# New Subclass Framework and Concrete Examples of Strongly Asymmetric Public Key Agreement
## 5 Generalization
### Notations and Definitions
We define the following functions of bit size that demonstrate the complexity of calculating each public key and SSK as well as the complexity for [PERSON] to obtain a certain key:  
* \(T_{y_{B,1}}:\mathbb{N}\to\mathbb{R}^{+}\): the complexity required for [PERSON] to construct \(y_{B,1}\);
* \(T_{y_{B,2}}:\mathbb{N}\to\mathbb{R}^{+}\): the complexity required for [PERSON] to construct \(y_{B,2}\);
* \(T_{y_{A}}:\mathbb{N}\to\mathbb{R}^{+}\): the complexity required for [PERSON] to calculate \(y_{A}\);
* \(T_{x_{A}}:\mathbb{N}\to\mathbb{R}^{+}\): the complexity required for [PERSON] to calculate her SSK \(\kappa_{A}\);
* \(T_{x_{B}}:\mathbb{N}\to\mathbb{R}^{+}\): the complexity required for [PERSON] to calculate their SSK \(\kappa_{B}\);
* \(T_{E\text{\tiny Eve }x_{A}}:\mathbb{N}\to\mathbb{R}^{+}\): the least complexity required for [PERSON] to obtain [PERSON]'s secret key \(x_{A}\) from public informations, where the input of \(T_{E\text{\tiny [PERSON] }x_{A}}\) is \(m\); and
* \(T_{f_{g}^{\prime-1}}:\mathbb{N}\to\mathbb{R}^{+}\): the complexity required to compute \(f_{g}^{\prime-1}\circ f_{g}^{\prime}(y)\) for map \(f_{g}^{\prime}\), where the input of \(T_{f_{g}^{\prime-1}}\) is \(m\) if \(y\in\underline{\mathcal{S}}\) and \(n\) if \(y\in\mathcal{S}\). Furthermore, for \(T_{E\text{\tiny [PERSON] }x_{A}}\), \(T_{f_{g}^{\prime-1}}\), we define the following functions to describe their order:  
* \(h_{E\text{\tiny [PERSON] }x_{A}}:\mathbb{N}\to\mathbb{R}^{+}\) to obtain the relation \(T_{E\text{\tiny [PERSON] }x_{A}}\in O(h_{E\text{\tiny Eve }x_{A}})\), and
* \(h_{f_{g}^{\prime-1}}:\mathbb{N}\to\mathbb{R}^{+}\) to obtain the relation \(T_{f_{g}^{\prime-1}}\in O(h_{f_{g}^{\prime-1}})\). **Example 1**. :
* _If_ \(T_{y_{B,1}}\)_,_ \(T_{y_{B,2}}\)_, and_ \(T_{x_{B}}\) _are defined as functions of_ \(n\)_,_ \(T_{y_{A}}\) _and_ \(T_{x_{A}}\) _are defined as functions of_ \(m\)_, and the polynomial_ \(s\) _is concretely determined, we can compare the total computational complexity for [PERSON] and [PERSON] as per Section_ 4.5_._
* _The least complexity for [PERSON] to obtain_ \(x_{A}\) _in Section_ 3.3 _is determined as_ \[T_{E\text{\tiny Eve }x_{A}}(m)\geq d^{2}2^{\frac{s(m)}{2}}.\]
* _The complexity of obtaining the_ \(d^{2}n\)_-bit_ \(f_{g}^{-1}\) _element, where_ \(f_{g}\) _is the non-easily invertible map of_ \(\text{C}_{SE}\) _as defined in Table_ 6_, is expressed as_ \[T_{f_{g}^{-1}}(n)=T_{f_{g}^{-1}}(s(m))=d^{2}2^{\frac{s(m)}{2}}.\]
* _The complexity of obtaining the_ \(d^{2}m\)_-bit_ \(f_{g}^{-1}\) _element is determined as_ \[T_{f_{g}^{-1}}(m)=d^{2}2^{\frac{m}{2}}.\]  
_In general, the complexity of obtaining the \(\text{km-bit}\ f_{g}^{\prime-1}\) element for a given non-easily invertible map \(f_{g}^{\prime}:\mathcal{S}\to\mathcal{S}\) is described by \(T_{f_{g}^{\prime-1}}(n)=T_{f_{g}^{-1}}(s(m))\), and that of the \(\text{km-bit}\ f_{g}^{\prime-1}\) element is \(T_{f_{g}^{\prime-1}}(m)\). We can compare not only the complexity but also the order of complexity of the two using the map \(h_{f_{g}^{\prime-1}}\) defined above, as per the following Remark 3._  
**Remark 3**.