Elliptic Curve Cryptography (ECC)

Elliptic curves are these beautiful mathematical objects - curves defined by equations like y² = x³ + ax + b. In cryptography, we do arithmetic on points on these curves. The security comes from this problem: given two points P and Q on the curve, find the number n where Q = nP (adding P to itself n times). This is called the elliptic curve discrete logarithm problem, and it's computationally hard for classical computers.

The beauty of ECC is efficiency - a 256-bit ECC key provides the same security as a 3072-bit RSA key, making it perfect for mobile devices and IoT. Bitcoin uses it. Your phone uses it. The problem: Shor's algorithm breaks it. A quantum computer with about 2000 good qubits can crack ECC-256, exposing Bitcoin wallets and breaking most mobile security. Unlike AES where you can just use longer keys, ECC doesn't have that option - you need to replace it entirely with post-quantum alternatives.

ECC is everywhere because it's so efficient. But that efficiency doesn't help against quantum computers - they break 256-bit ECC faster than 2048-bit RSA. Every smartphone, every cryptocurrency, every modern TLS connection uses ECC.

The migration challenge is enormous because you can't just "upgrade" - you need completely different math.

Cryptocurrency exchanges must migrate ECC to post-quantum signatures to protect billions in digital assets. Mobile security needs post-quantum alternatives that maintain efficiency on battery-powered devices.

The entire IoT ecosystem built on ECC needs firmware updates to post-quantum crypto before quantum computers arrive.