Parametrized System Of Diophantine Equations And Key Agreement Scheme

Solving Diophantine equation is an ancient and elegant problem in number theory, which has been properly applied in vital field of information security. In this thesis, we will discuss the parametrized Diophantine equation and key agreement.In Chapter4, we will consider solving the system of Diophantine equation for p is prime, a=c2-1, c=8s+5, s∈Z. We also obtain all of the integral points when fixing one parameter a for a parametrized Diophantine equation system.In Chapter5, we solve the parametrized Diophantine equation system where a, b are integers. Furthermore, we list all of the solutions for1≤a≤10,1≤b≤10.In Chapter6, we propose a method to solve the parametrized Diophantine equation system where a, b, c, A,B,C∈Z, aAC≠0. Moreover, computable examples are given to show the effectiveness of this method.Recently, the development of cryptography lead to wide application of number theory into information security. In Chapter7, from a view of application of number theory, a key agreement scheme with constant round of communication, XTR-CR, is designed and proposed, which is not based on bilinear pair of elliptic curves. This scheme is shown to be effective in communication and computation, as well as security.Generally, there are two kinds of methods to solve the Pell equation system; the elementary number theory method and logarithm linear form method developed by A. Baker. We mainly use classic algebraic number theory method and Skolem p-adic analysis to solve(5), and use Skolem p-adic analysis again to combine with formal group method to solve (6). Basicly, the prerequisite of using Skolem p-adic analysis is the fast computation of fundamental unit of number field. Henceforce, we discuss the LLL algorithm in Chapter3, so as to implement the fast computation of fundamental unit in total quartic field. On the other hand, before using formal group method to solve Diophantine equation based on elliptic curves, the computation of rational point group E(K), should be done. Therefore, we use2-descent method and professional math software, Magma, to implement it. Finally, the proposed key agreement schemd in this thesis, XTR-CR, is built on the basis of XTR cryptography, and we obtain its security proof under the XTR-DDH hypothesis.