 
Summary: CONSTRUCTING ELLIPTIC CURVES WITH A KNOWN NUMBER OF
POINTS OVER A PRIME FIELD
AMOD AGASHE, KRISTIN LAUTER, AND RAMARATHNAM VENKATESAN
Abstract. Elliptic curves with a known number of points over a given prime field Fn
are often needed for use in cryptography. In the context of primality proving, Atkin
and Morain suggested the use of the theory of complex multiplication to construct such
curves. One of the steps in this method is the calculation of a root modulo n of the
Hilbert class polynomial HD (X) for a fundamental discriminant D. The usual way of
doing this calculation is to first compute HD (X) over the integers and then to find
the root modulo n. We present a modified version of the Chinese remainder theorem to
compute HD (X) modulo n directly from the knowledge of HD (X) modulo enough small
primes. Our heuristic complexity analysis suggests that asymptotically our algorithm is
an improvement over previously known methods.
1. Introduction
In order to use elliptic curves in cryptography, one often needs to construct elliptic
curves with a known number of points over a given prime field. One way of doing this
is to randomly pick elliptic curves and then to count the number of points on the curve
over the prime field, repeating this until the desired number of points is found. Atkin and
Morain [AtMor] pointed out that instead, one can use the theory of complex multiplication
to construct elliptic curves with a known number of points. Although at present it may still
