Construction of Elliptic Curves over Finite Fields with a Point of Given Order

Abstract

The elliptic curves over a finite field with q elements are constructed. Let l be a prime, it is proved in this paper that if the equation U2-D(x)V2=e(x-a)l defined over GF(q)[x] has a primitive solution over GF(q)[x], where D(x)∈GF(q)[x] is a monic squarefree degree three polynomial, then the elliptic curve y2=D(x) has a point (a,b) with order l. This result provides an algorithm on constructing elliptic curves with a point of the prescribed order.