math concept
2 topics use this
Math concept
Elliptic Curves
Core equation
$$y^2 = x^3 + ax + b$$
Elliptic curves are smooth cubic curves with a group law that makes point addition the hard problem underlying ECC cryptography. They also appear in number theory, coding theory, and the proof of Fermat's Last Theorem.
The group law
An elliptic curve over a field $F$ is $E: y^2 = x^3 + ax + b$ (with $4a^3 + 27b^2 \neq 0$).
Points on $E$ form an abelian group under the chord-and-tangent rule: to add $P + Q$, draw the line through $P,Q$, find the third intersection with $E$, reflect over the $x$-axis. The identity is the point at infinity $\mathcal{O}$.
Elliptic Curve Discrete Logarithm
In $E(\mathbb{F}_p)$, given $P$ and $Q = kP$, finding $k$ is the elliptic curve discrete logarithm problem (ECDLP) — believed harder than the classical DLP, so ECC achieves equivalent security at smaller key sizes (256-bit ECC $\approx$ 3072-bit RSA).
Key curves in practice
| Curve | Field | Applications |
|---|---|---|
| P-256 (NIST) | $\mathbb{F}_p$ | TLS, ECDSA |
| Curve25519 | $\mathbb{F}_{2^{255}-19}$ | Signal, WireGuard |
| secp256k1 | $\mathbb{F}_p$ | Bitcoin, Ethereum |
Fields that use this concept
Engineering & CS
Cryptography