# New Subclass Framework and Concrete Examples of Strongly Asymmetric Public Key Agreement
## 5 Generalization
### Inclusion Relations between Subclasses of SAPKA and Necessity Condition  
**Lemma 1**. : _For \(SAPKA_{f_{g},A<B}\subset SAPKA_{f_{g}}\) and \(\mathit{SAPKA}_{f_{g},A=B}\subset SAPKA_{f_{g}}\), the following relation holds:_  
\[\mathit{SAPKA}_{f_{g},A<B}\cap SAPKA_{f_{g},A=B}=\emptyset.\]  
According to the attack introduced in Section 4.3, we obtain the following results. **Theorem 3**. : _For \(C_{SE}\) in Section 3.3, if Attack 2 of Section 4.3 and the exhaustive attack for \(x_{A}\in\underline{\mathcal{S}}\) are only attacks without calculating \(\log_{g}g^{\otimes N_{B}}=N_{B}\) or there exist other attacks but the order of complexity is equal to or greater than \(O(2^{dm})\),_  
\[C_{SE}\in SAPKA_{f_{g},A<B}.\]Proof. : According to Attack 3 in Section 4.3 and the descriptions of \(T_{Eve\ x_{A}}(m)\) and \(T_{f_{g}^{-1}}(s(m))\) in Example 1, we obtain  
\[T_{Eve\ x_{A}}(m)\geq d^{2}2^{\frac{i(m)}{2}}=T_{f_{g}^{-1}}(s(m)).\]  
Under this assumption, \(n=s(m)=2 dm\) holds, and thus, we obtain  
\[T_{y_{k,1}}(s(m))+T_{y_{k,2}}(s(m))+T_{x_{B}}(s(m))=4 md^{3}+2 md^{4}>T_{y_{A}}(m )+T_{x_{A}}(m)=2 md^{3}\]  
from Section 4.5. We introduce two necessary conditions for the \(SAPKA_{f_{g},A<B}\) class. **Theorem 4** (Necessary Condition 1). : _If the quintuple \(C:=(x_{1},x_{2},x_{3},x_{4},N_{1})\in SAPKA_{f_{g},A<B}\), the relation_  
\[C\
otin SAPKA_{f_{g}symmetry} \tag{52}\]  
_holds._  
Proof. : We prove the contrapositive. If \(C\in SAPKA_{f_{g},symmetry}\), [PERSON]'s public key \(y_{A}\) is described as:  
\[y_{A}=N_{1}^{-1}\circ x_{4}(x_{A})=f_{g}\circ N_{1}^{\prime-1}\circ x_{4}^{ \prime}(x_{A})\]  
according to (40), but \(N_{1}^{\prime}\) and \(x_{4}^{\prime}\) are identities; thus, \(y_{A}\) becomes  
\[y_{A}=f_{g}(x_{A}).\]  
Because \(x_{A}\in\underline{\mathcal{S}}\), \(x_{A}\) is a \(km\)-bit \(f_{g}^{-1}\) element for map \(f_{g}\), and we obtain  
\[T_{Eve\ x_{A}}(m)=T_{f_{g}^{-1}}(m).\]  
Thus, \(C\in SAPKA_{f_{g},A=B}\). Moreover, Lemma 1 indicates that \(C\
otin SAPKA_{f_{g},A<B}\). **Corollary 1**. : _The quintuple \(C_{DH}\) of the usual DH is_  
\[C_{DH}\in SAPKA_{f_{g},A=B},\]  
_and \(C_{SE,(N_{B}=E)}\) defined in Section 5.1 is_  
\[C_{SE,(N_{B}=E)}\in SAPKA_{f_{g},A=B}.\]  
**Theorem 5** (Necessary Condition 2). : _If the quintuple \(C:=(x_{1},x_{2},x_{3},x_{4},N_{1})\in SAPKA_{f_{g},A<B}\), then there does't exist a map \(f_{g}^{\prime}:\mathcal{S}\rightarrow\mathcal{S}\) such that to satisfy_  
\[C_{SE}\in SAPKA_{f_{g},A<B}.\]0  
_for all \(y\in\mathcal{S}\), \(m\in\mathbb{N}\)._  
Proof. : We prove the contrapositive. We assume that for \(C\in SAPKA_{f_{g},A<B}\), there exists a map \(f_{g}^{\prime}:\mathcal{S}\rightarrow\mathcal{S}\) to satisfy  
\[C_{SE}\in SAPKA_{f_{g},A<B}.\]1  
\[C_{SE}\in SAPKA_{f_{g},A<B}.\]2  
for all \(y\in\mathcal{S}\). According to (54), \(y_{A}\) can be described as  
\[C_{SE}\in SAPKA_{f_{g},A<B}.\]3This indicates that \(x_{A}\) is a \(km\)-bit \(f_{g}^{\prime-1}\) element for a map \(f_{g}^{\prime}\). In combination with (55), the relation  
\[C_{SE}\in SAPKA_{f_{g},A<B}.\]4  
holds. Thus, \(C\in SAPKA_{f_{g},A=B}\); that is, \(C\
otin SAPKA_{f_{g},A<B}\) is proven. **Corollary 2**. : \[C_{SE}\in SAPKA_{f_{g},A<B}.\]5  
Proof. : The attack of Section 4.2 indicates that  
\[C_{SE}\in SAPKA_{f_{g},A<B}.\]6  
where \(f_{g}^{\prime}(y):=f_{f_{g}(N_{1}^{\prime-1}\circ x_{4}^{\prime}(1))}(y)=f_{g^ {n_{B}}}(y)\), and (53) is satisfied. **Remark 4**. : _Theorem 1 of Section 4.3 implies that there does not exist a map \(f_{g}^{\prime}\) to satisfy \(f_{g}^{\prime}(y)=f_{g}\circ N_{1}^{\prime-1}\circ x_{4}^{\prime}(y)\) with \(T_{f_{g}^{\prime-1}}\leq T_{f_{g}^{-1}}\) as for algorithm of \(C_{SE}\)._  
According to the proof of Theorem 4 and Corollary 2, we obtain the following inclusion relation:  
**Corollary 3**. : \[C_{SE}\in SAPKA_{f_{g},A<B}.\]7  
Proof. : From the proof of Theorem 4,  
\[C_{SE}\in SAPKA_{f_{g},A<B}.\]8  
is obtained.