A long time ago I did a blog post on PIN coded lock boxes that you put on your house to hold a spare set of keys. Their main function is to give easy access for emergency services in case they need keys to get in if an elderly relative has a fall and can't get up. The point of my post was to demonstrate that although they have 10 buttons, look secure, and you can choose how many digits are in the PIN, they only really have 1024 PIN combinations. This is because the numbers can only be used once in each PIN.
|PIN coded key lock box|
The math is explained in my initial blog post but the results are below. I opined it would take about 4 hours to try all the combinations, and in this case it would require pressing the open button 1024 times, the clear button 1024 times, and pressing digits (0*1 + 1*10 + 2*45 + 3*120 + 4*210 + 5*252 + 6*210 + 7*120 + 8*45 + 9*10 + 10*1) = 5120 times. In total there are 7168 button presses. That works out at about one button press every 2 seconds.
|Number of PINs vs PIN length|
At some point in the 5 years since I first wrote about these locks (the time flies doesn't it) it occurred to me that I was being inefficient. Testing if the box opens doesn't clear the current code. So I can test multiple PINs at once by just chaining them together. It's similar to the De Bruijn sequence, a mathematical tool that can be used to brute force another type of PIN. In this case however you have to reset the lock after a few buttons are pressed. To illustrate the process I'm trying to explain, imagine that the clear button has been pressed and I then enter the numbers 0 through 9, testing if the lock will open in between numbers. I've actually just tested the codes (Null) (0) (01) (012) (0123) (01234) (012345) (0123456) (01234567) (012345678) (0123456789). I've tested 11 codes with one press of the clear button, 11 presses of the open button, and just 10 digit presses. The whole process won't be this efficient but it'll be better than nothing.
But where do we start? In a perfect world, you may see by looking at the chart below and table above, that the best we can do is 1 sequence testing PINs of length 0-10, followed by 9 sequences testing PINs of length 1-9, 35 sequences testing PINs of length 2-8, 75 sequences testing PINs of length 3-7, 90 sequences testing PINs of length 4-6, and 42 sequences testing PINs of length 5.
This will still result in 1024 presses of the test button, only 252 presses of the clear button, and (1*10)+(9*9)+(35*8)+(75*7)+(90*6)+(42*5) = 1646 presses of the number buttons. For a new total of 2922 button presses, or about 41% of the original estimate.
|Distribution of PIN lengths|
There's no guarantee that these sequences exist though and I had no deep understanding of how to create them, so I tried the first thing that came to mind and got lucky.
First, generate all possible combinations for each PIN length and then sort the numbers in each PIN. Then sort the PINs for each length comparing them element by element. Start in the middle with the PINs of length 5 as there are more of these than the others. Then take the PINs of length 4 and go through them one by one from the start and place them to the left of first 5 digit PIN that could follow it. For instance (1, 5, 6, 7) might go to the left of (1, 2, 5, 6, 7) as only a 2 would have to be pressed to get to the 5 digit PIN. Do the same for the 6 digit PINs and place them on the right of the 5 digits. In this case (1, 2, 3, 5, 6, 7) might go to the right of (1, 2, 5, 6, 7) as only a 3 needs to be pressed. Repeat this left right procedure until all PINs are processed. It will look like a mess, but if you sort the sequences by length you will get a spreadsheet that looks like the one below (zoomed and rotated to fit). Look familiar? It's reflects the distribution graph above, and it shows that the sequences can be generated.
|Rotated spreadsheet of the sequences|
I've placed my code in a Github Gist for you to generate your own sequences in case there are a different number of buttons on your lock. If you have a lock with 10 buttons I've already generated the file for you. It looks a little different from the output of the Python file because I've done some some find and replace operations in Notepad++. I've used the word test instead of unlock as well.
Update There is a MP3 file of the instructions in this blog post, Linux Text To Speech With Saved Audio.
So what about those locks you see in banks and other offices that have 14 buttons. I've done the maths so you don't have to, but they can be broken with 50316 button presses. So 4 extra buttons buys you an increase in security by a factor of about 17. Not that it matters, whenever I've seen these used I people aren't too discreet about entering the PIN.
|PIN Door Lock|