Novice in the area, so grain of salt (and if you can correct me where I'm wrong, that'd be great. All for learning): 1) I don't think the base changes nor the exponent deviates from the character approach (in reference to the 20,000, 40,000, and 171,000 base stated). If we're in a system that allows all uppercase letters, all lower case letters, all special characters, and all numbers, then the base is the sum of tho…
To your first point, I think it's about priors about password choices. To take an extreme example, even if a site allows up to 32 alphanumeric characters, we don't just say passwords on this site are uniformly secure under metric (26+26+10)^32 because theoretically a brute-force approach would have to go through up that many permutations. (Such a number is an upper bound on your password security.) In particular, sho…
What I was saying is if the rules of the site allow for all lower, all upper, special characters, and numerals, the sum of that is your base. If your password is purplepenguinparade, the expectation is you've cracked it by the time you've completed combined_base^1 through combined_base^19. They could artificially limit it (most people use lowercase characters and this site allows just lowercase characters, so let's try 26^n), but they'd run the risk of never cracking it because characters could've been added that deviate from their parameters.