I was thinking about using 2 keys for hashing each user password, obtaining 2 different hashes. This way, it would be (almost?) impossible to find a password that works, other than the actual password.
Is that right? Is it worth it?
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An important rule to learn is “never try to invent your own cryptography”. You are just wasting time at best and introducing security holes at worst.
If you are unsure whether you are an exception to this rule, then you are not an exception to this rule.
The designers of cryptographic hashes already worried about collisions so you do not have to. Just pick one (SHA-256 is a fine choice) and focus your efforts on the rest of your application.