# What is the difference between anomalous elliptic curve and ordinary elliptic curve?

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

## 1 Answer

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.