# New Subclass Framework and Concrete Examples of Strongly Asymmetric Public Key Agreement
## 1 Introduction
### Research Concept and Goals  
We define a function \(C:\mathbb{N}\times\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}\), which shows the computational costs for \(T\) calculation steps by a device with efficiency \(E_{D}\) (calculation steps per time) as follows:  
\[C(T,E_{D}):=T\times\frac{1}{E_{D}}.\]  
Thus, it is given by time (s, ms, or another unit). Let \(PKA\) be a set of PKA algorithms and let \(T_{A}:PKA\times\mathbb{N}\rightarrow\mathbb{N}\) be a function that shows the calculation steps required for [PERSON] to calculate the \(N\in\mathbb{N}\) bit length of the secret shared key (SSK) of an algorithm \(Alg\in PKA\), which increases monotonically for \(N\). \(T_{B}\) denotes the calculation steps for [PERSON] in a similar manner. Next, we consider \(Alg\in PKA\), which has the following relation:  
\[T_{A}(Alg,N)=T_{B}(Alg,N) \tag{1}\]  
for all \(N\in\mathbb{N}\). In this case, we suppose that the maximal computational cost that [PERSON] is allowed to incur for the SSK calculation of \(Alg\in PKA\), denoted by \(C^{B}_{max}\), is \(C^{B}_{max}=T_{B}(Alg,N_{0})\times\frac{1}{E_{B}}\), which is achieved when the bit length of the SSK is some \(N_{0}\in\mathbb{N}\), and [PERSON] can compute their SSK for all \(N\in\mathbb{N}\) to satisfy:  
\[C_{B}(T_{B}(Alg,N),E_{B}):=T_{B}(Alg,N)\times\frac{1}{E_{B}}\leq C^{B}_{max}: =T_{B}(Alg,N_{0})\times\frac{1}{E_{B}}, \tag{2}\]  
where \(E_{B}\) denotes the device efficiency of [PERSON]. As \(T_{B}\) is a monotonically increasing function for the bit size of the SSK, condition (2) is reduced to \(N\leq N_{0}\). Furthermore, if \(E_{B}=E_{A}\), where \(E_{A}\) denotes the device efficiency of [PERSON], the SSK can be computed for all \(N\leq N_{0}\). Thus, [PERSON] and [PERSON] can calculate the SSK of a bit size that is equal to or less than \(N_{0}\) within time \(C^{B}_{max}\). For the same \(Alg\in PKA\), we assume that \(E_{B}>E_{A}\) and the maximal computational cost that [PERSON] is allowed to incur \(C^{B}_{max}\) is the same as (2), where \(N_{0}\) is the smallest bit size of SSK to maintain security. Let [PERSON]'s computational cost for calculating her SSK of \(N\in\mathbb{N}\) bits be \(C_{A}(T_{A}(Alg,N),E_{A}):=T_{A}(Alg,N)\times\frac{1}{E_{A}}\), and if [PERSON] needs to calculate her SSK within the cost \(C^{B}_{max}\) as well as [PERSON], the following relation must be satisfied:  
\[C_{A}(T_{A}(Alg,N),E_{A})=T_{A}(Alg,N)\times\frac{1}{E_{A}}\leq C^{B}_{max}=T_ {B}(Alg,N_{0})\times\frac{1}{E_{B}},\]  
where the equality holds for some \(N_{1}\in\mathbb{N}\), but in this case, \(N_{1}<N_{0}\) must be satisfied because \(T_{A}\) is a monotonically increasing function for the bit size of the SSK and \(\frac{1}{E_{B}}<\frac{1}{E_{A}}\). This observation indicates that they must either use an \(N_{1}\)-bit SSK, which is obviously less secure than when using an \(N_{0}\)-bit SSK, or let [PERSON] incur a cost of \(T_{A}(Alg,N_{0})\times\frac{1}{E_{A}}\), which is larger than \(C^{B}_{max}\). Most PKA algorithms, including the DH algorithm, satisfy (1), and there are many cases in which \(E_{B}>E_{A}\) in modern society where IoT techniques are continually being developed; thus, this situation is inevitable in the near future, if not immediate. We consider determining an algorithm denoted by \(Alg_{A<B}\in PKA\), where  
\[T_{A}(Alg_{A<B},N)<T_{B}(Alg_{A<B},N) \tag{3}\]  
being satisfied for all \(N\in\mathbb{N}\) is one solution to the above undesirable situation. We denote the maximal computational cost that [PERSON] is allowed to incur as \(C^{B}_{max,A<B}\), which is defined as \(C^{B}_{max,A<B}:=T_{B}(Alg_{A<B},N_{0})\times\frac{1}{E_{B}}\), where \(N_{0}\in\mathbb{N}\) and it is the smallest bit size of SSK to maintain security. In addition to the above \(T_{A}=T_{B}\) case, we suppose that both [PERSON] and [PERSON] must calculate her SSK within the maximal computational cost that [PERSON] is allowed to incur \(C^{B}_{max,A<B}\).