# Part 1: What Are Elliptic Curves ? As stated in its name, elliptic curve cryptography (ECC) is based on a particular kind of curve called an elliptic curve. The aim of this first post on ECC is to explain what elliptic curves are. ## Definition of elliptic curves An elliptic curve is an abstract mathematical object generally defined as **"a smooth, projective, algebraic curve of genus one, with a specified point (O)."** This definition may seem complex, so let's break down the key terms: * **Smooth**: A smooth curve is a curve that is free of singularities, such as cusps or self-intersections. * **Projective**: A projective space is a fundamental geometric concept that generalizes the idea of euclidean space by adding "points at infinity," allowing for a unified treatment of lines, intersections, and perspectives. * **Algebraic**: The curve consists of points that satisfy a polynomial equation. * **Genus** : The genus of a curve is a topological invariant that, in intuitive terms, counts the number of "holes" in the curve. * **Specified point (O)**: This is the invariant point, the point at infinity of the curve where every lines intersect.

Illustration of the Point at Infinity: A Railway Analogy

This definition alone doesn’t provide a clear picture of what an elliptic curve looks like. In practice, elliptic curves are usually introduced via their algebraic equation, along with the criteria mentioned above. **Note:** Elliptic curves should not be confused with ellipses, which are a different type of curve known as conics.

Smooth curves (curves without singularities) in the cartesian plan with they algebraic equation.

## Familly of elliptic curves Elliptic curves can be defined by specific algebraic equations. There are several well-known families of elliptic curves, each with particular properties and applications. Let’s take a look at some of the most commonly used ones. ### Weirstrass curves $$ y^{2} + a\_{1}xy + a\_{3}y = x^{3} + a\_{2}x^{2} + a\_{4}x + a\_{6} \quad \text{(Weierstrass form)} $$ $$ y^{2} = x^{3} + ax + b \quad \text{(Weierstrass reduced form)} $$ where $$a\_i$$ belongs to a field and the discriminant $$\Delta$$ (given by $$4a^3 + 27b^2$$ in the reduced form) is nonzero in $$F\_p$$., defining an elliptic curve over $$F\_p$$. Every elliptic curve over $$F\_p$$. can be converted to a short Weierstrass equation if $$p$$. is larger than 3.

Weierstrass curve representation. The surrounding curves are not elliptic as they have a singularity point (cusp and a double point), i.e., Δ = 0.

### Montgomery curves $$ By^2 = x^3 + Ax^2 + x \quad \text{(Montgomery form)} $$ where $$B(A^2 - 4)$$ is nonzero in $$F\_p$$, defining an elliptic curve over $$F\_p$$. Substituting $$x = Bu - \frac{A}{3}$$ and $$y = Bv$$ produces the short Weierstrass equation: $$ v^2 = u^3 + au + b $$ where $$ a = \frac{3 - A^2}{3B^2}, \quad b = \frac{2A^3 - 9A}{27B^3}. $$ Montgomery curves were introduced by Montgomery in 1987.

Montgomery curve representation. The surrounding curves does not respect one of the Montgomery criteria B(A^2 - 4) ≠ 0.

### Edwards curves $$ y^{2} + x^{2} = 1 + dx^{2}y^{2} \quad \text{(Edwards form)} $$ where $$d(1 - d)$$ is nonzero in $$F\_p$$, defining an elliptic curve over $$F\_p$$. Substituting $$x = \frac{u}{v}$$ and $$y = \frac{u-1}{u+1}$$ produces the Montgomery equation: $$ Bv^2 = u^3 + Au^2 + u $$ where $$ A = \frac{2(1+d)}{1-d}, \quad B = \frac{4}{1-d}. $$ Edwards curves were introduced by Edwards in 2007 in the case that $$d$$ is a fourth power. [SafeCurves ](https://safecurves.cr.yp.to/)requires Edwards curves to be complete, i.e., for $$d$$ to not be a square; complete Edwards curves were introduced by Bernstein and Lange in 2007.

Edwards curve representation. The surrounding curves does not respesct an Edwards criteria d(1-d)≠ 0.

## Takeaways This article laid one of the mathematical foundations of elliptic curve cryptography: the **elliptic curve** itself. If you had to explain what an elliptic curve is, here are two ways to do it: 1. **(Mathematician) Formal definition**: Give the rigorous mathematical definition (which, while accurate, is hard to remember and not very intuitive). 2. **(Engineer) Practical approach**: * Show how these curves look graphically. * Describe the algebraic equations that define them. * Clarify that they are not ellipses and belong to a different mathematical family. In **Part 2** of this ECC series, we’ll explore **finite fields** and how elliptic curves are defined over them.