ECC group law and isogeny-based cryptography on supersingular curves — E8 Intelligence Research

FINDING: ECC rests on the group law of a genus-1 smooth cubic; isogeny-based crypto replaces the discrete-log assumption with the hardness of finding isogenies between supersingular curves. | MATH: Weierstrass form y² = x³ + ax + b, discriminant Δ = −16(4a³ + 27b²) ≠ 0; group law via chord–tangent; E(K̄) ≅ (torsion) ⊕ Z^r; Hasse bound |#E(F_q) − (q+1)| ≤ 2√q; Frobenius endomorphism π satisfies π² − tπ + q = 0 with t = q+1−#E; isogeny φ: E→E′ is a finite surjective morphism with deg φ = ker φ (separable case); supersingular iff p | t. | CONNECTION: The Hasse interval 2√q is the discriminant boundary of the characteristic polynomial of π — a quadratic lattice/root-system structure (rank-2 Z-module with positive-definite trace form). The j-invariant j = 1728·4a³/(4a³+27b²) is a modular function on the upper half-plane; its special values at CM points generate ring class fields (Kronecker's Jugendtraum). The constant 1728 = 12³ = (2·6)³ encodes the 12-fold symmetry of the modular curve X(1 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com