Computing the order of points on an elliptic curve modulo N is as difficult as factoring N

Computing the order of points on an elliptic curve modulo N is as difficult as factoring N

K e y w o r d s P u b l i c k e y cryptography, Elliptic curves, Costly computational problems.

Factoring integers and computing elliptic curve rational points

We conjecturally relate via a polynomial-time reduction, a subproblem of integer factoring to the problem of computing the MordellWeil group of an elliptic curve from a special family. This raises an interesting question about the growth of the height of the generators of the above group with respect to the discriminant of the elliptic curve. We gather numerical evidence to shed light on this b...

A Probabilistic Elliptic Curve Cryptosystem as Secure as Factoring

Equivalence of Counting the Number of Points on Elliptic Curve over the Ring Zn and Factoring n

1 I n t r o d u c t i o n Elliptic curves can be applied to public-key cryptosystems, and as such several schemes have been proposed [3, 4, 5, 6, 9, 11]. There are two typical elliptic curve cryptosystems: E1Gamal-type scheme [4, 11] and RSA-type schemes [3, 5, 6]. The security of the EIGamal-type elliptic curve cryptosystem is based on the difficulty of solving a discrete logarithm over ellipt...

ON THE ORDER OF a MODULO n, ON AVERAGE

Let a > 1 be an integer. Denote by la(n) the multiplicative order of a modulo integer n ≥ 1. We prove that there is a positive constant δ such that if x1−δ = o(y), then 1 y ∑