# New Subclass Framework and Concrete Examples of Strongly Asymmetric Public Key Agreement
## 1 Introduction  
Since [PERSON] proposed the concept of \"perfect secrecy\" in crypto-systems [1], in which he introduced a theoretically unbreakable system even against computational power, the distribution of secret keys between the sender ([PERSON]) and receiver ([PERSON]) via an insecure channel has been one of the greatest problems in cryptography.  
The Diffie-Hellman (DH) public key agreement (PKA) protocol that was proposed in 1976 [2] and the Rivest-Shamir-Adleman (RSA) crypto-system that was presented in 1978 [3] represented the most significant works in the area of cryptography, and it was previously believed that the problem of key distribution had been resolved.  
However, recent considerable developments in the computational power of eavesdroppers have introduced several potential (even if not immediate) threats against standard PKA algorithms and public key cryptographies, particularly for small key lengths [4]. To maintain security, users have been forced to select longer keys, and the increased key length has led to higher computational costs. Thus, the preparation of secure communication infrastructure, particularly for devices with limited memory and computational power, has become challenging. Furthermore, the threat of quantum computers that are currently under development and [PERSON]'s algorithm [5] cannot be underestimated.  
Considering the demand for algorithms that are resilient against any type of theoretical attack, including quantum algorithm-based attacks, the development and study of new PKA algorithms and public key cryptographies, namely post-quantum cryptography(PQC), has become widespread. PKAs and public key cryptographies based on lattice problems such as the shortest vector problem (SVP), closest vector problem, and learning with errors (LWE) are among the most well known methods. Among these, SVP-based PKA and public key cryptographies, including NTRU prime [6], NTRU-HRSS-KEM [7], and NTRU Encrypt [8], module LWE-based PKA such as CRYSTALS-Kyber [9], and ring LWE-based PKA including NewHope [10] are leading approaches in this research area and have been considered as candidates for the NIST (National Institute of Standards and Technology) standardization of PQC systems [11; 12]. When the parameters are properly selected, the above algorithms are considered to be resilient against attacks that use quantum computers and sufficiently computationally efficient to be used in practice.  
However, there has been substantial discussion regarding the security of such algorithms. For example, in certain LWE-based algorithms, even if sufficiently large parameters are selected, a possible weakness has been observed [13; 14; 15]. Moreover, the notion that the difficulty in solving ring LWE is equivalent to that of solving the LWE (the difficulty of LWE is discussed in [14; 16]) has not yet been proven; thus, other cases of weakened security [9; 10] may arise. Weak parameters for NTRU-type PKA algorithms and public key cryptography are also reported in [17]. Owing to these uncertainties in the parameter settings to maintain security even in an ideal situation (i.e., without assuming limited memory and computational power), the preparation of secure communication infrastructure with these new-generation algorithms for less capable devices has resulted in greater difficulty and insecurity. Although there is no doubt that these algorithms will offer significant benefits even after the post-quantum computer era, security analysis of these algorithms should continue until users can be provided with a \"guide\" that explains how to set parameters to ensure secure PKA and public key cryptography according to the needs and environments of users.