A catalog of elliptic curves. Region shown is [-3,3]² (For a=0 and b=0 it's not smooth and therefore not an elliptic curve. )

In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety In Algebraic geometry, an algebraic curve is an Algebraic variety of dimension one In Mathematics, genus has a few different but closely related meanings Topology Orientable surface 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. In Mathematics, particularly in Algebraic geometry, Complex analysis and Number theory, an Abelian variety is a projective algebraic variety An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the 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. In Algebraic geometry, an algebraic curve is an Algebraic variety of dimension one In Singularity theory a cusp is a singular point of a curve. Spinode is an alternative name but this is less commonly used today (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. In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's In Mathematics, a cubic plane curve is a Plane algebraic curve C defined by a cubic equation F ( x, y, ) The point O is actually the "point at infinity" in the projective plane. 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 See Real projective plane and Complex projective plane, for the cases met as manifolds of respective dimension 2 and 4 In Mathematics

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. In Mathematics, genus has a few different but closely related meanings Topology Orientable surface 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 Algebraic geometry, an algebraic curve is an Algebraic variety of dimension one In mathematics a quadric, or quadric surface, is any D -dimensional Hypersurface defined as the locus of zeros of a Quadratic

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. In Complex analysis, an elliptic function is a function defined on the Complex plane which is periodic in two directions (a Doubly-periodic Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar In Mathematics, the complex projective plane, usually denoted CP 2 is the two-dimensional Complex projective space. The torus is also an abelian group, and in fact this correspondence is also a group isomorphism. 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

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. 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 Sir Andrew John Wiles KBE FRS (born 11 April 1953 is a British Mathematician and a professor at Princeton University Richard Taylor (born Richard Lawrence Taylor 19 May 1962) is a British Mathematician working in the field of Number theory Fermat's Last Theorem is the name of the statement in Number theory that It is impossible to separate any power higher than the second into two like They also find applications in cryptography (see the article elliptic curve cryptography) and integer factorization. Cryptography (or cryptology; from Greek grc κρυπτός kryptos, "hidden secret" and grc γράφω gráphō, "I write" Elliptic curve cryptography (ECC is an approach to Public-key cryptography based on the algebraic structure of Elliptic curves over Finite fields The use

An elliptic curve is not an ellipse: see elliptic integral for the origin of the term. In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a In Integral calculus, elliptic integrals originally arose in connection with the problem of giving the Arc length of an Ellipse.

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 In Mathematics, the real numbers may be described informally in several different ways Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position

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

where a and b are real numbers. In mathematics a plane curve is a Curve in a Euclidian plane (cf This type of equation is called a Weierstrass equation.

The definition of elliptic curve also requires that the curve be non-singular. 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 Geometrically, this means that the graph has no cusps or self-intersections. In Singularity theory a cusp is a singular point of a curve. Spinode is an alternative name but this is less commonly used today Algebraically, this involves calculating the discriminant

Δ = − 16(4a³ + 27b²). In Algebra, the discriminant of a Polynomial with real or complex Coefficients is a certain expression in the coefficients of the

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. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element One can check that this turns the curve into an abelian group, and thus into an abelian variety. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Mathematics, particularly in Algebraic geometry, Complex analysis and Number theory, an Abelian variety is a projective algebraic variety It can be shown that the set of K-rational points (including the point at infinity) forms a subgroup of this group. In Number theory, a K - rational point is a point on an Algebraic variety where each coordinate of the point belongs to the field K. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of If the curve is denoted by E, then this subgroup is often written as E(K).

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. In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar In Mathematics, the complex projective plane, usually denoted CP 2 is the two-dimensional Complex projective space. In Mathematics, Weierstrass's elliptic functions are Elliptic functions that take a particularly simple form (cf Jacobi's elliptic functions) they are named These functions and their first derivative are related by the formula

Here, g₂ and g₃ are constants; is the Weierstrass elliptic function and its derivative. In Mathematics, Weierstrass's elliptic functions are Elliptic functions that take a particularly simple form (cf Jacobi's elliptic functions) they are named 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 . In Mathematics, a fundamental pair of periods is an Ordered pair of Complex numbers that define a lattice in the Complex plane. This torus may be embedded in the complex projective plane by means of the map

This map is a group isomorphism, carrying the natural group structure of the torus into the projective plane. 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 It is also an isomorphism of Riemann surfaces, and so topologically, a given elliptic curve looks like a torus. In Mathematics, particularly in Complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional 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 Mathematics, Klein's j -invariant, regarded as a function of a complex variable &tau is a Modular function defined on the

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. In Mathematics, Weierstrass's elliptic functions are Elliptic functions that take a particularly simple form (cf Jacobi's elliptic functions) they are named However, the complex numbers are the splitting field for polynomials, and so the elliptic curve may be written as

One finds that

and

so that the modular discriminant is

Here, λ is sometimes called the modular lambda function. In Abstract algebra, the splitting field of a Polynomial P ( X) over a given field K is a Field extension In Mathematics, Weierstrass's elliptic functions are Elliptic functions that take a particularly simple form (cf Jacobi's elliptic functions) they are named

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 Gaussian curvature. In Mathematics, particularly in Complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional

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 In Mathematics, genus has a few different but closely related meanings Topology Orientable surface

If the characteristic of K is neither 2 nor 3, then every elliptic curve over K can be written in the form

where p and q are elements of K such that the right hand side polynomial x³ − px − q does not have any double roots. In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's 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. In Mathematics, particularly Abstract algebra, an algebraic closure of a field K is an Algebraic extension of K that is Points of the curve whose coordinates both belong to K are called K-rational points.

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 Algebraic geometry, a branch of Mathematics, a morphism f X \rightarrow Y of schemes is a finite morphism, if Y has an In Mathematics, particularly in Algebraic geometry, Complex analysis and Number theory, an Abelian variety is a projective algebraic 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. In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or In Mathematics, a dual abelian variety can be defined from an Abelian variety A, defined over a field K. Every isogeny is an algebraic homomorphism and thus induces homomorphisms of the groups of the elliptic curves for k-valued points. In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element

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

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. In Mathematics, the Mordell–Weil theorem states that for an Abelian variety A over a Number field K, the group A ( In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, an algebraic number field (or simply number field) F is a finite (and hence algebraic) Field extension of the In Abstract algebra, an Abelian group ( G,+ is called finitely generated if there exist finitely many elements x 1 This means that the group can be expressed as the direct sum of a free abelian group and a finite torsion subgroup. The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction 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 In the theory of Abelian groups the torsion subgroup AT of an abelian group A is the Subgroup of A consisting of all elements 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. 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 over the A formula for this rank is given by the Birch and Swinnerton-Dyer conjecture. In Mathematics, the Birch and Swinnerton-Dyer conjecture relates the rank of the Abelian group of points over a Number field of an Elliptic

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. Fermat's Last Theorem is the name of the statement in Number theory that It is impossible to separate any power higher than the second into two like In Mathematics, the modularity theorem establishes an important connection between Elliptic curves over the field of Rational numbers and Modular forms In Mathematics, a modular form is a (complex Analytic function on the Upper half-plane satisfying a certain kind of Functional equation and

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

This fact can be understood and proven with the help of some general theory; see local zeta function, Étale cohomology. 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, Hasse's theorem on elliptic curves bounds the number of points on an Elliptic curve over a Finite field, above and below In Number theory, a local zeta-function is a Generating function Z ( t) for the number of solutions of a set of equations 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 The number of points on a specific curve can be computed with Schoof's algorithm. Schoof's 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

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. Cryptography (or cryptology; from Greek grc κρυπτός kryptos, "hidden secret" and grc γράφω gráphō, "I write" 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. In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation For more see also:

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

See also

• Riemann-Hurwitz formula
• Nagell–Lutz theorem
• Complex multiplication
• Arithmetic dynamics

References

Serge Lang, in the introduction to the book cited below, stated that "It is possible to write endlessly on elliptic curves. Elliptic curve cryptography (ECC is an approach to Public-key cryptography based on the algebraic structure of Elliptic curves over Finite fields The use 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 Elliptic Curve Primality Proving (ECPP is a method based on Elliptic curves to prove the primality of a number In Mathematics, the Riemann-Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics In Mathematics, the Nagell–Lutz theorem is a result in the diophantine geometry of Elliptic curves Suppose that the equation y In Mathematics, complex multiplication is the theory of Elliptic curves E that have an Endomorphism ring larger than the Integers and Arithmetic dynamics is a new field that is an amalgamation of two areas of mathematics Dynamical systems and Number theory. Serge Lang ( May 19, 1927 – September 12, 2005) was a French -born American Mathematician. (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.

• 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). Richard E Crandall is an American Computer scientist and physicist who has made contributions to Computational number theory, most notably the development Carl Pomerance (born in 1944 in Joplin, Missouri) is a well known number theorist. "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). Anthony W Knapp is a mathematician at State University of New York Stony Brook working on representation theory who classified the Tempered representations of a semisimple Elliptic Curves. Math Notes 40, Princeton Univ. Press.  
• Neal Koblitz (1984). Neal Koblitz (born December 24, 1948) is a Professor of Mathematics at the University of Washington in the Department of Mathematics Introduction to Elliptic Curves and Modular Forms. Springer.  
• Neal Koblitz (1994). Neal Koblitz (born December 24, 1948) is a Professor of Mathematics at the University of Washington in the Department of Mathematics "Chapter 6", A Course in Number Theory and Cryptography, 2nd edition, Springer. ISBN 0-387-94293-9.  
• Serge Lang (1978). Serge Lang ( May 19, 1927 – September 12, 2005) was a French -born American Mathematician. Elliptic Curves: Diophantine Analysis. Springer.  
• Joseph H. Silverman (1986). Joseph H Silverman (born 1955 is currently a professor of mathematics at Brown University. The Arithmetic of Elliptic Curves. Springer.  
• Joseph H. Silverman (1994). Joseph H Silverman (born 1955 is currently a professor of mathematics at Brown University. Advanced Topics in the Arithmetic of Elliptic Curves. Springer.  
• Joseph H. Silverman; John Tate (1992). Joseph H Silverman (born 1955 is currently a professor of mathematics at Brown University. John Torrence Tate Jr born March 13, 1925 in Minneapolis Minnesota, is an American Mathematician, distinguished for many fundamental Rational Points on Elliptic Curves. Springer.  
• Lawrence Washington (2003). Elliptic Curves: Number Theory and Cryptography. Chapman & Hall/CRC. ISBN 1-58488-365-0.  

External links

• The Mathematical Atlas: 14H52 Elliptic Curves
• Eric W. Weisstein, Elliptic Curves at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W

• 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. PlanetMath is a free, collaborative online Mathematics Encyclopedia.