Despite the name, an elliptic curve doesn't look anything like an ellipse (a squashed circle). The name comes from the fact that this curve shows up in "the calculation for finding an ellipse's circumference (elliptic integrals)." But this unfamiliar curve holds a curious rule for adding point to point — and let's see, step by step, how that rule became a cipher that protects Bitcoin and the HTTPS connections we use every day (this content includes high-school to early-college level material).
An elliptic curve is a curve drawn by the equation y² = x³ + ax + b. Two points on this curve can be "added" in a very particular way — and while repeating this addition over and over is easy, working backward from just the result to figure out how many times it was added is extremely hard. Where RSA uses "the difficulty of factoring," elliptic curve cryptography (ECC) uses "the difficulty of undoing this addition."
In y² = x³ + ax + b, change a and b and see how the curve's shape changes.
To add two points P and Q on an elliptic curve: draw the line connecting them, find the third point where it meets the curve, and flip that point across the x-axis. That's P+Q.
Real cryptography doesn't calculate with real numbers — it uses "remainder arithmetic (mod)." Check the multiples you get from repeatedly adding a base point P, on the small curve y² ≡ x³+2x+2 (mod 17), computed with remainders modulo the prime 17.
The computer picked a secret number k and computed kP to show you. See if you can figure out k from just P and kP — even on this toy curve with only 18 possible cases, notice that there's no better method than checking them one at a time.
Cryptocurrencies like Bitcoin and Ethereum use elliptic curve cryptography (ECDSA) for wallet signatures. It proves that a transaction was approved from your wallet, without ever revealing your private key (the secret number k).
HTTPS website connections mostly use elliptic-curve-based key exchange (ECDH) too, so your browser and the server can safely exchange encryption keys. The very moment that lock icon appears, the point addition you learned on this page is actually at work.
Smartphones and messenger apps (like Signal) favor ECC as well. It can achieve similar security with a much shorter key than RSA needs, which is especially useful for devices with limited battery and computing power.