I don't know why do we authenticate by prompting the user to enter both username and password. In my mental model, prompting password only suffices.
The reason is as follows:
Assume there are x valid characters to use.
Case 1 (prompting username and password)
Let the length of username and password be
n/2 characters each.
Since the username is exposed to the public, the probability of success to break the password is one over x^(n/2).
Case 2 (prompting password only)
Let the length of the password be n characters. The probability of success to break the password is one over x^n.
Why do we authenticate by prompting a user to enter both username and password? Does prompting the password only suffice?
For case 1: the username is unique.
For case 2: the password is unique.