A catalog of elliptic curves. Region shown is [-3,3]²

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 an abelian group—and O serves as the identity element. Often the curve itself, without O specified, is called an elliptic curve. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Informally, a differentiable manifold is a type of manifold (which is in turn a kind of topological space) that is locally similar enough to Euclidean space to allow one to do calculus. ... In mathematics, an algebraic variety is essentially a set of common zeroes of a set of polynomials. ... In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ... In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ... In mathematics, particularly in algebraic geometry, complex analysis and number theory, abelian variety is a term used to denote a complex torus that can be embedded into projective space as a projective variety. ... In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ...

Any elliptic curve can be written as a plane algebraic curve defined by an equation of the form In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. ...

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. A cusp, defined by x3+y2=0 In singularity theory a cusp is a singular point of a curve. ... In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ... In mathematics, a cubic curve is a plane curve C defined by a cubic equation F(X,Y,Z) = 0 applied to homogeneous coordinates [X:Y:Z] for the projective plane; or the inhomogeneous version for the affine space determined by setting Z = 1 in such an equation. ... The point at infinity, also called ideal point, is a point which when added to the real number line yields a closed curve called the real projective line, . Nota Bene: The real projective line is not equivalent to the extended real number line. ... Projective plane - Wikipedia, the free encyclopedia /**/ @import /skins-1. ...

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 three-dimensional quadric surfaces, is called an elliptic curve, provided that it has at least one rational point. In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ... In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ... Ellipsoid Elliptic Paraboloid Hyperbolic Paraboloid Hyperboloid of One Sheet Hyperboloid of Two Sheets Cone Elliptic Cylinder Hyperbolic Cylinder Parabolic Cylinder In mathematics a quadric, or quadric surface, is any D-dimensional (hyper-)surface represented by a second-order equation in spatial variables (coordinates). ...

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. In complex analysis, an elliptic function is, roughly speaking, a function defined on the complex plane which is periodic in two directions. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... A torus This article is about the surface and mathematical concept of a torus. ... In mathematics, the complex projective plane, usually denoted CP2, is the two-dimensional complex projective space. ... In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. ...

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. Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... For the French mathematician with work in the area of elliptic curves, see André Weil. ... Richard Taylor (born 19 May 1962) is a British mathematician working in the field of number theory. ... Pierre de Fermats conjecture written in the margin of his copy of Arithmetica proved to be one of the most intriguing and enigmatic mathematical problems ever devised. ... The German Lorenz cipher machine, used in World War II for encryption of very high-level general staff messages Cryptography (or cryptology; derived from Greek κρυπτός kryptós hidden, and the verb γράφω gráfo write or λεγειν legein to speak) is the study of message secrecy. ... Elliptic curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. ... Prime decomposition redirects here. ...

An elliptic curve is not the same as an ellipse: see elliptic integral for the origin of the term. For other uses, see Ellipse (disambiguation). ... In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Giulio Fagnano and Leonhard Euler. ...

Elliptic curves over the real numbers

Although the formal definition of an elliptic curve is fairly technical and requires some background in algebraic geometry, it is possible to describe some features of elliptic curves over the real numbers using only high school algebra and geometry. Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problematics of geometry. ... Please refer to Real vs. ... This article is about the branch of mathematics. ... For other uses, see Geometry (disambiguation). ...

Graphs of curves y² = x³ − x and y² = x³ − x + 1

In this context, an elliptic curve is a plane curve defined by an equation of the form example elliptic curves; by me for Wiki File links The following pages link to this file: Elliptic curve Categories: GFDL images ... In mathematics, the concept of a curve tries to capture our intuitive idea of a geometrical one-dimensional and continuous object. ...

where a and b are real numbers. This type of equation is called a Weierstrass equation.

The definition of elliptic curve also requires that the curve be non-singular. Geometrically, this means that the graph has no cusps or self-intersections. Algebraically, this involves calculating the discriminant In mathematics, a singular point of an algebraic variety V is a point P that is special (so, singular), in the geometric sense that V is not locally flat there. ... A cusp, defined by x3+y2=0 In singularity theory a cusp is a singular point of a curve. ... In algebra, the discriminant of a polynomial is a certain expression in the coefficients of the polynomial which equals zero if and only if the polynomial has multiple roots in the complex numbers. ...

Δ = − 16(4a³ + 27b²).

The curve is non-singular if the discriminant is not equal to zero. (Although the factor −16 seems irrelevant here, it turns out to be convenient in more advanced study of elliptic curves.)

The graph of a non-singular curve has two components if its discriminant is positive, and one component if it is negative. For example, in the graphs shown above, the discriminant in the first case is 64, and in the second case is −368.

The group law

By adding a "point at infinity", we obtain the projective version of this curve. If P and Q are two points on the curve, then we can uniquely describe a third point which is the intersection of the curve with the line through P and Q. If the line is tangent to the curve at a point, then that point is counted twice; and if the line is parallel to the y-axis, we define the third point as the point "at infinity". Exactly one of these conditions then holds for any pair of points on an elliptic curve.

It is then possible to introduce a group operation, "+", on the curve with the following properties: we consider the point at infinity to be 0, the identity of the group; and if a straight line intersects the curve at the points P, Q and R, then we require that P + Q + R = 0 in the group. One can check that this turns the curve into an abelian group, and thus into an abelian variety. It can be shown that the set of K-rational points (including the point at infinity) forms a subgroup of this group. If the curve is denoted by E, then this subgroup is often written as E(K). This picture illustrates how the hours on a clock form a group under modular addition. ... In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ... In mathematics, particularly in algebraic geometry, complex analysis and number theory, abelian variety is a term used to denote a complex torus that can be embedded into projective space as a projective variety. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group...

The above group can be described algebraically as well as geometrically. Given the curve y² = x³ − px − q over the field K (whose characteristic we assume to be neither 2 nor 3), and points P = (x_P, y_P) and Q = (x_Q, y_Q) on the curve, assume first that x_P ≠ x_Q. Let s = (y_P − y_Q)/(x_P − x_Q); since K is a field, s is well-defined. Then we can define R = P + Q = (x_R, y_R) by

If x_P = x_Q, then there are two options: if y_P = −y_Q, then the sum is defined as 0; thus, the inverse of each point on the curve is found by reflecting it across the x-axis. If y_P = y_Q ≠ 0, then R = P + P = 2P = (x_R, - y_R) is given by

If y_P = y_Q = 0, then P + P = 0.

Elliptic curves over the complex numbers

The formulation of elliptic curves as the embedding of a torus in the complex projective plane follows naturally from a curious property of Weierstrass's elliptic functions. These functions and their first derivative are related by the formula A torus This article is about the surface and mathematical concept of a torus. ... In mathematics, the complex projective plane, usually denoted CP2, is the two-dimensional complex projective space. ... In mathematics, Weierstrasss elliptic functions are a standard type of elliptic functions (the other is the Jacobis elliptic functions). ...

Here, g₂ and g₃ are constants; is the Weierstrass elliptic function and its derivative. It should be clear that this relation is in the form of an elliptic curve (over the complex numbers). The Weierstrass functions are doubly-periodic; that is, they are periodic with respect to a lattice Λ; in essence, the Weierstrass functions are naturally defined on a torus . This torus may be embedded in the complex projective plane by means of the map In mathematics, Weierstrasss elliptic functions are a standard type of elliptic functions (the other is the Jacobis elliptic functions). ... In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that define a lattice in the complex plane. ...

This map is a group isomorphism, carrying the natural group structure of the torus into the projective plane. It is also an isomorphism of Riemann surfaces, and so topologically, a given elliptic curve looks like a torus. If the lattice Λ is related to a lattice cΛ by multiplication by a non-zero complex number c, then the corresponding curves are isomorphic. Isomorphism classes of elliptic curves are specified by the j-invariant. In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. ... Riemann surface for the function f(z) = sqrt(z) In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional complex manifold. ... Real part of the j-invariant as a function of the nome q on the unit disk In mathematics, Kleins j-invariant, regarded as a function of a complex variable Ï„, is a modular function defined on the upper half-plane of complex numbers. ...

The isomorphism classes can be understood in a simpler way as well. The constants g₂ and g₃, called the modular invariants, are uniquely determined by the lattice, that is, by the structure of the torus. However, the complex numbers are the splitting field for polynomials, and so the elliptic curve may be written as In mathematics, Weierstrass introduced some particular elliptic functions that have become the basis for the most standard notations used. ... In abstract algebra, the splitting field of a polynomial P(X) over a given field K is a field extension L of K, over which P factorizes into linear factors X − ai, and such that the ai generate L over K. It can be shown that such splitting fields exist...

One finds that

and

so that the modular discriminant is In number theory, a cusp form is a particular kind of modular form, distinguished in the case of modular forms for the modular group by the vanishing in the Fourier series expansion of the constant coefficient a0. ...

Here, λ is sometimes called the modular lambda function.

Note that the uniformization theorem states that every compact Riemann surface of genus one can be represented as a torus. In mathematics, the uniformization theorem for surfaces says that any surface admits a Riemannian metric of constant Gauss curvature. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... Riemann surface for the function f(z) = sqrt(z) In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional complex manifold. ...

Elliptic curves over a general field

Elliptic curves can be defined over any field K; the formal definition of an elliptic curve is a non-singular projective algebraic curve over K with genus 1 with a given point defined over K. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ...

If the characteristic of K is neither 2 nor 3, then every elliptic curve over K can be written in the form In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ...

where p and q are elements of K such that the right hand side polynomial x³ − px − q does not have any double roots. If the characteristic is 2 or 3, then more terms need to be kept: in characteristic 3, the most general equation is of the form

for arbitrary constants b₂,b₄,b₆ such that the polynomial on the right-hand side has distinct roots (the notation is chosen for historical reasons). In characteristic 2, even this much is not possible, and the most general equation is

provided that the variety it defines is nonsingular. If characteristic were not an obstruction, each equation would reduce to the previous ones by a suitable change of variables.

One typically takes the curve to be the set of all points (x,y) which satisfy the above equation and such that both x and y are elements of the algebraic closure of K. Points of the curve whose coordinates both belong to K are called K-rational points. In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. ...

Isogeny

Let E and D be elliptic curves over a field k. An isogeny between E and D is a finite morphism of varieties that preserves basepoints (in other words, maps the given point on E to that on D). In mathematics, in algebraic geometry, a morphism of schemes is a finite morphism, if has an open cover by affine schemes such that for each , is an open affine subscheme , and the restriction of f to , which induces a map of rings makes a finitely generated module over . ... In mathematics, particularly in algebraic geometry, complex analysis and number theory, abelian variety is a term used to denote a complex torus that can be embedded into projective space as a projective variety. ...

The two curves are called isogenous if there is an isogeny between them. This is an equivalence relation, symmetry being due to the existence of the dual isogeny. Every isogeny is an algebraic homomorphism and thus induces homomorphisms of the groups of the elliptic curves for k-valued points. In mathematics, an equivalence relation is a binary relation between two elements of a set which groups them together as being equivalent in some way. ... Symmetry is a characteristic of geometrical shapes, equations and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces). ... This picture illustrates how the hours on a clock form a group under modular addition. ...

See also Abelian varieties up to isogeny. In mathematics, localization of a category consists of adding to a category inverse morphisms for some collection of morphisms, constraining them to become isomorphisms. ...

Connections to number theory

The Mordell-Weil theorem states that if the underlying field K is the field of rational numbers (or more generally a number field), then the group of K-rational points is finitely generated. This means that the group can be expressed as the direct sum of a free abelian group and a finite torsion subgroup. While it is relatively easy to determine the torsion subgroup of E(K), no general algorithm is known to compute the rank of the free subgroup. A formula for this rank is given by the Birch and Swinnerton-Dyer conjecture. In mathematics, the Mordell-Weil theorem states that for an abelian variety A over a number field K, the group A(K) of K-rational points of A is a finitely-generated abelian group. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days... In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x1,...,xs in G such that every x in G can be written in the form x = n1x1 + n2x2 + ... + nsxs with integers n1,...,ns. ... In abstract algebra, the direct sum is a construction which combines several modules into a new, bigger one. ... In abstract algebra, a free abelian group is an abelian group that has a basis in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. ... In group theory, the torsion subgroup of an abelian group A is the subgroup of A consisting of all elements that have finite order. ... In mathematics, the rank, or torsion-free rank, of an abelian group measures how large a group is in terms of how large a vector space one would need to contain it; or alternatively how large a free abelian group it can contain as a subgroup. ... In mathematics, the Birch and Swinnerton-Dyer conjecture relates the rank of the abelian group of points over a number field of an elliptic curve E to the order of zero of the associated L-function L(E, s) at s = 1. ...

The recent proof of Fermat's last theorem proceeded by proving a special case of the deep Taniyama-Shimura conjecture relating elliptic curves over the rationals to modular forms; this conjecture has since been completely proved. Pierre de Fermats conjecture written in the margin of his copy of Arithmetica proved to be one of the most intriguing and enigmatic mathematical problems ever devised. ... The Taniyama-Shimura theorem establishes an important connection between elliptic curves, which are objects from algebraic geometry, and modular forms, which are certain periodic holomorphic functions investigated in number theory. ... Modular form - Wikipedia /**/ @import /skins-1. ...

While the precise number of rational points of an elliptic curve E over a finite field F_p is in general rather difficult to compute, Hasse's theorem on elliptic curves tells us In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements. ... In mathematics, Hasses theorem on elliptic curves bounds the number of points on an elliptic curve over a finite field. ...

This fact can be understood and proven with the help of some general theory; see local zeta function, Étale cohomology. The number of points on a specific curve can be computed with Schoof's algorithm. In number theory, a local zeta-function is a generating function Z(t) for the number of solutions of a set of equations defined over a finite field F, in extension fields Fk of F. The analogy with the Riemann zeta function comes via consideration of the logarithmic derivative . ... In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures. ... Schoofs algorithm, first described by R. Schoof in 1985, allows one to calculate the number of points on an elliptic curve over a finite field and is used mostly in elliptic curve cryptography. ...

For further developments see 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. ...

Algorithms that use elliptic curves

Elliptic curves over finite fields are used in some cryptographic applications as well as for integer factorization. Typically, the general idea in these applications is that a known algorithm which makes use of certain finite groups is rewritten to use the groups of rational points of elliptic curves. For more see also: The German Lorenz cipher machine, used in World War II for encryption of very high-level general staff messages Cryptography (or cryptology; derived from Greek κρυπτός kryptós hidden, and the verb γράφω gráfo write or λεγειν legein to speak) is the study of message secrecy. ... Prime decomposition redirects here. ... Flowcharts are often used to graphically represent algorithms. ...

• Elliptic curve cryptography
• Elliptic Curve DSA
• Lenstra elliptic curve factorization
• Elliptic curve primality proving

Elliptic curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. ... Elliptic Curve DSA (ECDSA) is a variant of the Digital Signature Algorithm (DSA) which operates on elliptic curve groups. ... The Lenstra elliptic curve factorization or the elliptic curve factorization method (ECM) is a fast, sub-exponential running time algorithm for integer factorization which employs elliptic curves. ... Elliptic Curve Primality Proving is a method based on elliptic curves to prove the primality of a number. ...

References

Serge Lang, in the introduction to the book cited below, stated that "It is possible to write endlessly on elliptic curves. (This is not a threat.)" The following short list is thus at best a guide to the vast expository literature available on the theoretical, algorithmic, and cryptographic aspects of elliptic curves. Serge Lang (May 19, 1927–September 12, 2005) was a French-born American mathematician. ...

• I. Blake; G. Seroussi, N. Smart, N.J. Hitchin (2000). Elliptic Curves in Cryptography. Cambridge Univ. Press. ISBN 0-521-65374-6. 
• Richard Crandall; Carl Pomerance (2001). "Chapter 7: Elliptic Curve Arithmetic", Prime Numbers: A Computational Perspective, 1st edition, Springer, 285–352. ISBN 0-387-94777-9. 
• John Cremona (1992). Algorithms for Modular Elliptic Curves. Cambridge Univ. Press. 
• Dale Husemöller (2004). Elliptic Curves, 2nd edition, Springer. 
• Kenneth Ireland; Michael I. Rosen (1990). "Chapters 18 and 19", A Classical Introduction to Modern Number Theory, 2nd edition, Springer. 
• Anthony Knapp (1992). Elliptic Curves. Math Notes 40, Princeton Univ. Press. 
• Neal Koblitz (1984). Introduction to Elliptic Curves and Modular Forms. Springer. 
• Neal Koblitz (1994). "Chapter 6", A Course in Number Theory and Cryptography, 2nd edition, Springer. ISBN 0-387-94293-9. 
• Serge Lang (1978). Elliptic Curves: Diophantine Analysis. Springer. 
• Joseph H. Silverman (1986). The Arithmetic of Elliptic Curves. Springer. 
• Joseph H. Silverman (1994). Advanced Topics in the Arithmetic of Elliptic Curves. Springer. 
• Joseph H. Silverman; John Tate (1992). Rational Points on Elliptic Curves. Springer. 
• Lawrence Washington (2003). Elliptic Curves: Number Theory and Cryptography. Chapman & Hall/CRC. ISBN 1-58488-365-0. 

Richard E. Crandall is an American computer scientist who has made contributions to computational number theory. ... One of the top number theorists of our time, Carl Pomerance received his PhD from Harvard University in 1972 and immediately joined the faculty at the University of Georgia, becoming full professor in 1982. ... Neal Koblitz is a Professor of Mathematics in the University of Washington in the Department of Mathematics. ... Neal Koblitz is a Professor of Mathematics in the University of Washington in the Department of Mathematics. ... Serge Lang (May 19, 1927–September 12, 2005) was a French-born American mathematician. ... Joseph H. Silverman is currently a professor of mathematics at Brown University, where he has taught since 1988. ... Joseph H. Silverman is currently a professor of mathematics at Brown University, where he has taught since 1988. ... Joseph H. Silverman is currently a professor of mathematics at Brown University, where he has taught since 1988. ... You may be looking for John Tate (boxer) John Torrence Tate, born March 13, 1925 in Minneapolis, Minnesota, is an American mathematician, distinguished for many fundamental contributions in algebraic number theory and related areas in algebraic geometry. ... Major Lawrence Washington (1659-1698) was the grandfather of George Washington. ...

External links

Wikimedia Commons has media related to:

Elliptic curve

• The Mathematical Atlas: 14H52 Elliptic Curves
• Eric W. Weisstein, Elliptic Curves at MathWorld.

• Matlab code for implicit function plotting - Can be used to plot elliptic curves.

This article incorporates material from Isogeny on PlanetMath, which is licensed under the GFDL. Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is an encyclopedist who created and maintains MathWorld and Eric Weissteins World of Science (ScienceWorld). ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ... PlanetMath is a free, collaborative, online mathematics encyclopedia. ...