What is a Nonce in Digital Signatures?

Nonce (short for "number used once") is a random, unique value used in the signing process.

In Schnorr signatures, each signature requires a fresh nonce. Reusing the same nonce is a dangerous mistake that allows an attacker to calculate your private key.

Let's start by breaking down the Schnorr signature equation to see how this happens.

If the nonce k remains the same across different transactions, an attacker can isolate the private key d by comparing two signatures.

Imagine you want to sign two separate transactions but mistakenly use the same nonce for both. Here's how it plays out:

Here’s where things go wrong when the nonce is reused!

Since the only variable difference between S1 and S2 is the hash of the transactions, an attacker can calculate k and subsequently derive your private key d. This exposes your Bitcoin funds to theft.

5. Calculating the Private Key from Reused Nonces

With both S1 and S2 known, an attacker can use the following equation to isolate d (private key):

Since all these values are publicly accessible (S1, S2, and the hashed transactions), reusing the nonce makes it trivial to solve for d. This leads to a complete compromise of the private key.

6. How to Prevent Nonce Reuse Attacks

The solution is straightforward:

Never reuse a nonce!
Never reuse a nonce!
Never reuse a nonce!
Never reuse a nonce!
Never reuse a nonce!
Never reuse a nonce!
Never reuse a nonce!
Never reuse a nonce!
Never reuse a nonce!
Never reuse a nonce!
Never reuse a nonce!

Always ensure that your cryptographic library generates a unique, random nonce for each signature.

7. Final Warning: Don’t Reuse Nonces!

To safeguard your Bitcoin, remember:

DON’T REUSE NONCES

This simple rule keeps your private key secure and your transactions safe from attackers.