# New Subclass Framework and Concrete Examples of Strongly Asymmetric Public Key Agreement
## 5 Generalization
### Notations and Definitions  
Prior to constructing the main subclass (Definition 9), we present several definitions and notations below. Definition 7 provides one \(SAPKA_{f_{g}}\) subclass in which the algorithms are symmetric and include DH. In Section 5.2, we investigate how this subclass relates to the main subclass of Definition 9. Definition 8 provides the name for the input \(y\) of a given map \(f_{g}^{\prime}:\mathcal{S}\rightarrow\mathcal{S}\) according to the set of \(y\). We use this definition to emphasize the difference in complexity for calculating \(f_{g}^{\prime-1}\circ f_{g}^{\prime}(y)\) of a given map \(f_{g}^{\prime}\) when \(y\in\mathcal{S}\) and when \(y\in\mathcal{S}\). The notations directly below Definition 8 are functions that demonstrate the complexity for calculating each key. As these notations and Definition 8 allow us to express the breaking and computational complexity for each key quantitatively, the main subclass can be defined by equalities and inequalities. **Definition 7**. : _For the quintuple \(C\) such that \(C\in SAPKA_{f_{g}}\), if \(N_{1}^{\prime}=x_{4}^{\prime}=id\), we state that \(C\) is a member of the symmetric \(SAPKA_{f_{g}}\) class, and we express this relation as_  
\[C\in SAPKA_{f_{g},symmetry}.\]  
If \(C\in SAPKA_{f_{g},symmetry}\), one of the public keys of [PERSON] \(y_{B,1}\) is described as  
\[y_{B,1}(y)=N_{1}^{-1}\circ x_{4}(y)=f_{g}\circ N_{1}^{\prime-1}\circ x_{4}^{ \prime}(y)=f_{g}(y),\]  
where \(y\in\mathcal{S}\). This means that [PERSON] sending \(y_{B,1}\) is equivalent to simply informing [PERSON] of the construction rule of \(y_{A}\). [PERSON] can calculate \(y_{A}\) without any secret/public information from [PERSON]. Symmetry means that the number of public keys that both [PERSON] and [PERSON] construct is essentially one, and they can construct their public keys without any public/secret information from the other. For \(C_{NDH}\) in Section 3.2, if the parameter \(n_{B}\) is equal to \(1\), \(C_{NDH}\) describes the usual DH. We denote this case of \(C_{NDH}\) as \(C_{DH}\), and we obtain  
\[C_{DH}\in SAPKA_{f_{g},symmetry}.\]  
Moreover, \(C_{SE}\) in the case of \(N_{B}=E\), denoted by \(C_{SE,(N_{B}=E)}\), is  
\[C_{SE,(N_{B}=E)}\in SAPKA_{f_{g},symmetry}.\] \begin{table}
\begin{tabular}{l l} \hline \hline Public keys of [PERSON] &
\begin{tabular}{l} \(y_{B,1}(y)=N_{1}^{-1}\circ x_{4}(y)=g^{\circ N_{B}y}=f_{g}\circ N_{1}^{\prime -1}\circ x_{4}^{\prime}(y)\ (y\in\mathcal{S})\), \\ \(y_{B,2}(y)=x_{1}\circ x_{2}(y)=g^{\circ x_{B}y}=f_{g}\circ x_{1}^{\prime}\circ x _{2}^{\prime}(y)\ (y\in\mathcal{S})\) \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Public keys of Bob for \(C_{SE}\). **Definition 8**. : _For a given non-easily invertible map \(f_{g}^{\prime}:\mathcal{S}\to\mathcal{S}\), if the set to which the input \(y\) belongs is \(\underline{\mathcal{S}}\), we refer to the input \(y\) as a \(\text{km-bit}\ f_{g}^{\prime-1}\)_**element**_. If the set to which the input \(y\) belongs is \(\mathcal{S}\), we refer to the input \(y\) as a \(\text{km-bit}\ f_{g}^{\prime-1}\)_**element**_._  
The complexity of obtaining the \(\text{km-bit}\ f_{g}^{\prime-1}\) element means the complexity of calculating \(f_{g}^{\prime-1}\circ f_{g}(y)\) when \(y\in\mathcal{S}\) (for the \(\text{km-bit}\ f_{g}^{\prime-1}\) element, \(y\in\underline{\mathcal{S}}\)).