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The elliptic curve $y^{2}=x^{3}-x^{2}+{\frac {1}{2}}$.

Introduction

Elliptic curves over $\mathbb {R}$, the field of real numbers, are visually appealing and readily depicted as plots of cubic curves, or curves of degree three on x-y planes with Cartesian coördinates, which serve as a graphical aid for understanding algebraic operations on them. The terminology is somewhat vague and confusing to the uninitiated, because actual ellipses are conic sections or quadratic curves, having a degree of two, whereas quartic or quintic curves, of degree four or five, or curves of even higher degree are often called “hyperelliptic,” with respect to algebraic degree rather than “genus” or other topological properties.

And again, without respect of “gender” as such, this is an area of high school algebra level “math jocks,” girls chewing bubble gum and saying “Math is hard!” etc., etc. and then we have to deal with overeducated college “frat boys” and “sorry girls.” In other words, there is a great deal of deliberate stupidity that needs to be confronted head-on.

Rational points

Finding the rational points on elliptic curves determined by equations with rational coefficients in the third degree in two variables has long been the object of much pure mathematical study for the sake of its own beauty.

Mordell’s theorem, that all the rational points on an elliptic curve, even infinitely many of them, may be generated by only a finite number of them with a certain algebraic point group operation, is the starting point for this study.

Finite fields

Quotient groups among the rational points on an elliptic curve have led naturally to the study of elliptic curves over finite fields. The idea is akin to finding a large prime number to serve as a “least common denominator” of sorts for a group of rational points, and then considering only the numerators of proper fractions with respect to that denominator, using modular arithmetic, with the extended Euclidean algorithm among other multiple precision arithmetic operations on big integers.

Helmut Hasse and Emil Artin proved that the number of points on an elliptic curve over a finite field of q elements, [i.e., modulo the prime q or the finite field GF(q=p^k),] is between $q+1-2{\sqrt {q}}$ and $q+1+2{\sqrt {q}}$ inclusive. (André Weil generalized the result to range between $q+1-2g{\sqrt {q}}$ and $q+1+2g{\sqrt {q}}$ inclusive for hyperelliptic curves of genus g>1.)

It is in general a very difficult problem to calculate the exact number of points on an algebraic curve over a finite field with this range, and the security of all elliptic curve cryptographic schemes, based on the difficulty of the discrete logarithm problem with respect to the point group operation, is in turn supposed to depend on this difficulty, if it is not defeated by the use of weak or trivially reducible curves in cryptographic applications.

Cryptographic applications

Elliptic curves over finite fields have serious applications to public key schemes of strong cryptography, the first widely implemented example being Ed25519, still in use today despite being somewhat controversial because of intellectual property patent claims and a perceived association with communism, communist spies, busybodies in general, and people who don’t mind their own business; hence the very need for strong cryptography rather than bad curves and weak crypto.

Unsolved problems

Dirichlet L-functions are the elliptic curve finite-field analogs of the Riemann ζ-function , closely related to two of CMI’s Millennium problems, the Birch and Swinnerton-Dyer conjecture and the Riemann hypothesis.