I thought the strength of a password depended on the total number of possible combinations. Therefore, allowing longer passwords should be safer because you then have more possibilities. But why do some systems set a minimum number? Doesn't this simply reduce the number of possible combinations?
Even if you allow 128-byte passwords, most people will choose something 10 chars or less. Some even will choose something as short as 1-3 chars. Hackers know this and brute-force those possibilities first. Indeed, in most brute-force hacking programs, they naturally come up first as they start with shortest possibilities and build to longer ones. Therefore, a minimum.
To your point, it does minimize the total possibilities, but not by much, as the LSB's don't amount to much compared to the MSB's.
Doesn't this simply reduce the number of possible combinations?
yes, but not significantly.
As an analogy, consider passwords that are combinations of uppercase letters A-Z only, with max length of 8. Total # of possibilities = 268 + 267 + ... + 26 + 1 = 217.2 billion. If you restrict this to a minimum length of 6, that eliminates passwords of length 5 or less, which have a total # of possibilities of 265 + 264 + ... + 26 + 1 = 12.4 million, a tiny fraction of the larger password space.
No, because the longer the password, the more time it takes to crack. By setting a minimum length the designer is forcing the user to make a longer password.