Why Is a Random Number (K) Critical for the Security of Each ECDSA Signature?

A unique, securely random number 'k' is essential for every ECDSA signature. If the same 'k' is used to sign two different messages, an attacker can easily use the two resulting signatures to compute the private key.

This vulnerability, known as a 'fault attack' or 'k-reuse attack', has been exploited in the past. To mitigate this, modern implementations use a deterministic method (RFC 6979) to generate 'k' based on the private key and the message hash.