[benaubin]

> However, as the number of members in the set increases, the number of possible IDs increases exponentially. Case-sensitive: 53^8 = 62,259,690,411,361 Case-insensitive: 22^8 = 54,875,873,536

Nitpick, but isn't this polynomial to the members of the set?

[pxx]

aⁿ grows (a/b)ⁿ as quickly as bⁿ. The multiplicative difference still grows exponentially in n.

[afiori]

a^n is polynomial in a and exponential in n.

This is why longer password are more efficient than complex passwords: to gain the same security effect of doubling the password length you would need to square the alphabet

[pxx]

You've the proper definitions but are missing the context. An exponential with larger base still has an exponential multiplicative difference compared to an exponential with a smaller base.

We're comparing the growth rate of of two exponentials representing variable-length identifiers. We're not looking at a constant-length identifier (which is what you're doing with only looking at a^n). Notice the context of where exponential is used in the article: we are changing n from 5 to 8.