The Trapdoor Geometry

Course Overview

A trapdoor function is easy to compute in one direction yet computationally infeasible to invert without secret information. The trapdoor geometry of an elliptic curve $E$ over a finite field $\mathbb{F}_p$ underwrites modern public-key cryptography: scalar multiplication $k \mapsto kP$ is cheap, while the discrete logarithm $P \mapsto k$ is presumed hard. This tome will guide the scholar from group law axioms to ECDSA signatures and the post-quantum landscape.

Planned Sections

• ⬡ The group law on an elliptic curve — geometric & algebraic forms
• ⬡ Curves over $\mathbb{F}_p$ and the Hasse-Weil bound
• ⬡ Scalar multiplication via double-and-add
• ⬡ The Elliptic Curve Discrete Logarithm Problem (ECDLP)
• ⬡ ECDSA signature construction and verification
• ⬡ Pairing-based cryptography (Weil & Tate pairings)
• ⬡ Post-quantum considerations — why ECC will eventually fall