You can also throw some private bits into the data stream before hashing.
If user supplies PW, which gets hashed to PW', and a 3rd party gets PW', the weakness allows them to discover another password that also hashes to PW'.
If you add additional bits to the supplied PW -- PWpri, and hash that to PWpri', which the attacker gets, and using the techniques described, comes up with some bits that also hash to PWpri', they still can't come up with a valid PW that when pri gets added to it also produce PWpri'.
Of course, if your security is such that an attacker can discover PWpri', they can probably find pri out anyway, and you're probably owned at that point anyway, so discovering PW is the least of your concerns.
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