# 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 in fact an abelian variety — that is, it has a multiplication defined algebraically, with respect to which it is a (necessarily commutative) group — and O serves as the identity element. Often the curve itself, without O specified, is called an elliptic curve.

Any elliptic curve can be written as a plane algebraic curve defined by an equation of the form:

which is non-singular; that is, its graph has no cusps or self-intersections. (When the characteristic of the coefficient field is equal to 2 or 3, the above equation is not quite general enough to comprise all non-singular cubic curves; see below for a more precise definition.) The point O is actually the "point at infinity" in the projective plane.

If y2 = P(x), where P is any polynomial of degree three in x with no repeated roots, then we obtain a nonsingular plane curve of genus one, which is thus also an elliptic curve. If P has degree four and is squarefree this equation again describes a plane curve of genus one; however, it has no natural choice of identity element. More generally, any algebraic curve of genus one, for example from the intersection of two quadric surfaces embedded in three-dimensional projective space, is called an elliptic curve, provided that it has at least one rational point.

Using the theory of elliptic functions, it can be shown that elliptic curves defined over the complex numbers correspond to embeddings of the torus into the complex projective plane. The torus is also an abelian group, and in fact this correspondence is also a group isomorphism.

Elliptic curves are especially important in number theory, and constitute a major area of current research; for example, they were used in the proof, by Andrew Wiles (assisted by Richard Taylor), of Fermat's Last Theorem. They also find applications in cryptography (see the article elliptic curve cryptography) and integer factorization.

An elliptic curve is not an ellipse: see elliptic integral for the origin of the term. Topologically, an elliptic curve is a torus.

### Other articles related to "elliptic curve, elliptic, curves, curve":

Schoof's Algorithm - The Frobenius Endomorphism
... Given the elliptic curve defined over we consider points on over, the algebraic closure of i.e ... The Frobenius endomorphism of over extends to the elliptic curve by ... Thus we have for all that, where + denotes addition on the elliptic curve and and denote scalar multiplication of by and of by ...
Szpiro's Conjecture
... Szpiro's conjecture concerns a relationship between the conductor and the discriminant of an elliptic curve ... The conjecture states that given ε > 0, there exists a constant C(ε) such that for any elliptic curve E defined over Q with minimal discriminant Δ and conductor f, we have The ...
Algebraic-group Factorisation Algorithm - Methods Corresponding To Particular Algebraic Groups
... If the algebraic group is an elliptic curve, the one-sided identities can be recognised by failure of inversion in the elliptic-curve point addition procedure, and the result is the elliptic ... algebraic groups—higher-order extensions of N or groups corresponding to algebraic curves of higher genus—is occasionally proposed, but almost always impractical ...
Key Size - Asymmetric Algorithm Key Lengths
... One of the asymmetric algorithm types, elliptic curve cryptography, or ECC, appears to be secure with shorter keys than those needed by other asymmetric key algorithms ... A message encrypted with an elliptic key algorithm using a 109-bit long key has been broken by brute force ... The NSA specifies that "Elliptic Curve Public Key Cryptography using the 256-bit prime modulus elliptic curve as specified in FIPS-186-2 and SHA-256 are appropriate for protecting ...
Alternative Representations of Elliptic Curves
... Hessian curve Edwards curve Twisted curve Twisted Hessian curve Twisted Edwards curve Doubling-oriented Doche–Icart–Kohel curve Tripling-oriented Doche–Icart–Kohel curve Jacobian ...

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