The main objective of this paper is to propose a new theory of cryptography based on the problem of solving the discrete logarithm problem under the definite integral calculus in the multiplicative group of the finite field. This is the first work in the cryptographic field, where we found the mathematical relation between the discreteness and continuous mathematics. This is very difficult to solve, because three distinct discrete logarithm problems will be solved at the same time under the definite integral in the multiplicative group of the finite field. This is the motivation for giving the new public key cryptography. Thus, we used this hard mathematical problem for designing a secure public key cryptosystem, because we strongly claim that there are no algorithms and methods in the cryptographic world for solving the proposed mathematical problem. This public key cryptosystem is also very practical in various cryptographic applications, because the encryption and decryption algorithm is very simple with small key length.

In this paper, we propose the discrete logarithm problem under definite integral calculus in the multiplicative group of the finite field. We also design a public key cryptosystem, whose security is based on the problem of solving the discrete logarithm problem under the definite integral in the multiplicative group of the finite field. We claim that there are no algorithms in existence for solving this discrete logarithm problem under the definite integral calculus in the multiplicative group of the finite field. To solve the proposed problem, one has to solve the three distinct discrete logarithm problems at the same time, which is not simple; therefore, this makes the proposed public key cryptosystem more secure as compared to the traditional discrete logarithm problem-based public key cryptosystems. This research is based on the works of Diffie and Hellman (1976), Pohling and Hellman (1978), Elgamal (1985), Stinson (1995), and Kashyap et al. (2006).

Discrete logarithm problem was first used for providing the security of public key cryptography by Diffie and Hellman (1976). Elgamal (1985) presented the first secure and practical public key cryptosystem, whose security was based on the problem of solving the discrete logarithm problem in the multiplicative group of the finite field. Kashyap et al. (2006) gave another advanced secure and efficient public key cryptosystem, whose security is based on some variants of the discrete logarithm problem in the multiplicative group of the finite field. The mathematical structure of the discrete logarithm problem in the multiplicative group of the finite field is defined.

Computer Sciences Journal, Theory of Cryptography, Discrete Logarithm Problem , Definite Integral Calculus, Public key Cryptography, Discrete Logarithm Problem, Multiplicative Group, Encryption and Decryption, Index-Calculus Algorithm, Literature Review.