Elliptic Curves Over Number Fields And Integer Factoring

The paper consists of two parts. One part is on elliptic curve defined over ratio-nal number field and integer factoring; the other part is on elliptic curve defined over imaginary quadratic number fields and integer factoring.Firstly, we study elliptic curve E2rD: y2=x3-2rDx and its dual curve E’2rD:y2=x3+8rDx with D is a product of two distinct odd primes p and q, which are defined over rational number field. Under the Parity Conjecture, by choosing parameter r∈Z, we construct infinitely many elliptic curves E2rD, which rank is equal to one and the Φ-part of the Shafarevich-Tate group TS(E’2rD/Q) is trivial, tat is TS(E’2rD/Q)[Φ]={0} and prove that vp(x([k]Q))≠vq(x([k]Q)) with Q is the generator of the free part of Mordell Weil group E2rD(Q) for our constructed elliptic curves E2rD and k is any odd integer. Furthermore, under the generalized Riemann Hypothesis, the parameter r can be chosen such that r≤c log4D, where c is an absolute constant. As an application, we can recover p or q by computing the x-coordinate of nontrivial rational point.At last, we want to extend the integer factoring method using elliptic curve over rational number field to imaginary quadratic number field K, that is, we try to establish connection between integer factoring and computing K-rational points of infinity order of elliptic curve. As an attempt, we firstly consider establishing connection between rational prime factoring and computing K-rational points of infinity order of elliptic curve. On this basis, we study elliptic curve Ep:y2=x3+px and its dual curve E’p:y2=x3-4px with p rational odd prime over K=Q((?)), Q((?)), Q((?)), Q((?)), where q is congruent to3modulo8and gcd(p,q)=1, and we determine the explicit structure of S (Φ)(EP/K) and S(?)(E/K)), according to the ramification of p over K. Specially, when K=Q((?)),p=7,11mod16and (p/7)=1. If the rank of Ep/K is equal to one, then v(x[Q])≠v(a[Q]), where v∈MQ((?)),v|p and Q the generator of the free part of Mordell-Weil group EP(Q((?))).