A Research On The Diophantine Equation And The Elliptic Curve Related Computing Problems

The Diophantine equation is the oldest branch in number theory, whose content is extremely abundant, and it has close connections with the algebraic number theory, the algebraic geometry, the combinatorics and so on. In the recent30years, the number theory cryptography has developed too much in the modern cryptography, the information encoding theory and computer science, especially in number theory cryptography.The pubic key cryptography based on the discrete logarithm problem(DLP) of Elliptic curve abelian group is hottest right now. With quantum computers becoming the topic, of more and more people, the task that how to design a new cryptosystem to resist the attack of quantum computers more and more receives takes. As well as a research tool than the abstract elliptic curve, elliptic curve isogenies itself can also be used by cryptography. Isogenies computation between the elliptic curves given can be used to construct new cryptography modules and cryptosystems, in particular the problem seems to be hard for solving with a quantum computer.The paper focuses on the solving of a class of Diophantine equations and the elliptic curve related computing problems.The main study objects include:Benjamin and weger proved that x=59is the largest positive integer for which the fouthpowerfree part of x2+2is at most100. In this paper we prove two results about the Diophantine equation x2+2=Dy4, including that when D is even,the least solution of the equation is explicit,what’s more,for D is odd we can get the structure of solutions and find out that a class of the equations x2+2=Dy4have no solutions effectively. Our proof is very elementary and not complicated. Chen jianhua perfectly solve the condition of D=3. In the paper, let D=5, we completely solve the equations, which have only two non-trivial integer solutions x=49, y=10, z=3, p=11and x=485, y=99, z=70, p=2.The proof utilizes elementary analysis combining some properties of quadratic and quartic Diophantine to get the structure of the solutions, A.Baker’theory to estimate the upper bound of solutions and LLL algorithm to reduce the upper bound.About the elliptic curve related computing problems, at first the author gives a new proof to the Hasse theorem. Hasse theorem has an important application in cryptography. Many engineers make use of Hasse theorem but cannot totally understand the proof. So it is important and quite interesting to give an elementary proof. Then from mathematic theory, the paper research the computing of isogeny between elliptic curves. The author gives a new explicit and effective algorithm, which reach the superlative complexity right now. The conclusions and produces in the thesis are very valuable to the applications of elliptic curve isogenies cryptosystems.