Elliptic curve
In mathematics, elliptic curves are the graphs of certain cubic (third degree) equations. They have been used in the proof of Fermat's last theorem and they also find applications in cryptography and integer factorization.
Elliptic curves are non-singular, meaning they don't have cusps or self-intersections, and a binary operation can be defined for their points in a natural geometric fashion, thus turning the set of points into an abelian group.
Typical elliptic curves over the field of real numbers are given by the equations
- y2 = x3 - x
and
- y2 = x3 - x + 1
(images of the graphs would be nice)
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.
If the characteristic of K is neither 2 nor 3, then every elliptic curve over K can be written in the form
- y2 = x3 - px - q
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.
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.
By adding a point "at infinity", we obtain the projective version of this curve; every straight line intersects this curve in three points (if the line is tangent to the curve at a point, then that point is counted twice). It is then possible to introduce a group operation on the curve with the following property: if a straight line intersects the curve at the points P, Q and R, then 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. The set of K-rational points forms a subgroup of this group.
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. The recent proof of Fermat's last theorem proceeded by proving a special case of the deep Taniyama-Shimura conjecture about elliptic curves over the rationals; the conjecture has since been completely proved.
If the underlying field K is the field of complex numbers, then every elliptic curve can be parametrized by a certain elliptic function and its derivative. Specifically, to every elliptic curve E there exists a lattice L and a Weierstrass elliptic function p, such that the map
- φ : C/L → E
with
- φ (z) = (p(z), p '(z))
is a group isomorphism and an isomorphism of Riemann surfaces. This shows in particular that E topologically looks like a torus (since C/L is a torus).
Elliptic curves over finite fields are used in some cryptographic applications (elliptic curve cryptography) as well as for integer factorization (Lenstra Elliptic Curve Factorization). Typically, the general idea in these applications is that a known algorithm which makes use of a certain group is rewritten to use the group arising from an elliptic curve.