An elliptic curve is a curve defined by $y^2 = x^3 + ax + b$ (over a finite field). ECC provides the same security as RSA with much smaller key sizes — 256-bit ECC ≈ 3072-bit RSA.

* A non-singular elliptic curve is smooth and mirror-symmetric across the x-axis; a point P and its reflection −P are additive inverses. *

Mathematical definition: An elliptic curve over a field $\mathbb{F}$ is the set of points $(x, y)$ satisfying: $$y^2 = x^3 + ax + b$$ plus a special "point at infinity" $\mathcal{O}$ (the neutral element).

The discriminant $4a^3 + 27b^2 \neq 0$ must hold (ensures no singularities).

Why ECC for cryptography:

• Much shorter keys: 256-bit ECC ≈ 3072-bit RSA security
• Faster operations: Especially important for constrained devices (smart cards, IoT)
• No subexponential attacks known: Unlike RSA (where number field sieve exists), the best attack on ECC is still exponential (Pollard's rho)
• Used everywhere: TLS 1.3, Signal, Bitcoin, SSH, mobile devices

The group operation: Points on the curve form a group under "point addition" — geometrically, draw a line through two points, find the third intersection, and reflect over the x-axis. This replaces multiplication in RSA with addition on the curve.

Go deeper:

• Elliptic-curve cryptography — Wikipedia — curve equation, group law, and the RSA key-size parity in one place.
• Corbellini — ECC: a gentle introduction (part 1) — the clearest visual walk-through of curves and the group operation.