Prime Numbers

Chapter 7
ELLIPTIC CURVE ARITHMETIC
The history of what are called elliptic curves goes back well more than
a century. Originally developed for classical analysis, elliptic curves have
found their way into abstract and computational number theory, and now sit
squarely as a primary tool. Like the prime numbers themselves, elliptic curves
have the wonderful aspects of elegance, complexity, and power. Elliptic curves
are not only celebrated algebraic constructs; they also provide considerable
leverage in regard to prime number and factorization studies. Elliptic curve
applications even go beyond these domains; for example, they have an
increasingly popular role in modern cryptography, as we discuss in Section
8.1.3.
In what follows, our primary focus will be on elliptic curves over fields
Fp, withp>3 an odd prime. One is aware of a now vast research field—
indeed even an industry—involving fields F p k where k>1 or (more prevalent
in current applications) fields F 2 k. Because the theme of the present volume
is prime numbers, we have chosen to limit discussion to the former fields of
primary interest. For more information in regard to the alternative fields, the
interested reader may consult references such as [Seroussi et al. 1999] and
various journal papers referenced therein.
7.1 Elliptic curve fundamentals
Consider the general equation of a degree-3 polynomial in two variables, with
coefficients in a field F , set equal to 0:
ax 3 + bx 2 y + cxy 2 + dy 3 + ex 2 + fxy + gy 2 + hx + iy + j =0. (7.1)
To ensure that the polynomial is really of degree 3, we assume that at least
one of a, b, c, d is nonzero. We also assume that the polynomial is absolutely
irreducible; that is, it is irreducible in F [x, y], where F is the algebraic closure
of F . One might consider the pairs (x, y) ∈ F × F that satisfy (7.1); they
are called the affine solutions to the equation. Or one might consider the
projective solutions. For these we begin with triples (x, y, z) ∈ F × F × F
(with x, y, z not all zero) that satisfy
ax 3 + bx 2 y + cxy 2 + dy 3 + ex 2 z + fxyz+ gy 2 z + hxz 2 + iyz 2 + jz 3 =0. (7.2)
Note that (x, y, z) is a solution if and only if (tx, ty, tz) is also a solution, for
t ∈ F , t = 0. Thus, in the projective case, it makes more sense to talk of