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When I change my password on some web servers like email, cloud, and social networks and try to use my previous password, the server denies it with message "Don't use your previous passwords".
Does that mean that servers store my previous passwords? How is that secure?
Hash functions are deterministic; the same input always results in the same output. They can store the previous hash values of your old passwords and compare them with your new proposed password's hash. To achieve this, they also need to store at least the salt values, too. This means the salt and other parameters has to be the same for you in this site.
As pointed out by Micheal
The salt for the old passwords does not have to be the same as the new one.
In this way, you can use different salts and other parameters. This can be considered more secure than reusing the salt with an increase on the storage.
From here, without the server side code, we cannot say more than like this.
it aims to maximize the probability of a “collision” for similar items This is a not a good property for Cryptographic hash functions.
Most of the time you have to reauthenticate in order to change your password. In this case, the backend could
Compare the clear text (old) password with the clear text (new) password with metrics like Levenshtein in order to prevent a password change like
S3cur3Pa55w0rd_1 -> S3cur3Pa55w0rd_2
Possibly yes, unknown in practice, assume that they do.
There are many reasons why a site might store one or more of your old passwords (or, if they have three working brain cells, password hash).
The most common thing is that it will store the last password/hash before the change, in case you come calling and say "that password change wasn't me". In that case they can restore the previous password, which can be preferable over resetting to a new password.
The second common thing is the case you describe - a password policy that states your password can't be equal to the last n passwords you had used. To validate that, those last n passwords or their hashes need to be stored, probably in a seperate database table (so they may or may not be included in a data breach).
Finally, passwords/hashes might be in all kinds of logfiles, debug dumps, backups and other secondary data stores.
As you rarely have insights into the implementation details of websites you use, you will most likely never know. You should therefore assume that they probably do.
