Note: I'm not asking about this password scheme is the best or not (of course it isn't); I'm asking about its theoretical or practical secureness relative to the optimal password scheme based on the algorithms commonly used in authentication and encryption. As I already say in my question, mathematically this scheme (as I've defined it here) is as secure as what people are told to do (but rarely do; though irrelevant here), but I don't know whether there are weird aspects of authentication or encryption that invalidate the mathematical analysis based on entropy alone.
Well so I had this interesting thought. Since password-protected systems often require users to change their password frequently, many people just increment a counter appended to some prefix, such as:
I wonder whether using such a family of passwords is as secure as choosing a completely random new password each time, as long as the prefix is uniformly randomly chosen and as long as each random new password plus about 10 bits. I ignored the extra bits from the counter since that might be guessable within a small error margin given the frequency that users are required to change their password. Mathematically this password scheme is good enough because the extra 10 bits lets the attacker try 1024 times as long to crack the password as compared to the completely random one, assuming one does not change a password more than 1024 times. (Here I'm assuming the attacker somehow has a hash of the password, including any salt needed.)
Now I know one obvious disadvantage to this, which is that an attacker who somehow gets hold of the current password will know all previous passwords, nevertheless this seems completely irrelevant in most practical situations, where once logged in the user has full access to his/her account and never needs the old passwords.
Hence my question; is there any concrete reason why users should not use this scheme for changing passwords? I'm guessing that if the answer is positive, it will depend on the authentication or encryption algorithms involved themselves, in which case I'm fine with restricting the question to the most commonly used algorithms.
To be clear, I'm only asking about the situation where:
Users have no choice but to change their passwords at some fixed frequency, and do not have the option to use a password manager (such as on a workplace computer). They therefore have to decide on some scheme. This question is about whether one particular scheme is better than another.
Choose a fixed constant n. I have in mind n ≥ 64 at minimum, if that matters.
a. The first scheme is to choose a new independent random string with n-bit entropy every time one needs to change the password.
b. The second scheme is to choose a fixed random string with (n+10)-bit entropy and append a counter that is incremented every time one needs to change the password.
The question is whether (b) is at least as secure as (a) for common types of password-protected systems.