# Elliptic Curves

Generally speaking, elliptic curves are geometric objects in [projective planes](https://allfi.gitbook.io/zk-notes/algebra/field/projective-planes) over a given [field](https://allfi.gitbook.io/zk-notes/algebra/field), made up of points that satisfy certain equations. One of their key features from the point of view of cryptography is that, if the underlying field is of positive characteristic, elliptic curves are finite, cyclic [groups](https://allfi.gitbook.io/zk-notes/algebra/group). Further, it is believed that, in this case, [the Discrete Logarithm Problem](https://allfi.gitbook.io/zk-notes/cryptography/assumptions/discrete-logarithm-assumption) on many elliptic curve groups is hard, given that the underlying characteristic is large enough.
