Study Of Supersingular Isogeny Graphs And Congruent Equations

In this thesis,we study the problems of supersingular isogeny graphs and counting the solutions of congruent equation ?1x1k1+…+?txtkt?c mod n.In Chapter 1,we first briefly review the backgrounds and known results of su-persingular isogeny graphs,and introduce our results about the local information at Fp-vertices in isogeny graph.Next,we recall the problem of counting solutions of congruent equations,and introduce our main results.In Chapter 2,we recall some basic theories on number theory,elliptic curves and complex multiplication,and Deuring's correspondence.In Chapter 3,we introduce the definition of supersingular elliptic curve isogeny graph gl(Fp)and two key exchange protocols based on isogenies.Moreover,we intro-duce some algorithms for computing isogenies and their computation complexities.In Chapter 4,we introduce some relevant properties of supersingular isogeny graphs,and prove two main theorems.To study the number of neighbors of Fp-vertices in isogeny graph gl(Fp)and the edges of Fp-isogeny,we use the descriptions of en-domorphism rings of supersingular ellptic curves over Fp by Ibukiyama,and combine Deuring's correspondence,we reduce the problems that we concerned to the problems of some Diophantine equations.In this way,we prove our first theorem.Then we prove our second theorem by using some basic theories of class field theory and Chebotarev density theorem.These two theorems play important roles in the study of isogeny graphs.In Chapter 5,we concern the solutions of congruent equations.First,we introduce some preliminaries.Then we prove our three main theorems.As the applications of our results,we give an effective algorithm for counting the solutions of congruent equations,and we describe the number of solutions more precisely in some special cases.Our results in this part extend greatly and contain the known results about these problems.In the last chapter,we make a summary and prospect of the two problems studied in this paper.