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.
Who discovered elliptic curves?
Victor Miller
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].
What’s so 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 ECC used for?
What is ECC? ECC is a mathematical method that can be used for all sorts of stuff – creating encryption keys, providing secure digital signatures, and more. When it comes to ECC’s use with SSL certificates – it’s a very flexible tool.
Which of the following constructions use for elliptic curves?
Arches, bridges, dams, monuments, man-holes, glands and stuffing boxes etc use elliptical 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.
Why is an elliptic curve called elliptic?
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 are elliptic curves called?
Anyway, since Jacobi’s functions started off with ellipse arc length problem, they are called elliptic functions. These curves are called elliptic curves. So elliptic curves are the set of points that are obtained as a result of solving elliptic functions over a predefined space.
Where are elliptic curves used?
Elliptic curves are applicable for encryption, digital signatures, pseudo-random generators and other tasks. They are also used in several integer factorization algorithms that have applications in cryptography, such as Lenstra elliptic-curve factorization.
How do elliptic curves work?
An elliptic curve for current ECC purposes is a plane curve over a finite field which is made up of the points satisfying the equation: y²=x³ + ax + b. In this elliptic curve cryptography example, any point on the curve can be mirrored over the x-axis and the curve will stay the same.
How many propositions are there in elliptic modular form?
The text contains 29 numbered “Propositions” whose proofs are given or sketched and 20 unnumbered “Theorems” which are results quoted from the literature whose proofs are too difficult (in many cases,muchtoo difficult) to be given here, though in many cases we have tried to indicate what the main ingredients are.
How did the modular group of elliptic curves get its name?
4 D. Zagier The modular group takes its name from the fact that the points of the quotient spaceΓ1\\H aremoduli(= parameters) for the isomorphism classes of elliptic curves over C. To each pointz∈H one can associate the lattice
Why is the space of modular forms computable?
First of all, the space of modular forms of a given weight onΓis finite dimen- sional and algorithmically computable, so that it is a mechanical procedure to prove any given identity among modular forms. Secondly, modular forms occur naturally in connection with problems arising in many other areas of mathematics.