[kurikuri]

What? You’ve managed to mangle so many terms in so few words… Signatures can refer to two things: integrity checks on a file or authentication checks for a recieved file. In the integrity check situation a hash function (e.g., SHA) is often used. In the authentication check situation, we usually use a public/private keypair for asymmetric encryption; the hash function is only part of the process. The key material used to make this keypair (should) comes from some random number generator…

The ‘hash’ function is a deterministic transform, not a source of randomness.

[Tyyps]

He is technically not wrong, most signatures can be seen has a public coin interactive proof system where you prove knowledge of a private key. They are then compiled into an non-interactive proof system via the Fiat-Shamir transform that uses a random oracle concretely instantiated using a hash function (easy to see in Schnorr signature). So at the end you are using a Hash function to generate your random coin.