Factoring Integers Using Elliptic Curves over Q
نویسندگان
چکیده
For the integer D = pq of the product of two distinct odd primes, we construct an elliptic curve E2rD : y 2 = x3 − 2rDx over Q, where r is a parameter dependent on the classes of p and q modulo 8, and show, under Birch and Swinnerton-Dyer conjecture, that the elliptic curve has rank one and vp(x([k]Q)) 6= vq(x([k]Q)) for odd k and a generator Q of the free part of E2rD(Q). Thus we can get p or q from D by computing GCD(D, x([k]Q)). Furthermore, under the Generalized Riemann hypothesis, we prove that one can take r < c logD such that the elliptic curve E2rD has these properties, where c is an absolute constant.
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