**1. Introduction**

Apart from a rich history of Message encryption, the cryptosystem became more popular in the twentieth century upon the evolution of information technology. Until the last part of the 1970s, all cryptographic message was sent by the symmetric key. This implies somebody who has sufficient data to encode messages likewise has enough data to decode messages. Consequently, the clients of the framework must have to impart the secret key furtively. As a result of an issue stealthily key sharing, Diffie and Hellman [1] developed a totally new sort of cryptosystem called public key cryptosystem.

In a Public key cryptosystem, both parties (in a two-party system) have a pair of public enciphering and secret deciphering keys [2, 3]. Any party can send encrypted messages to an assigned party using a public enciphering key. However, only the assigned party can decrypt the message utilizing their corresponding secret deciphering key [4]. After that various public key cryptosystems were introduced based on tricky mathematical problems. Among these, RSA is the longest reasonable use of cryptography. Since its design, in spite of all effort, it has not been broken yet. The security of the RSA is acknowledged to be established on the issue of the factorization of an enormous composite number. Be that as it may, there are some practical issues in RSA execution. The fundamental issue is the key arrangement time that is absurdly long for computationally restricted processors used in certain applications. Another issue is the size of the key. It was demonstrated that the time

required to factor an n-bit integer by *index calculus factorization* technique is of order 2*n*1*=*2þ*<sup>δ</sup>* , *δ*>0 [5]. In 1990's, J. Pollard [6] demonstrated that it was possible in time bounded by 2*n*1*=*3þ*<sup>δ</sup>* , *δ*>0. The reduction of the exponent of *n* has significant outcomes over the long run. It should likewise be expanded each year as a result of upgrades in the factorization calculations and computational power. Until 2015, it was prescribed the base size of the RSA key should be 1024 bits and subsequently increases to 4096 & 8192 bits by 2015 & 2025 respectively [7]. While trying to remedy these issues, Discrete logarithm problem (DLP) has been utilized (to reduce key setup time and size of the key).

Discrete logarithm problem (DLP) is a mathematical problem that occurs in many settings and it is hard to compute exponent in a known multiplicative group [8]. Diffie-Hellman [1], ElGamal [9], Digital Signature Algorithm [10], Elliptic curve cryptosystems [11, 12] are the schemes evolved under the Discrete logarithm algorithm. The security of Diffie-Hellman relied upon the complexity of solving the discrete logarithm problem. However, the scheme has some disadvantages. It has not been demonstrated that breaking the Diffie-Hellman key exchange has relied upon DLP and also the scheme is vulnerable to a man-in-the-middle attack. For the security aspect, cryptosystem [9] was proposed, to introduce a digital signature algorithm (DSA) that's primarily based on Diffie-Hellman DLP and key distribution scheme. It was demonstrated that DSA is around multiple times littler than the RSA signature and later DSA has been supplanted by the elliptic curves digital signature algorithm (ECDSA). Nonetheless, it has some practical implementation problems [13–15]. The length of the smallest signature is of 320 bits, which is still being too long for computationally restricted processors. Another issue emerged is as a correlation with RSA in a field with prime characteristics, which is forty times slower than RSA [16].

There are some other designs for public-key cryptosystems based on some extensive features of matrices. However, there were some practical implementation problems. Thus it had never achieved wide popularity in the cryptographic community. McElice [17] come up with a public key cryptosystem rooted on the Goppa codes Hamming metric. The scheme has the advantage that it has two to three orders of magnitude faster than RSA. Despite its advantage, it has some drawbacks. It was demonstrated that the length of the public key is 2<sup>19</sup> bits and the data expansion is too large. Some other extensions of the scheme can also be found in [18–20]. Unfortunately, the scheme & its variants has been broken in [21–23]. Later, Gabidulin [24] come up with the rank metric & the Gabidulin codes over a finite field with *<sup>q</sup>* elements, where *<sup>q</sup>* <sup>¼</sup> *<sup>p</sup><sup>r</sup>* i.e. **<sup>F</sup>***q*, as an alternative for the Hamming metric. The efficiency of the scheme relied on same set of parameters and the complexity of the decoding algorithm for random codes in rank metric is tons higher than the Hamming metric [17, 25–27]. Numerous fruitful attacks were utilized on the structure of the public code [28–30]. To prevent these attacks, numerous alterations of the cryptosystems were made, consequently drastically increases the size of the key [31–33]. Lau and Tan [34] proposed new encryption with a public key matrix by considering the addition of a random distortion matrix over **F***<sup>q</sup>* of full column rank *n*. There are also many other design on matrices, which are not cited here, but none of them gain wide popularity in the cryptographic community due to lack of efficient implementation problems in one and another way.

Thinking about these inadequacies, it would be desirable to have a cryptosystem dependent on other than the presumptions as of now being used. Thus, we propose a cyclotomy asymmetric cryptosystem (CAC) based on strong assumptions of DLP that have to reduce the key size and faster the computational process.
