Why Must the Nonce (K) Be Unique and Secret for Every Signature?

The nonce (k) must be unique and secret because it is a variable in the ECDSA signing equation that, if known or reused, allows for the private key (d) to be algebraically solved. Reusing the same 'k' for two different messages creates a system of two equations with two unknowns (d and k), which can be solved simultaneously to find 'd'.

This is a critical failure mode that compromises the entire key.