ECDH Key Exchange + ECDH Discrete Logarithm Problem

How discrete logarithm works on an elliptic curve group:

Let’s take a point G and add it to itself x times to produce another point P via the addition operation we defined. We can write that as P = G + … + G (x times) or just write that as P = [x]G, which reads x times G.

The elliptic curve discrete logarithm problem (ECDLP)

is to find the number x from knowing just P and G.

Step 1

All the participants agree on an elliptic curve equation, a finite field (most likely a prime number), and a generator G (usually called a base point in elliptic curve cryptography).

Step 2

Each participant generates a random number x, which becomes their private key.

Step 3

Each participant derives their public key as [x]G.

Because the elliptic curve discrete logarithm problem is hard,

no one should be able to recover your private key just by looking at your public key.

If x is small

the attacker can find the private key x