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Introduction
Elliptic curves over , the field of real numbers, are visually appealing and readily depicted as plots of curves on x-y planes with Cartesian coördinates, which serve as a graphical aid for understanding algebraic operations on them.
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 without much practical application in mind.
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.
Now all of a sudden, elliptic curves 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. Necessity is the mother of invention.
