ECDSA Signatures

What’s a Digital Signature?

Alice wants to send a message to Bob. But how does Bob know it’s really from Alice?

A digital signature proves two things:

The message came from Alice (not an imposter)
The message wasn’t tampered with

Bitcoin and Ethereum use ECDSA to sign every transaction.

The Setup

Alice has her ECC key pair:

Private key: $d$ (a secret number only she knows)
Public key: $Q = dG$ (a point everyone can see)

She wants to sign a message so anyone can verify it came from her.

How Signing Works

Alice has a message. Here’s what she does:

1. Hash the message

Any message becomes a fixed-size number $h$ . This is the message’s fingerprint.

2. Pick a random number $k$

This is a one-time secret. It must be random and never reused.

3. Compute $r$

Multiply: $R = kG$ (a point on the curve)

Take just the x-coordinate: $r$ = x-coordinate of $R$

4. Compute $s$

$s = \frac{h + d \cdot r}{k}$

This mixes together the message hash ( $h$ ), her private key ( $d$ ), and the random value ( $k$ ).

5. The signature is $(r, s)$

Two numbers. Alice sends these along with her message.

How Verification Works

Bob receives the message and signature $(r, s)$ . He also knows Alice’s public key $Q$ .

1. Hash the message

Bob computes the same hash $h$ from the message.

2. Compute a check point $R'$

Using only public information:

$R' = \frac{h \cdot G + r \cdot Q}{s}$

3. Compare x-coordinates

If the x-coordinate of $R'$ equals $r$ , the signature is valid.

Bob never needed Alice’s private key. He verified using only public information.

Why Only Alice Can Sign

The signature formula uses Alice’s private key $d$ :

$s = \frac{h + d \cdot r}{k}$

To forge a signature, an attacker would need to find $d$ from $Q = dG$ .

That’s ECDLP. Practically impossible.

The Random $k$ is Critical

If Alice ever uses the same $k$ for two different messages, an attacker can compute her private key.

In 2010, Sony’s PlayStation 3 signing key was stolen because they used the same $k$ every time.

The random number $k$ must be truly random and never repeated.