# Supersingular vs non-singular (Same or different)

I have a frustration for the term "supersingular" elliptic curve and "non-singular" elliptic curve.

In some paper, performance evaluation is done using supersingular elliptic curve and in some cases, they use non-singular elliptic curve. Are they same or different curves?

Do not confuse the notions of singularity and supersingularity. A supersingular elliptic curve is, by definition, an elliptic curve, so it is nonsingular. The origin of the potentially confusing terminology is as follows. Historically, elliptic curves defined over $\mathbb{C}$ whose endomorphism rings are larger than $\mathbb{Z}$ were called singular, where "singular" was used in the sense of "unusual" or "rare". However, in this sense, all elliptic curves defined over $\bar{\mathbb{F}}_p$ are singular! The endomorphism rings of most elliptic curves over $\bar{\mathbb{F}}_p$ are order in imaginary quadratic fields. It is only the rare and unusual curve whose endomorphism ring is an order in a quaternion algebra, whence the term "supersingular".