10,753\) and \(Q=15,569\) and sets \(N=P\cdot Q=167,413,457\). She generates a large integer \(d=19,973<N\) at random.
2. [PERSON] computes \(A=T_{d}(x)(modN)=131,403,701\). The public key is \((x,\ A,\ N)=(112,233,\ 131,403,701,\ 167,413,457)\), and the private key is \(d=19,973\).
3. [PERSON] randomly selects an integer \(r=13,297<N\) and computes \(T_{r}(A)(modN)=T_{13297}(131,403,701)(mod\ 167,413,457)=96,703,620\). According to Equation (20), he selects the value of \(k_{i}\). For clarity, let us assume \(\delta=N=167,413,457\), \(n=7\) and select \(k_{i}=3i+1\) as an arithmetic sequence. Let us calculate coefficient \(k_{i}\) shown in Equation (24) by applying Equation (20), and since \(T_{r}(A)(modN)=T_{r}((T_{d})(x))(modN)=96,703,620\), we choose \(k_{i}=k_{5}=16\).  
\[k_{i}=\left\{\begin{array}{ll}4,&0\leq T_{r}(T_{d}(x))modN\leq 23,916,208.1\\ 7,&23,916,208.1\leq T_{r}(T_{d}(x))modN\leq 47,832,416.3\\ 10,&47,832,416.3\leq T_{r}(T_{d}(x))modN\leq 71,748,624.4\\ 13,&71,748,624.4\leq T_{r}(T_{d}(x))modN\leq 95,664,832.6\\ 16,&95,664,832.6\leq T_{r}(T_{d}(x))modN\leq 119,581,040.7\\ 19,&119,581,040.7\leq T_{r}(T_{d}(x))modN\leq 143,497,248.9\\ 22,&143,497,248.9\leq T_{r}(T_{d}(x))modN\leq 167,413,457.0\end{array}\right.. \tag{24}\]
4. Calculate \(B=T_{r}(x)(modN)=38,698,867\) and send it to [PERSON].
5. [PERSON] computes \(T_{d}(B)(modN)=T_{r}((T_{d})(x))(modN)=96,703,620\) based on the private key \(d=19,973\). Similarly, according to Equation (20) and the assumptions made in Section 3.1, \(k_{i}\) can still be selected as \(k_{5}=16\).  
Through the above process, [PERSON] and [PERSON] can select the same value of \(k_{i}\) while preserving their private key, which will support the subsequent improvement of the Chebyshev polynomial multi-chaos public key algorithm.  
Assume that, in the case of \(N=167,413,457\), \(x_{0}=112,233\) and \(n=7\), the value of the coefficient \(k_{i}\) is randomly selected through arbitrary transformations of \(r\) and \(d\). Here, the generation method of two large prime numbers in the Rivest-Shamir-Adleman algorithmcan be used to select \(r\) and \(d\). This not only ensures strong randomness but also avoids unpredictable troubles caused by arbitrary value selection. Table 2 presents five sets of experimental results under arbitrary transformations of \(r\) and \(d\).  
From the values in Table 2, it is easy to see that when \(n=7\), the choice of \(k_{i}\) already exhibits preliminary randomness. In practical applications, as long as \(k_{n}<N\), theoretically, the value of \(N\) can be arbitrarily large and the values of \(k_{n}\) and \(n\) can also be arbitrarily large. This means that the choice of coefficient \(k_{i}\) has strong randomness in practical applications.