## 4 Privacy-Preserving Polynomial Model Generation  
In this section, we use the Chebyshev polynomials to demonstrate the effectiveness of our introduced method (LAPP) mathematically. Consider a dataset \(\{(x_{i},y_{i})|\leq i\leq n\}\), and let  
\[\hat{f}(x)=a_{1}\varphi_{1}(x)+a_{2}\varphi_{2}(x)+\cdots+a_{m}\varphi_{m}(x) \tag{21}\]  
where, \(a_{1},a_{2},\ldots,a_{m}\) are coefficients and \(\varphi_{1}(x),\varphi_{2}(x),\ldots,\varphi_{m}(x)\) are Chebyshev polynomials of the first kind,  
\[\varphi_{1}(x)=T_{0}(x)=1\]  
\[\varphi_{2}(x)=T_{1}(x)=x\]  
\[\varphi_{n}(x)=T_{n+1}(x)=2 xT_{n}(x)-T_{n-1}(x)\]  
Assume that the data \(\{x_{i}\}\), are chosen from an interval \([\alpha,\beta]\). The Chebyshev polynomials can be modified as  
\[\varphi_{k}(x)=T_{k-1}\bigg{(}\frac{2x-\alpha-\beta}{\beta-\alpha}\bigg{)} \tag{22}\]  
The approximated function \(\hat{f}\) of degree \((m-1)\) can be given by Equation (22), where the degree of \((\varphi_{k})\) is \(k-1\). We will assume the interval \([\alpha,\beta]=[0,1]\) and construct the model accordingly. According to Equation (22) when \([\alpha,\beta]=[0,1]\), we get Equation (23)  
\[\varphi_{k}(x)=T_{k-1}\bigg{(}\frac{2x-\alpha-\beta}{\beta-\alpha}\bigg{)}=T_ {k-1}(2x-1) \tag{23}\]  
We can obtain the following equations for \(m=4\). \[\varphi_{1}(x)=T_{0}(2x-1)=1\]  
\[\varphi_{2}(x)=T_{1}(2x-1)=2x-1\]  
\[\varphi_{3}(x)=T_{2}(2x-1)=8x^{2}-8x+1\]  
\[\varphi_{4}(x)=T_{3}(2x-1)=32x^{3}-48x^{2}+18x-1\]  
To determine the \(\hat{f}(x)\) when \(m-4\). \[\varphi_{1}(x)=T_{0}(x)=1\]0  
\[\varphi_{1}(x)=T_{0}(x)=1\]1  
Let the actual input be \(y_{i}\), where \(i=1\) to \(n\). The error of the approximated input can be determined by Equation (25)  
\[\varphi_{1}(x)=T_{0}(x)=1\]2  
We need to determine the values of \(a_{1},a_{2},a_{3}\), and \(a_{4}\), we use the root mean square error. \[\varphi_{1}(x)=T_{0}(x)=1\]3  
Take the least squares fitting of \(\hat{f}(x_{i})\) of the class of functions \(C\) which minimizes \(E\) as \(\hat{f}_{*}(x)\)We can obtain \(\hat{f}*(x)\) by minimizing \(E\). Thus, we seek to minimize \(M(a_{1},a_{2},a_{3},a_{4})\) which is given in Equation (26)  
\[ \begin{array}{ll}M(a_{1},a_{2},a_{3},a_{4})&=\sum\limits_{i=1}^{n}\Big{[}a_{1} +a_{2}(2x-1)+a_{3}\Big{(}8x^{2}-8x+1\Big{)}\\ &+a_{4}\Big{(}32x^{3}-48x^{2}+18x-1\Big{)}-y_{i}\Big{]}^{2}\end{array}  \tag{26}\]  
The values of \(a_{1},a_{2},a_{3}\), and \(a_{4}\) that minimize \(M(a_{1},a_{2},a_{3},a_{4})\) will satisfy the expressions given from Equations (27)-(30). \[ \begin{array}{ll}\frac{\partial M(a_{1},a_{2},a_{3},a_{4})}{ \partial a_{1}}\\ =\frac{\partial\Big{(}\sum\limits_{i=1}^{n}\big{[}a_{1}+a_{2}(2x-1)+a_{3} \big{(}8x^{2}-8x+1\big{)}+a_{4}\big{(}32x^{3}-48x^{2}+18x-1\big{)}-y_{i}\big{]} ^{2}\Big{)}}{\partial a_{1}}=0\\ =\frac{\partial M(a_{1},a_{2},a_{3},a_{4})}{\partial a_{2}}\\ =\frac{\partial\Big{(}\sum\limits_{i=1}^{n}\big{[}a_{1}+a_{2}(2x-1)+a_{3} \big{(}8x^{2}-8x+1\big{)}+a_{4}\big{(}32x^{3}-48x^{2}+18x-1\big{)}-y_{i}\big{]} ^{2}\Big{)}}{\partial a_{2}}=0\\ =\frac{\partial M(a_{1},a_{2},a_{3},a_{4})}{\partial a_{3}}\\ =\frac{\partial\Big{(}\sum\limits_{i=1}^{n}\big{[}a_{1}+a_{2}(2x-1)+a_{3} \big{(}8x^{2}-8x+1\big{)}+a_{4}\big{(}32x^{3}-48x^{2}+18x-1\big{)}-y_{i}\big{]} ^{2}\Big{)}}{\partial a_{3}}=0\\ =\frac{\partial M(a_{1},a_{2},a_{3},a_{4})}{\partial a_{4}}\\ =\frac{\partial\Big{(}\sum\limits_{i=1}^{n}\big{[}a_{1}+a_{2}(2x-1)+a_{3} \big{(}8x^{2}-8x+1\big{)}+a_{4}\big{(}32x^{3}-48x^{2}+18x-1\big{)}-y_{i}\big{]} ^{2}\Big{)}}{\partial a_{4}}=0\end{array}  \tag{30}\]  
A summary of the used notation is given in Table 1.