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Through this research paper, authors construct a public key cryptosystem which works in the finite algebraic extension field Fpn of the finite field Fp. The security of this system is based on difficulty of solving DLP in Fpn. The primitive polynomials are used in the construction of algebraic extension fields. In this system all users select commonly a primitive polynomial f (x) of degree n over the finite field Fp.The prime number p, the primitive polynomial f (x), encryption rule and decryption rule are given to the public, and all other features kept secret. In this system each character is treated as a polynomial of degree less than n over Fp. After the encryption the character divided into two parts. The first part is sent to the other and the second part is used for the decryption. In this system we use similar procedure of ElGamal Exchange Cryptosystem. But our system has used more parameters than the ElGamal Exchange Cryptosystem. Hence our proposed system is more secure than ElGamal Exchange PKC.

A design of public key Cryptosystem in an algebraic extension field over a finite field using the difficulty of solving DLP