New Discrete Logarithm Problem over Finite Field by applying One Way Hash Function and Public Key Cryptography SUNIL KUMAR KASHYAP, BIRENDRA KUMAR SHARMA 1 and AMITABH BANERJEE 2 Department of Mathematics Rungta College of Engineering and Technology, Chhattisgarh Swami Vivekanand Technical University, Bhilai, Chhattisgarh, 491024 (INDIA) 7sunilkumarkashyap@gmail.com 1 School of Studies in Mathematics, Pt. Ravishankar Shukla University, Raipur, Chhattisgarh, 492010 (INDIA) 2 Department of Mathematics, Govt. DBGA Post Graduate College, Pt. Ravishankar Shukla University, Raipur Raipur, Chhattisgarh 492001, (INDIA) ABSTRACT We found the new result as “New Discrete Logarithm Problem over Finite Field by applying One Way Hash Function and Public Key Cryptography” . In this paper, we found the distinct discrete logarithm problem as compare to the traditional exist discrete logarithm problem. We also applied the one way hash function in the sense of Cryptographic Applications. The main objective of this paper is to present a new theory of the Discrete Logarithm Problem over Finite Field by applying One Way Hash Function and design the New Public Key Cryptography, whose security is impossible to break, because w e claim that there are no algorithm are in the existence which can be compute the proposed new mathematical problem. 2000 AMS Mathematics Subject Classification Number: 94A60. Key Words: Discrete Logarithm Problem, Finite Field, One Way Hash Function, Public Key Cryptography. J. Comp. & Math. Sci. Vol. 1(2), 193-196 (2010). Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)