Elliptic Functions

ECC (P-256) protects 2 billion Apple devices. The group of points on an elliptic curve is isomorphic to the torus C/Lambda -- a genus-1 Riemann surface parametrized by the Weierstrass p-function.

• **Cryptography:** ECC (P-256, Curve25519) secures TLS 1.3, Signal, WhatsApp. Security rests on ECDLP.
• **Number theory:** elliptic curves over Q used by Wiles (1995) to prove Fermat's Last Theorem via the Frey curve modularity.
• **Physics:** elliptic integrals (periods) describe pendulum motion at large amplitudes and the catenary chain shape.

The Weierstrass p-Function

Elliptic curve cryptography (ECC) secures TLS 1.3 -- the backbone of HTTPS. Apple uses ECC with the NIST P-256 curve to protect 2 billion devices. The group of points on the curve is algebraically isomorphic to a torus -- a genus-1 Riemann surface parametrized by the Weierstrass p-function.

Elliptic Curves and Cryptography

NIST P-256 is the standard elliptic curve y^2=x^3-3x+b over F_p (p~2^256), securing TLS 1.3, Signal, and WhatsApp. Key generation: choose random k, compute Q=kG -- multiply base point G by scalar k using the curve group law.

Key ideas

• **Weierstrass p-function:** doubly periodic with poles at lattice Lambda. Equation: (p')^2=4p^3-g2*p-g3.
• **Torus C/Lambda:** genus-1 Riemann surface. Parametrization: x=p(z), y=p'(z) -- isomorphism C/Lambda -> E(C).
• **ECDLP:** Q=kG in O(log k) steps. Inverting takes O(sqrt(p)). Foundation of ECC security.