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Elliptic Curves And Finite Fields

7 articles inside this collection.

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.

Articles in this collection

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  1. 01
    Introduction To Elliptic CurvesJune 11, 2026 · 3 min read

    A high-level orientation to elliptic curves and the mathematical path from smooth real curves to finite-field point groups.

  2. 02
    Elliptic Curves Over Real NumbersJune 11, 2026 · 8 min read

    Build visual intuition for elliptic curves using real-number curves, Weierstrass form, smoothness, and curve points.

  3. 03
    Point Addition And The Elliptic Curve GroupJune 11, 2026 · 10 min read

    Learn the chord-and-tangent point addition rule, point doubling, inverse points, the point at infinity, and why curve points form a group.

  4. 04
    Scalar MultiplicationJune 11, 2026 · 5 min read

    Understand scalar multiplication as repeated point addition and the efficient doubling methods that make large finite-field calculations practical.

  5. 05
    Finite Fields And Modular ArithmeticJune 11, 2026 · 6 min read

    Learn the prime-field arithmetic behind elliptic curves: modular operations, inverses, and division modulo p.

  6. 06
    Elliptic Curves Over Finite FieldsJune 11, 2026 · 15 min read

    Construct a finite-field elliptic curve point set, add its points with modular formulas, and connect the resulting group to cryptographic parameters.

  7. 07
    Subgroups, Generators, Orders, And CofactorsJuly 22, 2026 · 8 min read

    Follow scalar multiples around a finite curve to distinguish curve order, point order, generated subgroups, generators, base points, and cofactors.