cryptography

ECC in Real Life: What Happens When You Open a Website

When you open a website over HTTPS, send a WhatsApp or Signal message, or make a Bitcoin transaction, elliptic curves are working in the background. This page follows one complete example, opening https://example.com, using the tiny curve from the step-by-step walkthrough so you can check every number. Then it shows what changes at real size.

In short:

  1. Your device picks a random secret number (the private key).
  2. It computes secret × G (the public key) and sends that.
  3. Both sides combine the other’s public key with their own secret and reach the same shared point.
  4. That shared point becomes the AES key that encrypts everything.

Contents

The setup (public, the same for everyone)

Both sides already know:

Real browsers use a standard curve such as X25519, which is built into every browser and server.

Step 1: Each side picks a random secret (private key)

  Your browser example.com server
Random secret 3 9
Who knows it only your browser only the server

A fresh secret is picked every time you connect, and thrown away afterwards.

Step 2: Each side computes secret × G (public key) and sends it

  Your browser Server
Computes 3 × G = 3G = (10, 6) 9 × G = 9G = (7, 6)
Sends “Hello, my key is (10, 6)” → ← “Hello, my key is (7, 6)”

This exchange is the start of the TLS handshake, and it happens in plain sight.

👀 Eve, who is watching the Wi-Fi, sees: the curve, G = (5, 1), (10, 6) and (7, 6). She does not see 3 or 9.

Step 3: Each side combines the other’s public key with its own secret

  Your browser Server
Takes the server’s key (7, 6) = 9G your key (10, 6) = 3G
Multiplies by its own secret 3 × 9G = 27G 9 × 3G = 27G
Result (13, 7) (13, 7) ✓

Both get the same point without ever sending it, because 3 × 9 = 9 × 3. (The cycle has 19 steps, so 27G = 8G = (13, 7). The full cycle is in Step 8 of the walkthrough.)

Eve is stuck. To get (13, 7) she needs 3 or 9. Getting 3 from (10, 6) is the elliptic-curve discrete logarithm problem. On this toy curve she could try all 19 values, but on a real curve there are about 10⁷⁷.

Step 4: The shared point becomes the AES key

The browser and server take the x-coordinate, 13, and run it through a hash function (a key-derivation function, HKDF) to get a proper key:

shared point x = 13
        ↓  HKDF (hashing)
AES key = 7f3a…c91e   (256 bits of random-looking data; illustrative)

From now on, everything is encrypted with that AES key:

You type:   password = "hunter2"
Sent as:    8f1c09e4b2a7…   ← Eve sees only gibberish (illustrative)
Server decrypts it with the same AES key → "hunter2"

Why switch to AES? ECC is good at agreeing on a secret. AES is much faster at encrypting large amounts of data, such as web pages, videos and messages. So ECC sets up the key, and AES does the heavy lifting.

The same thing at real size

  Toy example Real HTTPS (X25519)
Prime p 17 2²⁵⁵ − 19 (77 digits)
Number of points 19 ≈ 2²⁵² ≈ 10⁷⁶
Private key 3 a random 32-byte number
Public key (10, 6) 32 bytes
Time to compute instant about 0.05 ms
Eve trying every key 19 tries longer than the age of the universe

How this changes in each app

🔒 HTTPS: exactly the steps above. There’s one extra piece: the server also signs its public key with its certificate (using ECDSA or a similar scheme). This proves you’re talking to the real example.com and not Eve pretending to be it, which stops a “man-in-the-middle” attack.

💬 WhatsApp / Signal: the same key exchange, but repeated constantly. Every message gets a new key derived from fresh ECC exchanges (the “double ratchet”). Even if one key leaked, older and newer messages would stay safe.

₿ Bitcoin works differently. It uses ECC to sign, not to share an AES key, and nothing is encrypted:

So “the shared point becomes the AES key” applies to HTTPS and messaging. Bitcoin uses the same curve maths, but for proving ownership.

The example on one screen

Public:   curve y² = x³ + 2x + 2 (mod 17),  G = (5, 1)

Browser:  secret 3  →  sends 3G = (10, 6)
Server:   secret 9  →  sends 9G = (7, 6)

Browser:  3 × (7, 6)  = 27G = (13, 7)  ┐
Server:   9 × (10, 6) = 27G = (13, 7)  ┘ same, never sent

AES key = HKDF(13)  →  encrypts the whole page
Eve saw (10, 6) and (7, 6), but can't get 3, 9 or (13, 7).

Try it in the lab

You can reproduce these exact numbers in the Elliptic Curve Lab:

  1. In section 2, click Tiny p = 17, then click the dot at (5, 1) to make it G.
  2. In section 3, set Alice (the browser) = 3 and Bob (the server) = 9. The shared point is (13, 7).
  3. In section 4, press Try to find Alice’s secret to see Eve break the toy key instantly.

Read next: Step-by-step walkthrough · Full elliptic curve guide