[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