Elliptic Curves And Finite Fields
This collection builds the mathematical foundation behind elliptic-curve cryptography. It begins with the familiar shape of an elliptic curve over the real numbers, using geometry to make point addition and repeated addition feel concrete. It then replaces continuous coordinates with exact finite-field arithmetic, where the curve becomes a finite set of points that can safely support cryptographic operations. By the end, you will understand the public base point, its large cyclic subgroup, the subgroup order, and the cofactor—the ingredients that later protocols use for keys, key agreement, signatures, encryption, and threshold systems.
Chapters
- Introduction To Elliptic Curves
A high-level orientation to elliptic curves and the mathematical path from smooth real curves to finite-field point groups.
June 11, 2026 · 3 min read - Elliptic Curves Over Real Numbers
Build visual intuition for elliptic curves using real-number curves, Weierstrass form, smoothness, and curve points.
June 11, 2026 · 8 min read - Point Addition And The Elliptic Curve Group
Learn the chord-and-tangent point addition rule, point doubling, inverse points, the point at infinity, and why curve points form a group.
June 11, 2026 · 12 min read - Scalar Multiplication
Understand scalar multiplication as repeated point addition and the efficient doubling methods that make large finite-field calculations practical.
June 11, 2026 · 5 min read - Finite Fields And Modular Arithmetic
Learn the prime-field arithmetic behind elliptic curves: modular operations, inverses, and division modulo p.
June 11, 2026 · 6 min read - Elliptic Curves Over Finite Fields
Construct a finite-field elliptic curve point set, add its points with modular formulas, and connect the resulting group to cryptographic parameters.
June 11, 2026 · 15 min read - Subgroups, Generators, Orders, And Cofactors
Follow scalar multiples around a finite curve to distinguish curve order, point order, subgroups, base points, cofactors, and the base and scalar fields.
July 22, 2026 · 13 min read