What is special about elliptic curves?
The definition of an elliptic curve is an equation in the form: Moreover, the curve must be non-singular, i.e. its graph has no cusps or self-intersections. This seems like an awfully specific definition for a family of functions.
What is meant by elliptic curve?
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O. An elliptic curve is defined over a field K and describes points in K2, the Cartesian product of K with itself.
What is the equation for an elliptic curve?
This equation defines an elliptic curve. y2 = x3 + Ax + B, for some constants A and B. Below is an example of such a curve. An elliptic curve over C is a compact manifold of the form C/L, where L = Z + ωZ is a lattice in the complex plane.
Why is it called an elliptic curve?
So elliptic curves are the set of points that are obtained as a result of solving elliptic functions over a predefined space. I guess they didn’t want to come up with a whole new name for this, so they named them elliptic curves.
Why is an elliptic curve a torus?
After adding a point at infinity to the curve on the right, we get two circles topologically. Since these parameterizing functions are doubly periodic, the elliptic curve can be identified with a period parallelogram (in fact a square in this case) with the sides glued together i.e. a torus.
Is elliptic curve a function?
an integer is known as a Mordell curve. Whereas conic sections can be parameterized by the rational functions, elliptic curves cannot. The simplest parameterization functions are elliptic functions.
Who discovered elliptic curves?
Elliptic curve cryptography was introduced in 1985 by Victor Miller and Neal Koblitz who both independently developed the idea of using elliptic curves as the basis of a group for the discrete logarithm problem. [16, 20].
Is Elliptic Curve a function?
What does elliptic curve mean?
Informally, an elliptic curve is a type of cubic curve whose solutions are confined to a region of space that is topologically equivalent to a torus.
Why is there a group law on an elliptic curve?
Group Law The group law on an elliptic curve is what makes the theory of elliptic curves so special and interesting. In particular, it provides a way to generate points on the curve from other points. (For an introduction to group theory, see the wiki.)
How does elliptic curve cryptography encryption work?
How does elliptic curve cryptography work? An elliptical curve can simply illustrated as a set of points defined by the following equation: y 2 = x 3 + ax + b . Based on the values given to a and b, this will determine the shape of the curve. Elliptical curve cryptography uses these curves over finite fields to create a secret that only the private key holder is able to unlock. The larger the key size, the larger the curve, and the harder the problem is to solve.
What is the math behind elliptic curve cryptography?
Elliptic curve cryptography is based on discrete mathematics. In discrete math, elements can only take on certain discrete values. Boolean algebra is an example of discrete math where the possible values are zero and one. These values are usually interpreted as true and false.