The Equivalence For Variations Of Diffie-Hellman Problem

It has a very important significance to solve the equivalence for variations ofDiffie-Hellman Problem for the improvement of cryptograpy. This research is basedon this point to extend. Firstly, the paper presents the background of cryptograpy, thedescription and application of cryptograpy, the feature of cryptograpy, researchmethods of cryptograpy and the current research status. Then this paper introducesthe definition of the discrete logarithms in finite fields and computationalDiffie-Hellman problem. We descibe several public key cryptosystems and digitalsignature and zero-knowledge protocols that are based on the computational difficultyof solving the discrete logarithm problem in finite fields.Furthermore, we shall brieflydiscuss the present state of algorithms to solve the discrete logarithm problem in finitefields.The Diffie-Hellman problem is a golden mine for cryptographic purposes and ismore and more studied.This problem is very closely related to the difficult ofcomputing the discrete logarithm problem over a cyclic group. We will show that allfour variations of computational Diffie-Hellman problem: square Diffie-Hellmanproblem, inverse Diffie-Hellman problem, divisible Diffie-Hellman problem andsquare root Diffie-Hellman problem, are equivalent.Also, we are consideringvariations of the decisional Diffie-Hellman problem.The computational Diffie-Hellman problem is the most well-known hardproblem and there are many variants of it. So it is a waste of time to show theequivalence for variations of Diffie-Hellman problem one by one. We will show theequivalence for other variations of Diffie-Hellman problem under a certain condition.It is a abstract and efficiency method. Later, given a definition of equivalence classbased on the discrete logarithms in finite fields and a statement. We are not able toprove or disprove this statement, thus leave an interesting open problem.Finally, the work is summarized and the direction of future research is discussed.