Elliptic curve cryptography

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What is it?

Public-key cryptography on the group of points of $y^2 = x^3 + ax + b$ over a finite field. The group law was derived geometrically over the reals (the tangent slope by implicit differentiation); security rests on the discrete logarithm problem, which has nothing to do with calculus.

Formulas

$\lambda = \frac{3x_1^2 + a}{2y_1}, \quad x_3 = \lambda^2 - 2x_1, \quad y_3 = \lambda(x_1 - x_3) - y_1$
point doubling $P + P$ (computed mod $p$ in practice)

The mathematics behind it

Polynomial functions★★★★★indirect
The curve is the zero set of the polynomial $y^2 - x^3 - ax - b$ , but over a finite field, where calculus does not apply.
Implicit differentiation★★★★★historical
The point-doubling formula uses the slope $\frac{3x^2 + a}{2y}$ , derived over the reals and reused verbatim over finite fields.

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.