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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?

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Every elliptic curve is non-singular. Some elliptic curves are supersingular.

A quote from "The Arithmetic of Elliptic Curves" by Silverman, Remark V.3.2.2:

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".

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  • $\begingroup$ It means all non-singular curve is not supersingular? $\endgroup$
    – myat
    Commented Jan 31, 2017 at 12:09
  • $\begingroup$ You beat me to it. I knew I had read a remark about that terminology somewhere... $\endgroup$
    – yyyyyyy
    Commented Jan 31, 2017 at 12:11
  • $\begingroup$ @CurveEnthusiast. In some performance evaluation, they describe two using supersingular and another with non-singular. I don't know they are same or not. $\endgroup$
    – myat
    Commented Jan 31, 2017 at 12:13
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    $\begingroup$ @myat Read my statement carefully. An elliptic curve is non-singular by definition. There do not exist non-singular elliptic curves. Perhaps they meant non-supersingular, and just made a typo. $\endgroup$
    – CurveEnthusiast
    Commented Jan 31, 2017 at 12:29
  • $\begingroup$ @CurveEnthusiast I have difficulties with the singular definition. E.g. an Edwards curve have singularities if I embed it in the projective plane. Should I just have to look for singularities in the affine plane ? $\endgroup$
    – Ruggero
    Commented Jan 31, 2017 at 12:48

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