For elliptic-curve-based protocols, it is assumed that finding the discrete logarithm of a random elliptic curve element with respect to a publicly known base point is infeasible: this is the "elliptic curve discrete logarithm problem" (ECDLP). The security of elliptic curve cryptography depends on the ability to compute a point multiplication and the inability to compute the multiplicand given the original and product points. The size of the elliptic curve determines the difficulty of the problem.
We can formally define an elliptic curve (over a field k) is a smooth projective curve of genus 1 (defined over k) with a distinguished (k-rational) point. However, not every smooth projective curve of genus 1 corresponds to an elliptic curve, it needs to have at least one rational point. The definition of the elliptic curve also requires that the curve be non-singular. Geometrically, this means that the graph has no cusps, self-intersections, or isolated points.
There is an infinite number of elliptic curves, but a small number that is used in elliptic curve cryptography (ECC), and these special curves have names. Apparently there are no hard and fast rules for how the names are chosen, but there are patterns. IETF has published a report on the list of alternate elliptic curve representations recently. You can refer to the IETF report here. Following are some of the reference elliptic curve families. A few of them are missing in the diagram.
- Curves in short Weierstrass form
- Montgomery Curves
- Twisted Edwards Curves
- Supersingular Edwards Curve
- Hessian Curve
- Twisted Hessian Curve
- Jacobi Quartic Curve
- Doubling-oriented Doche–Icart–Kohel Curve
Elliptic Curves were originally written in Weierstrass form. Edwards elliptic curves have their own advantages: addition, doubling and tripling can be done faster on Edwards curves than on curves given by a Weierstrass equation. This is because the addition law on Edwards curves does not have exceptions, while the addition on Weierstrass curves
distinguishes several special cases. The Montgomery curve is suggested for application in elliptic curve cryptography because arithmetic in this curve representation is faster and needs less memory than arithmetic in standard Weierstrass form.
Please find an insightful article on the names of some the prominent elliptic curves used in cryptography.