EC Cryptography Tutorials - Herong's Tutorial Examples - v1.00, by Dr. Herong Yang
Elliptic Curves with Singularities
This section describes elliptic curves with singularities where curves are not smooth.
If we review the elliptic curve equation again, we will see that for certain combination of coefficients a and b, the elliptic curve will a singularity point where the curve is not smooth.
y2 = x3 + ax + b
Here are two examples of elliptic curves with singularities (source: andrea.corbellini.name):
To exclude curves with singularities, we need to add an extra condition on coefficients a and b:
y2 = x3 + ax + b 4a3 + 27b2 != 0
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