# New Subclass Framework and Concrete Examples of Strongly Asymmetric Public Key Agreement
## 5 Generalization  
In this section, all investigations are conducted under the following assumptions:  
* the numbers \(n,m\in\mathbb{N}\) and a not linear and monic polynomial \(s\), where \(n=s(m)>m\) for all \(m\in\mathbb{N}\);* a number \(p\), where \(|p|=n\);
* a semi-group \(\mathcal{S}\), where \(|y|_{\mathcal{S}}\leq kn\ (y\in\mathcal{S},k\in\mathbb{N})\);
* a subset \(\underline{\mathcal{S}}:=\{y\in\mathcal{S}\ :\ |y|_{\mathcal{S}}\leq km\}\subset \mathcal{S}\); and
* \(x_{A}\in\underline{\mathcal{S}}\). In the above, \(|\cdot|_{\mathcal{S}}\) is the same notation as that in Section 4.1. Furthermore, the constant \(k\) is uniquely determined by the properties of \(\mathcal{S}\), such as the dimensions of the matrices and vectors. For example, \(k=1\) when \(\mathcal{S}=\mathbb{Z}_{p}\) and \(k=d^{2}\) when \(\mathcal{S}=\mathcal{M}(d,\mathbb{Z}_{p})\). As mentioned in Section 2.3, we first construct a SAPKA subclass, in which the security of each algorithm is ensured by a non-easily invertible map (Definition 1). Thereafter, we attempt to construct the main SAPKA subclass mentioned in Section 2.2. We introduce inclusion relations between other subclasses and necessary conditions for algorithms to belong to the main subclass in Section 5.2. We present the definition of the subclass based on the non-easily invertible map. Definition 6: For a multiplicative semi-group \(\mathcal{S}\) and the quintuple \(C:=(x_{1},x_{2},x_{3},x_{4},N_{1})\), where the maps \(x_{1},x_{2},x_{3},x_{4}\) satisfy (13) and \(N_{1}\) is an easily invertible map of \(N_{1}:\mathcal{S}\to\mathcal{S}\), if maps \(x_{1}^{\prime},x_{2}^{\prime},x_{3}^{\prime},x_{4}^{\prime},N_{1}^{\prime}: \mathcal{S}\to\mathcal{S}\) exist, where \(N_{1}^{\prime}\) is easily invertible and a non-easily invertible map \(f_{\mathcal{S}}:\mathcal{S}\to\mathcal{S}\) exists such that the following equations:  
\[x_{1}^{\prime}\circ x_{2}^{\prime}(y)=x_{3}^{\prime}\circ x_{4}^{\prime}(y) \tag{40}\]  
\[x_{1}\circ x_{2}(y)=f_{\mathcal{S}}\circ x_{1}^{\prime}\circ x_{2}^{\prime}(y) \tag{41}\]  
\[N_{1}^{-1}\circ x_{4}(y)=f_{\mathcal{S}}\circ N_{1}^{\prime-1}\circ x_{4}^{ \prime}(y) \tag{42}\]  
\[x_{1}^{\prime}\circ x_{2}^{\prime}(e)\
ot\in\underline{\mathcal{S}}\ \vee\ N_{1}^{\prime-1}\circ x_{4}^{\prime}(e)\
ot\in\underline{\mathcal{S}} \tag{43}\]  
are satisfied for all \(y\in\mathcal{S}\), where \(e\) is an identity element of \(\mathcal{S}\), the quintuple \(C\) is a member of the \(SAPKA_{f_{\mathcal{S}}}\) class, and we express this relation as  
\[C\in SAPKA_{f_{\mathcal{S}}}.\]  
From (40) and (41), and the compatibility condition of (13), the equation  
\[x_{1}\circ x_{2}(y)=f_{\mathcal{S}}\circ x_{1}^{\prime}\circ x_{2}^{\prime}(y) =f_{\mathcal{S}}\circ x_{3}^{\prime}\circ x_{4}^{\prime}(y)=x_{3}\circ x_{4}(y) \tag{44}\]  
for all \(y\in\mathcal{S}\) is automatically satisfied. Using (42), the equation  
\[f_{\mathcal{S}}\circ x_{3}^{\prime}\circ N_{1}^{\prime}\circ N_{1}^{\prime-1} \circ x_{4}^{\prime}(y)=f_{\mathcal{S}}\circ x_{3}^{\prime}\circ x_{4}^{ \prime}(y)\]  
\[=x_{3}\circ x_{4}(y)=x_{3}\circ N_{1}\circ N_{1}^{-1}\circ x_{4}(y)=x_{3} \circ N_{1}\circ f_{\mathcal{S}}\circ N_{1}^{\prime-1}\circ x_{4}^{\prime}(y) \tag{45}\]  
holds for all \(y\in\mathcal{S}\). Thus, the public keys and SSK of Alice and Bob for the algorithms in this class can be described as Table 4.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{2}{c}{**Key**} & **Parameter** \\ \hline \multirow{4}{*}{Bob} & Public keys & \( \begin{matrix}y_{B,1}(y)=N_{1}^{-1}\circ x_{4}(y)=f_{\mathcal{S}}\circ N_{1}^{ \prime-1}\circ x_{4}^{\prime}(y)\ \ \ (y\in\mathcal{S}),\\ y_{B,2}(y)=x_{1}\circ x_{2}(y)=f_{\mathcal{S}}\circ x_{1}^{\prime}\circ x_{2}^{ \prime}(y)\ \ (y\in\mathcal{S})\end{matrix}