Group theory
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Algebraic curve
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Analytic number theory
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Abelian variety
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Elliptic function
Elliptic curve
In mathematics, an elliptic curve (EC) is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety – that is, it has a ...
Elliptic curve - Wikipedia
Birch and Swinnerton-Dyer conjecture
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Modularity theorem
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Isogeny
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Arithmetic of abelian varieties
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Elliptic curve cryptography
Elliptic curve - Wikipedia
Birch and Swinnerton-Dyer conjecture
In mathematics, the Birch and Swinnerton-Dyer conjecture is an open problem in the field of number theory. It is widely recognized as one of the most challenging mathematical problems; the conjecture ...
Modularity theorem
In mathematics, the modularity theorem (formerly called the Taniyama–Shimura–Weil conjecture and several related names) states that elliptic curves over the field of rational numbers are related to m...
Modularity theorem - Wikipedia
Isogeny
In mathematics, an isogeny is a morphism of algebraic groups that is surjective and has a finite kernel.If the groups are abelian varieties, then any morphism f : A → B of the unde...
Arithmetic of abelian varieties
In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or family of those. It goes back to the studies of Fermat on what are now recognised as ell...
Elliptic curve cryptography
Elliptic curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. One of the main benefits in comparison with non-ECC ...
Mordell–Weil theorem
XDH assumption
The external Diffie–Hellman (XDH) assumption is a mathematical assumption used in elliptic curve cryptography. The XDH assumption holds that there exist certain subgroups of elliptic curves whi...
XDH assumption - Wikipedia
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...
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 ...
Poncelet's closure theorem
In geometry, Poncelet's porism (sometimes referred to as Poncelet's closure theorem) states that whenever a polygon is inscribed in one conic section and circumscribes another one, the polygon must be...
Poncelet's closure theorem - Wikipedia
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...
Tripling-oriented Doche–Icart–Kohel curve - Wikipedia
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...
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 pairin...
Igusa variety
Congruent number
In mathematics, 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 t...
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, and ...
Sato–Tate conjecture
In mathematics, the Sato–Tate conjecture is a statistical statement about the family of elliptic curves Ep over the finite field with p elements, with p a prime number, obtained from an elliptic curve...
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...
Hessian form of an elliptic curve - Wikipedia
Mordell curve
In algebra, a Mordell curve is an elliptic curve of the form y = x + n, where n is a fixed non-zero integer. These curves were closely studied by Louis Mordell, from the point of view of determining t...
Mordell curve - Wikipedia
Grothendieck–Ogg–Shafarevich formula
Grothendieck–Ogg–Shafarevich formula - Wikipedia
Jacobian curve
In mathematics, the Jacobi curve is a representation of an elliptic curve different from the usual one (Weierstrass equation). Sometimes it is used in cryptography instead of the Weierstrass form beca...
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...
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 varie...
KCDSA
KCDSA (Korean Certificate-based Digital Signature Algorithm) is a digital signature algorithm created by a team led by the Korea Internet & Security Agency (KISA). It is an ElGamal variant, simil...
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.
Suppose that the e...
Hesse configuration
In geometry, the Hesse configuration, introduced by Colin Maclaurin and studied by Hesse (1844), is a configuration of 9 points and 12 lines with three points per line and four lines through eac...
Hesse configuration - Wikipedia
Twists of curves
In 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 o...
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...
Elliptic curve primality proving
In mathematics elliptic curve primality testing techniques are among the quickest and most widely used methods in primality proving. It is an idea forwarded by Shafi Goldwasser and Joe Kilian in 1986 ...