In 1976, Diffie and Hellman discovered a theory of Public Key Cryptography (PKC) as the application of Discrete Logarithm Problem (DLP). They presented a Key Exchange Protocol (KEP). DLP was first explained by the Prince of Mathematics, Carl Fredrick Johannan Gauss. Taher ElGamal was the first person to design a public key cryptosystem whose security was based on DLP. Basically, this DLP involves a very special type of setting mathematically. DLP is only an element which can be represented in index form. But this element should be a member of an appropriate group. For cryptographic example, DLP is the member of a multiplicative group of the finite field in the basic cryptosystem. Till now, several different group settings have come into existence. Against this backdrop, we propose a new type of DLP, which is found in Cardon and Murty’s research work (Ram Murty, 1998 and 1999), and also PKC based on the proposed DLP.

In our opinion, cryptography provides the required physical security to thoughts. Physical security can be classified into two types: electronic and non-electronic. In this paper, we present only the first type.

2. Finite Field

A finite field is a field with a finite field order (i.e., number of elements), also called as Galois field. The order of a finite field is always a prime (p) or a power of a prime (p). For each prime power, there exists exactly one up to an isomorphism finite field GF(p), often written as pn , F or Fq where q=pn (in the current paper). For a finite set X, |X| denotes its cardinality. By f << g for x eX, or f = O(g) for x e X, where X is an arbitrary set on which f is defined, we mean that synonymously there exists a constant C  0 such that |f(x)|<= Cg(x) for all xe X. The ‘implied constant’ is any admissible value of C. It may depend on the set X which is always specified or clear in the context. We use elementary scheme-theoretic language for our algebraic geometry.

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