Probably this is a silly question but an anomalous curves and ordinary curves are the same things?


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Let $E$ be an elliptic curve defined over a field $K$. Let $n$ be a positive integer.

$$E[n] = \{P \in E(\overline{K}) \,|\, [n]P = \infty\}$$

Where $\overline{K}$ is the algebraic closure of $K$ and $E[n]$ include the point with coordinate $\overline{K}$, not just from $K$. The above is the set of $n$ torsion points on the curve, namely the points of finite order $n$. In Elliptic curves over a finite field, all points are torsion points.

  • An elliptic curve $E$ in characteristic $p$ is called ordinary curves if $ E[p] \backsimeq \mathbf{Z}/p\mathbf{Z}$
  • An elliptic curve $E$ in characteristic $p$ is called supersingular curves if $ E[p] \backsimeq 0$, and we know that those curves are not secure.
  • An elliptic curves $E$ over $F_q$ with $\#E(\mathbf{F}_q ) = q$ are called anomalous curves. This is suggested to use against the MOV attack. Unfortunately, the discrete log problem for group $E(\mathbf{F}_q )$ can be solved quickly.

Ref: A good textbook: Elliptic Curves Number Theory and Cryptography, Second Edition, by Lawrence Washington.


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