Constructing Supersingular Elliptic Curves Depending on the Coefficients of Weierstrass Equation

Abstract

In the use of elliptic curves in its applications (especially in cryptography),
one often needs to construct elliptic curves with a known type (number of
points) over a given finite field Fp, where some types are more secure than
others. So we introduce a simple way to construct a supersingular elliptic curve
by computing the coefficient of xp-1 in f(x)(p-1)/2 where
f(x)= x3
+ax+b. And we give an algorithm that compute that coefficient and
construct the required curve.