Password Constraint Puzzler
I was creating a Java snippet for password generation for programming.guide and stumbled across a fun puzzler. Suppose you have the following scenario:
You’re a user trying to register a new account. The site says that the password must be at least 8 characters, so you enter
"qurcoehx"
. The site responds withPassword must contain at least 1 digit
You’re lazy and don’t want to type more characters than necessary, so you stick to 8 characters and replace the
c
with a7
:"qur7oehx"
. The site accepts the password.
The puzzler : Is the second version “safer” than the first version? Or, put differently: Are there more 8 character passwords with exactly 1 digit, than there are 8 character passwords with only letters?
On one hand you can reason…
a mix of characters and digits must be better than just characters
…but on the other hand…
being forced to ‘downgrade’ a letter to a digit must surely be bad.
So which one is it?
Let’s first focus on lower case character. Without a digit, there are 26 8 8 character passwords. When forced to include a digit, we have:
- a 7 letter password (26 7 choices),
- 1 digit (10 choices), and
- a digit position (8 choices)
We see that 26 7 ×10×8 is greater than 26 8 , so replacing a character with a digit actually helps. What “saves” us here, intuitively, is the fact that the digit has many possible positions.
If we simplify the inequality by dividing both sides with 26 7 , we get
10×8 > 26
In other words, inserting a digit is better as long as
10×[number of digit positions] > [number of letters].
Even with just 2 characters we’re better off “downgrading” one of them from a letter to a digit.
If we on the other hand consider upper case letters as well, i.e. 52 alternatives, we need more than ⌊52/10⌋ = 5 characters for the downgrade to make sense.
Conclusion : The tongue in cheek security analysis here is, as long as you have a reasonable long password, it’s an improvement to replace a letter with a digit.