# New Subclass Framework and Concrete Examples of Strongly Asymmetric Public Key Agreement
## 1 Introduction
### Research Concept and Goals
In this case, [PERSON] can calculate all \(N\) bits of the SSK to satisfy  
\[C_{A}(T_{A}(Alg_{A<B},N),E_{A})=T_{A}(Alg_{A<B},N)\times\frac{1}{E_{A}}\leq C ^{B}_{max,A<B}=T_{B}(Alg_{A<B},N_{0})\times\frac{1}{E_{B}}.\]  
The equality holds when \(T_{A}(Alg_{A<B},N_{1})\times\frac{1}{E_{A}}=T_{B}(Alg_{A<B},N_{0})\times\frac {1}{E_{B}}\) holds for some \(N_{1}\in\mathbb{N}\). In this case, it should be noted that \(N_{1}=N_{0}\) is achieved; that is, [PERSON] calculating the SSK of \(N_{0}\) bits within time \(C^{B}_{max,A<B}\) is possible, provided that  
\[\frac{T_{A}(Alg_{A<B},N_{0})}{T_{B}(Alg_{A<B},N_{0})}\leq\frac{E_{A}}{E_{B}} \tag{4}\]  
holds, which is impossible when (1), because in this case, the left-hand side is equal to 1, but the right-hand side is less than 1. As \(\frac{E_{A}}{E_{B}}\) is given (we may say that \(\frac{E_{A}}{E_{B}}\) is a communication environment in which the algorithm is used), (4) is not always achieved for some \(Alg_{A<B}\in PKA\) and \(N_{0}\in\mathbb{N}\). Conversely, we can determine the minimal environment \(\frac{E_{A}}{E_{B}}\) where [PERSON] and [PERSON] can calculate \(N_{0}\) bits SSK within \(C^{B}_{max,A<B}\) time using \(Alg_{A<B}\in PKA\) by simply calculating the left-hand side of (4). Based on the above considerations, our research goals are as follows:  
1. Constructions of \(Alg_{A<B}\in PKA\). 2. For \(Alg_{A<B}\in PKA\), the determination of \(T_{A}(Alg_{A<B},N^{\prime}_{0})\) and \(T_{B}(Alg_{A<B},N^{\prime}_{0})\) for any \(N^{\prime}_{0}\in\mathbb{N}\). 3. The construction of the PKA class to which \(Alg_{A<B}\in PKA\) belongs and the introduction of conditions for PKA algorithms to be members of this class. As mentioned above, goal 2 provides a lower bound of \(\frac{E_{A}}{E_{B}}\) to calculate the SSK of \(N_{0}\) bits within time \(C^{B}_{max,A<B}\). Goal 3 provides instructions on how to construct PKA algorithms to possess the relation (3). Thus, improving algorithms such as those of [9,10] to possess this property may be possible by attempting to fix their parameters according to the class conditions. We do not attempt to improve these algorithms in this study, but this subject is worthy of consideration and will be one of our most important future works. We consider that these goals are achievable by fully utilizing the characteristics of the PKA framework known as strongly asymmetric public key agreement (SAPKA) [18]. The characteristics, high level of generality, and asymmetry of the key agreement process of SAPKA are explained in Section 1.2, along with its definition, and concrete methods that are derived from the characteristics are explained in Section 2. Note that this study is not focused on how to construct secure PKA algorithms against any types of theoretical attacks; rather, it investigates how to reduce Alice's computational complexity while maintaining the security of one given PKA algorithm. Our main theorems (in Section 5) do not provide any instructions on how to enhance the security of PKA algorithms, and resilience against attack such as man-in-the-middle (MITM) attack is not discussed in this paper (we consider that these topics should be discussed after existence of \(Alg_{A<B}\) is proven and mentioned in Section 7 ).