ECDSA Signatures: The Proof of Intent

In Bitcoin, a Digital Signature is the cryptographic "Seal" that proves you authorize a transaction. It allows you to spend your funds without ever revealing your Private Key to the network.

1. The Signing Equation

The algorithm used for these signatures is ECDSA (Elliptic Curve Digital Signature Algorithm). To create a signature, you need three pieces of data:

1. Transaction Hash ($z$): A unique ID of the message you want to sign.

2. Private Key ($d$): Your secret 256-bit number.

3. Ephemeral Key ($k$): A temporary, random number used only once (the "Nonce").

2. The Result: (r, s)

The output of the ECDSA algorithm is not a single number, but a Pair of Coordinates known as $r$ and $s$.

• $r$: Represents the X-coordinate of a point on the curve derived from the ephemeral key $k$.

• $s$: Represents the actual signature proof, combining the private key, the nonce, and the message hash.

3. The Magic of Verification

When a Bitcoin node receives your transaction, it looks at your Public Key, the Transaction Hash, and the $(r, s)$ Signature.

• It performs a mathematical calculation to see if the signature "Fits" the public key.

• If the math checks out, the node knows that ONLY the person with the matching private key could have created that signature.

4. One-Time Use

A signature is only valid for One specific transaction. If you try to use the same signature for a different transaction, the verification will fail because the Transaction Hash ($z$) will be different.

Component	Role	Publicly Visible?
Private Key	The "Master Secret"	NO
Public Key	Your Identity	Yes
(r, s)	The Proof of Spending	Yes
Hash (z)	The "Check" or "Contract"	Yes

In the next section, we will analyze the r and s components in detail.