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A question about passwords | characters used in them
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Hashing on client side is both more private, and secure. All the user ever submits is a combined hash (auth/pubkey) of their username + password.
If the server has that hash? Check the DB if it requires 2FA, and if the user sent a challenge response. If not, fail the login.
Registering is pretty much the same. User submits hash, server checks DB against it, fail if exists.
Edit: If data is also encrypted properly in the DB, it doesn't even matter if the entire DB is completely public, leaked, or secured on their own servers.
This means that the submitted hash is effectively a password. You get a minor benefit in that it obscures the original password in case it contains sensitive info or is reused. But the DB is now storing the hash password in plain text. This means that if the DB leaks anyone can just log in by sending the hash.
If you want to do something like this you would need some sort of challenge to prevent replay attacks.
This scheme would also benefit from some salt. Although the included username does act as a form of weak salt.
Per your edit, the DB being "encrypted properly" just means "hashing server side". There's little benefit (though not necessarily zero) to encrypting the entire database, since the key has to live in plaintext somewhere on the same system. It's also making the slowest part of most systems even slower.