# New Subclass Framework and Concrete Examples of Strongly Asymmetric Public Key Agreement
## 1 Introduction
### SAPKA Framework  
We provide a brief definition of SAPKA (the explicit definition is presented in Section 2.4) and its characteristics in this section.  
First, [PERSON] prepares a multiplicative semi-group \(\mathcal{S}\) with 1. Subsequently, he selects five maps:  
\[x_{1},x_{2},x_{3},x_{4},N_{1}:\mathcal{S}\rightarrow\mathcal{S},\]  
where \(N_{1}\) must be an easily invertible map. In this case, \"easily\" means that the calculation of \(N_{1}^{-1}\circ N_{1}(y)\) for all \(y\in\mathcal{S}\) can be performed in polynomial time. Furthermore, \(x_{1},x_{2},x_{3}\), and \(x_{4}\) must satisfy the following equation, which is known as the compatibility condition:  
\[x_{1}\circ x_{2}(y)=x_{3}\circ x_{4}(y) \tag{5}\]  
for all \(y\in\mathcal{S}\), where \(\circ\) denotes the map composition. Equation (5) is a condition for [PERSON] and [PERSON] to calculate the same SSK (see the key agreement process in Figure 1). The key agreement process of SAPKA can be described as in Figure 1, and every secret/public key is displayed in Table 1.  
Figure 1: Key agreement process of SAPKA.  
As can be observed from Figure 1 and Table 1, [PERSON]'s public keys are described by the map compositions and not by the element of \(\mathcal{S}\). Sending a map means sending the calculation rule of the map in combination with a set of parameters, which is the domain of the map. Thus, [PERSON] simply follows the rules of \(y_{B,1}\) and \(y_{B,2}\) to calculate \(y_{A}\) and \(\kappa_{A}\), and to calculate these, she must first receive \(y_{B,1}\) and \(y_{B,2}\) from [PERSON]. Regardless of the \(x_{A}\) that [PERSON] selects (provided that \(x_{A}\in\mathcal{S}\)), the equality of \(\kappa_{A}\) and \(\kappa_{B}\) holds, because the compatibility condition (5) holds for all elements of \(\mathcal{S}\). The generality mentioned above arises from the fact that there are only several restrictions for the secret keys of [PERSON], namely \(x_{1},x_{2},x_{3},x_{4},N_{1}\), and semi-group \(\mathcal{S}\). As the restrictions are only those in (5) and invertible regarding \(N_{1}\), [PERSON] has substantial freedom in terms of the choices of these maps and the algebraic structure. By fixing these maps and \(\mathcal{S}\) concretely, various PKA algorithms can be described, including the most well known of these, namely the DH algorithm (presented in [18]). In this study, we do not attempt to describe new-generation algorithms such as [7; 9; 10] in the form of SAPKA. However, we are optimistic that these can be described because \(\mathcal{S}\) can be selected as not only scalars but also matrices, for example, with numerous options for \(x_{1},x_{2},x_{3},x_{4},N_{1}\).  
Another notable characteristic of SAPKA is the asymmetry of the key agreement process. In this case, the asymmetry means that the number of public keys calculated by [PERSON] and [PERSON] differ, and thus, the two perform essentially different operations. Owing to this characteristic, an eavesdropper ([PERSON]) must attempt attacks against a maximum of two public keys to obtain the secret information of either [PERSON] or [PERSON]. This may allow [PERSON] to select her secret key from a set of small bit sizes and to reduce her computational complexity in certain cases. In Section 2, we explain [PERSON]'s strategies for recovering the SSK from public keys, an observation from her strategies, and the research method derived from this observation.