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Password Constraint Puzzler
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Andreas Lundblad
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Dec 11, 2016
</div>
<h1 class="post-title">
Password Constraint Puzzler
</h1>
</header>
<article>
<p>
I was creating a Java snippet for password generation for
<a>
programming.guide
</a>
and stumbled across a fun puzzler. Suppose you have the following scenario:
</p>
<blockquote>
<p>
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
<code class="language-plaintext highlighter-rouge">
"qurcoehx"
</code>
. The site responds with
</p>
<p>
<em>
<font style="color: red">
Password must contain at least 1 digit
</font>
</em>
</p>
<p>
You’re lazy and don’t want to type more characters than necessary, so you stick to 8 characters and replace the
<code class="language-plaintext highlighter-rouge">
c
</code>
with a
<code class="language-plaintext highlighter-rouge">
7
</code>
:
<code class="language-plaintext highlighter-rouge">
"qur7oehx"
</code>
. The site accepts the password.
</p>
</blockquote>
<p>
<strong>
The puzzler
</strong>
: 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?
</p>
<p>
On one hand you can reason…
</p>
<p>
<em>
a mix of characters and digits must be better than just characters
</em>
</p>
<p>
…but on the other hand…
</p>
<p>
<em>
being forced to ‘downgrade’ a letter to a digit must surely be bad.
</em>
</p>
<p>
So which one is it?
</p>
<p>
Let’s first focus on lower case character. Without a digit, there are 26
<sup>
8
</sup>
8 character passwords. When forced to include a digit, we have:
</p>
<ul>
<li>
a 7 letter password (26
<sup>
7
</sup>
choices),
</li>
<li>
1 digit (10 choices), and
</li>
<li>
a digit position (8 choices)
</li>
</ul>
<p>
We see that 26
<sup>
7
</sup>
×10×8 is greater than 26
<sup>
8
</sup>
, so replacing a character with a digit actually helps. What “saves” us here, intuitively, is the fact that the digit has many possible positions.
</p>
<p>
If we simplify the inequality by dividing both sides with 26
<sup>
7
</sup>
, we get
</p>
<p>
10×8 > 26
</p>
<p>
In other words, inserting a digit is better as long as
</p>
<p>
10×[number of digit positions] > [number of letters].
</p>
<p>
Even with just 2 characters we’re better off “downgrading” one of them from a letter to a digit.
</p>
<p>
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.
</p>
<p>
<strong>
Conclusion
</strong>
: 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.
</p>
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