Category: Elliptic curves

Doubling-oriented Doche–Icart–Kohel curve
In mathematics, the doubling-oriented Doche–Icart–Kohel curve is a form in which an elliptic curve can be written. It is a special case of Weierstrass form and it is also important in elliptic-curve c
Poncelet's closure theorem
In geometry, Poncelet's closure theorem, also known as Poncelet's porism, states that whenever a polygon is inscribed in one conic section and circumscribes another one, the polygon must be part of an
Hasse's theorem on elliptic curves
Hasse's theorem on elliptic curves, also referred to as the Hasse bound, provides an estimate of the number of points on an elliptic curve over a finite field, bounding the value both above and below.
Congruent number
In number theory, a congruent number is a positive integer that is the area of a right triangle with three rational number sides. A more general definition includes all positive rational numbers with
Brumer bound
In mathematics, the Brumer bound is a bound for the rank of an elliptic curve, proved by Brumer.
Mordell–Weil theorem
In mathematics, the Mordell–Weil theorem states that for an abelian variety over a number field , the group of K-rational points of is a finitely-generated abelian group, called the Mordell–Weil group
Grothendieck–Ogg–Shafarevich formula
In mathematics, the Grothendieck–Ogg–Shafarevich formula describes the Euler characteristic of a complete curve with coefficients in an abelian variety or constructible sheaf, in terms of local data i
Tate pairing
In mathematics, Tate pairing is any of several closely related bilinear pairings involving elliptic curves or abelian varieties, usually over local or finite fields, based on the Tate duality pairings
Heegner point
In mathematics, a Heegner point is a point on a modular curve that is the image of a quadratic imaginary point of the upper half-plane. They were defined by Bryan Birch and named after Kurt Heegner, w
Mestre bound
In mathematics, the Mestre bound is a bound on the analytic rank of an elliptic curve in terms of its conductor, introduced by Mestre.
In cryptography, Curve25519 is an elliptic curve used in elliptic-curve cryptography (ECC) offering 128 bits of security (256-bit key size) and designed for use with the elliptic curve Diffie–Hellman
Twisted Hessian curves
In mathematics, the Twisted Hessian curve represents a generalization of Hessian curves; it was introduced in elliptic curve cryptography to speed up the addition and doubling formulas and to have str
Tate curve
In mathematics, the Tate curve is a curve defined over the ring of formal power series with integer coefficients. Over the open subscheme where q is invertible, the Tate curve is an elliptic curve. Th
Elliptic cohomology
In mathematics, elliptic cohomology is a cohomology theory in the sense of algebraic topology. It is related to elliptic curves and modular forms.
Gauss lemniscate function
No description available.
Moduli of abelian varieties
Abelian varieties are a natural generalization of elliptic curves, including algebraic tori in higher dimensions. Just as elliptic curves have a natural moduli space over characteristic 0 constructed
Conductor of an elliptic curve
In mathematics, the conductor of an elliptic curve over the field of rational numbers, or more generally a local or global field, is an integral ideal analogous to the Artin conductor of a Galois repr
Height function
A height function is a function that quantifies the complexity of mathematical objects. In Diophantine geometry, height functions quantify the size of solutions to Diophantine equations and are typica
Néron–Ogg–Shafarevich criterion
In mathematics, the Néron–Ogg–Shafarevich criterion states that if A is an elliptic curve or abelian variety over a local field K and ℓ is a prime not dividing the characteristic of the residue field
Counting points on elliptic curves
An important aspect in the study of elliptic curves is devising effective ways of counting points on the curve. There have been several approaches to do so, and the algorithms devised have proved to b
Rank of an elliptic curve
In mathematics, the rank of an elliptic curve is the rational Mordell–Weil rank of an elliptic curve defined over the field of rational numbers. Mordell's theorem says the group of rational points on
Hessian form of an elliptic curve
In geometry, the Hessian curve is a plane curve similar to folium of Descartes. It is named after the German mathematician Otto Hesse.This curve was suggested for application in elliptic curve cryptog
Table of costs of operations in elliptic curves
Elliptic curve cryptography is a popular form of public key encryption that is based on the mathematical theory of elliptic curves. Points on an elliptic curve can be added and form a group under this
Mordell–Weil group
In arithmetic geometry, the Mordell–Weil group is an abelian group associated to any abelian variety defined over a number field , it is an arithmetic invariant of the Abelian variety. It is simply th
Tripling-oriented Doche–Icart–Kohel curve
The tripling-oriented Doche–Icart–Kohel curve is a form of an elliptic curve that has been used lately in cryptography; it is a particular type of Weierstrass curve. At certain conditions some operati
Igusa variety
In mathematics, an Igusa curve is (roughly) a coarse moduli space of elliptic curves in characteristic p with a level p Igusa structure, where an Igusa structure on an elliptic curve E is roughly a po
Semistable abelian variety
In algebraic geometry, a semistable abelian variety is an abelian variety defined over a global or local field, which is characterized by how it reduces at the primes of the field. For an abelian vari
Mordell curve
In algebra, a Mordell curve is an elliptic curve of the form y2 = x3 + n, where n is a fixed non-zero integer. These curves were closely studied by Louis Mordell, from the point of view of determining
Nagell–Lutz theorem
In mathematics, the Nagell–Lutz theorem is a result in the diophantine geometry of elliptic curves, which describes rational torsion points on elliptic curves over the integers.It is named for Trygve
Raynaud's isogeny theorem
In mathematics, Raynaud's isogeny theorem, proved by Raynaud, relates the Faltings heights of two isogeneous elliptic curves.
Modular elliptic curve
A modular elliptic curve is an elliptic curve E that admits a parametrisation X0(N) → E by a modular curve. This is not the same as a modular curve that happens to be an elliptic curve, something that
Montgomery curve
In mathematics the Montgomery curve is a form of elliptic curve introduced by Peter L. Montgomery in 1987, different from the usual Weierstrass form. It is used for certain computations, and in partic
Tate's algorithm
In the theory of elliptic curves, Tate's algorithm takes as input an of an elliptic curve E over , or more generally an algebraic number field, and a prime or prime ideal p. It returns the exponent fp
Sato–Tate conjecture
In mathematics, the Sato–Tate conjecture is a statistical statement about the family of elliptic curves Ep obtained from an elliptic curve E over the rational numbers by reduction modulo almost all pr
Jacobian curve
In mathematics, the Jacobi curve is a representation of an elliptic curve different from the usual one defined by the Weierstrass equation. Sometimes it is used in cryptography instead of the Weierstr
Hesse configuration
In geometry, the Hesse configuration, introduced by Colin Maclaurin and studied by Hesse, is a configuration of 9 points and 12 lines with three points per line and four lines through each point. It c
Period mapping
In mathematics, in the field of algebraic geometry, the period mapping relates families of Kähler manifolds to families of Hodge structures.
Twisted Edwards curve
In algebraic geometry, the twisted Edwards curves are plane models of elliptic curves, a generalisation of Edwards curves introduced by Bernstein, Birkner, Joye, Lange and Peters in 2008. The curve se
Twists of elliptic curves
In the mathematical field of algebraic geometry, an elliptic curve E over a field K has an associated quadratic twist, that is another elliptic curve which is isomorphic to E over an algebraic closure
In the study of the arithmetic of elliptic curves, the j-line over a ring R is the coarse moduli scheme attached to the moduli problem sending a ring to the set of isomorphism classes of elliptic curv
Hesse pencil
In mathematics, the syzygetic pencil or Hesse pencil, named for Otto Hesse, is a pencil (one-dimensional family) of cubic plane elliptic curves in the complex projective plane, defined by the equation
Edwards curve
In mathematics, the Edwards curves are a family of elliptic curves studied by Harold Edwards in 2007. The concept of elliptic curves over finite fields is widely used in elliptic curve cryptography. A
In mathematics, and in particular the study of Weierstrass elliptic functions, the equianharmonic case occurs when the Weierstrass invariants satisfy g2 = 0 and g3 = 1.This page follows the terminolog
Elliptic Gauss sum
In mathematics, an elliptic Gauss sum is an analog of a Gauss sum depending on an elliptic curve with complex multiplication. The quadratic residue symbol in a Gauss sum is replaced by a higher residu
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,
In cryptography, Curve448 or Curve448-Goldilocks is an elliptic curve potentially offering 224 bits of security and designed for use with the elliptic-curve Diffie–Hellman (ECDH) key agreement scheme.
Weil pairing
In mathematics, the Weil pairing is a pairing (bilinear form, though with multiplicative notation) on the points of order dividing n of an elliptic curve E, taking values in nth roots of unity. More g
Néron differential
In mathematics, a Néron differential, named after André Néron, is an almost canonical choice of 1-form on an elliptic curve or abelian variety defined over a local field or global field. The Néron dif
Schoof's algorithm
Schoof's algorithm is an efficient algorithm to count points on elliptic curves over finite fields. The algorithm has applications in elliptic curve cryptography where it is important to know the numb
Elliptic curve point multiplication
Elliptic curve scalar multiplication is the operation of successively adding a point along an elliptic curve to itself repeatedly. It is used in elliptic curve cryptography (ECC) as a means of produci
Supersingular elliptic curve
In algebraic geometry, supersingular elliptic curves form a certain class of elliptic curves over a field of characteristic p > 0 with unusually large endomorphism rings. Elliptic curves over such fie