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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, the Cartesian product of K with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions (x, y) for:

Not to be confused with Ellipse.

for some coefficients a and b in K. The curve is required to be non-singular, which means that the curve has no cusps or self-intersections. (This is equivalent to the condition 4a3 + 27b2 ≠ 0, that is, being square-free in x.) It is always understood that the curve is really sitting in the projective plane, with the point O being the unique point at infinity. Many sources define an elliptic curve to be simply a curve given by an equation of this form. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular cubic curves; see § Elliptic curves over a general field below.)


An elliptic curve is an abelian variety – that is, it has a group law defined algebraically, with respect to which it is an abelian group – and O serves as the identity element.


If y2 = P(x), where P is any polynomial of degree three in x with no repeated roots, the solution set is a nonsingular plane curve of genus one, an elliptic curve. If P has degree four and is square-free 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 the intersection of two quadric surfaces embedded in three-dimensional projective space, is called an elliptic curve, provided that it is equipped with a marked point to act as the identity.


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 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 Andrew Wiles's proof of Fermat's Last Theorem. They also find applications in elliptic curve cryptography (ECC) and integer factorization.


An elliptic curve is not an ellipse in the sense of a projective conic, which has genus zero: see elliptic integral for the origin of the term. However, there is a natural representation of real elliptic curves with shape invariant j ≥ 1 as ellipses in the hyperbolic plane . Specifically, the intersections of the Minkowski hyperboloid with quadric surfaces characterized by a certain constant-angle property produce the Steiner ellipses in (generated by orientation-preserving collineations). Further, the orthogonal trajectories of these ellipses comprise the elliptic curves with j ≤ 1, and any ellipse in described as a locus relative to two foci is uniquely the elliptic curve sum of two Steiner ellipses, obtained by adding the pairs of intersections on each orthogonal trajectory. Here, the vertex of the hyperboloid serves as the identity on each trajectory curve.[1]


Topologically, a complex elliptic curve is a torus, while a complex ellipse is a sphere.

A is defined as an odd square-free integer n which is the area of a right triangle with rational side lengths. It is known that n is a congruent number if and only if the elliptic curve has a rational point of infinite order; assuming BSD, this is equivalent to its L-function having a zero at s = 1. Tunnell has shown a related result: assuming BSD, n is a congruent number if and only if the number of triplets of integers (x, y, z) satisfying is twice the number of triples satisfying . The interest in this statement is that the condition is easy to check.[14]

congruent number

In a different direction, certain analytic methods allow for an estimation of the order of zero in the center of the for certain L-functions. Admitting BSD, these estimations correspond to information about the rank of families of the corresponding elliptic curves. For example: assuming the generalized Riemann hypothesis and BSD, the average rank of curves given by is smaller than 2.[15]

critical strip

Elliptic curve cryptography

key exchange

Elliptic-curve Diffie–Hellman

Supersingular isogeny key exchange

Elliptic curve digital signature algorithm

digital signature algorithm

EdDSA

random number generator

Dual EC DRBG

Lenstra elliptic-curve factorization

Elliptic curve primality proving

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:

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 curve

Montgomery curve

Arithmetic dynamics

Elliptic algebra

Elliptic surface

Comparison of computer algebra systems

Isogeny

j-line

Level structure (algebraic geometry)

Modularity theorem

Moduli stack of elliptic curves

Nagell–Lutz theorem

Riemann–Hurwitz formula

Wiles's proof of Fermat's Last Theorem

I. Blake; G. Seroussi; N. Smart (2000). Elliptic Curves in Cryptography. LMS Lecture Notes. Cambridge University Press.  0-521-65374-6.

ISBN

Brown, Ezra (2000), "Three Fermat Trails to Elliptic Curves", The College Mathematics Journal, 31 (3): 162–172, :10.1080/07468342.2000.11974137, S2CID 5591395, winner of the MAA writing prize the George Pólya Award

doi

; Carl Pomerance (2001). "Chapter 7: Elliptic Curve Arithmetic". Prime Numbers: A Computational Perspective (1st ed.). Springer-Verlag. pp. 285–352. ISBN 0-387-94777-9.

Richard Crandall

Cremona, John (1997). (2nd ed.). Cambridge University Press. ISBN 0-521-59820-6.

Algorithms for Modular Elliptic Curves

Darrel Hankerson, and Scott Vanstone (2004). Guide to Elliptic Curve Cryptography. Springer. ISBN 0-387-95273-X.

Alfred Menezes

; Wright, E. M. (2008) [1938]. An Introduction to the Theory of Numbers. Revised by D. R. Heath-Brown and J. H. Silverman. Foreword by Andrew Wiles. (6th ed.). Oxford: Oxford University Press. ISBN 978-0-19-921986-5. MR 2445243. Zbl 1159.11001. Chapter XXV

Hardy, G. H.

Hellegouarch, Yves (2001). Invitation aux mathématiques de Fermat-Wiles. Paris: Dunod.  978-2-10-005508-1.

ISBN

(2004). Elliptic Curves. Graduate Texts in Mathematics. Vol. 111 (2nd ed.). Springer. ISBN 0-387-95490-2.

Husemöller, Dale

; Michael I. Rosen (1998). "Chapters 18 and 19". A Classical Introduction to Modern Number Theory. Graduate Texts in Mathematics. Vol. 84 (2nd revised ed.). Springer. ISBN 0-387-97329-X.

Kenneth Ireland

(2018) [1992]. Elliptic Curves. Mathematical Notes. Vol. 40. Princeton University Press. ISBN 9780691186900.

Knapp, Anthony W.

(1993). Introduction to Elliptic Curves and Modular Forms. Graduate Texts in Mathematics. Vol. 97 (2nd ed.). Springer-Verlag. ISBN 0-387-97966-2.

Koblitz, Neal

(1994). "Chapter 6". A Course in Number Theory and Cryptography. Graduate Texts in Mathematics. Vol. 114 (2nd ed.). Springer-Verlag. ISBN 0-387-94293-9.

Koblitz, Neal

(1978). Elliptic curves: Diophantine analysis. Grundlehren der mathematischen Wissenschaften. Vol. 231. Springer-Verlag. ISBN 3-540-08489-4.

Serge Lang

Henry McKean; (1999). Elliptic curves: function theory, geometry and arithmetic. Cambridge University Press. ISBN 0-521-65817-9.

Victor Moll

Ivan Niven; Herbert S. Zuckerman; (1991). "Section 5.7". An introduction to the theory of numbers (5th ed.). John Wiley. ISBN 0-471-54600-3.

Hugh Montgomery

(1986). The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics. Vol. 106. Springer-Verlag. ISBN 0-387-96203-4.

Silverman, Joseph H.

(1994). Advanced Topics in the Arithmetic of Elliptic Curves. Graduate Texts in Mathematics. Vol. 151. Springer-Verlag. ISBN 0-387-94328-5.

Joseph H. Silverman

; John Tate (1992). Rational Points on Elliptic Curves. Springer-Verlag. ISBN 0-387-97825-9.

Joseph H. Silverman

(1974). "The arithmetic of elliptic curves". Inventiones Mathematicae. 23 (3–4): 179–206. Bibcode:1974InMat..23..179T. doi:10.1007/BF01389745. S2CID 120008651.

John Tate

Lawrence Washington (2003). . Chapman & Hall/CRC. ISBN 1-58488-365-0.

Elliptic Curves: Number Theory and Cryptography

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.

LMFDB: Database of Elliptic Curves over Q

, Encyclopedia of Mathematics, EMS Press, 2001 [1994]

"Elliptic curve"

"Elliptic Curves". MathWorld.

Weisstein, Eric W.

from PlanetMath

The Arithmetic of elliptic curves

and over Zp – web application that requires HTML5 capable browser.

Interactive elliptic curve over R

This article incorporates material from Isogeny on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.