LEO: Great. Brent Nesbitt wonders if Off The Grid goes both ways. Is it possible to decipher a message if you have the grid that was used to encipher it? If so, what’s the process?

STEVE: Okay. This is such a neat question, I am posing it to our listeners.

LEO: A stumper.

STEVE: Can you, and under what conditions can you, decrypt from the password back to the domain name? I mean, it’s a perfect little puzzle for everyone to think about.

LEO: I like it.

STEVE: So I’m not going to answer Brent’s question. I’m going to say, well, what does everybody think?

It is fairly easy to show that the Off the Grid system can be made into a many-to-one mapping, so that it become non-invertible. Consider:

This Latin Square for Steve Gibson's Off-the-Grid is not invertible

Following the unmodified instructions described by Steve to hash the domain names “AMAZON” and “PAYPAL”, beginning on the first line we find the start point:

Tracing out the letters of either domain name to find the grid location to begin encryption

Steve designed the system so that before any encrypted letters are generated, the starting grid location will differ from domain to domain as shown above. To generate the password, we proceed with the next phase:

Using Steve's overshoot by 2 method to derive password characters yields the same password for both domains

Thus, this particular Latin Square so happens to produce the same password “DECIPHERTHIS” from two different domains “AMAZON” and “PAYPAL”. Hence, the process is not invertible. A more difficult question is to describe the conditions under which a given grid is invertible or not — a decidability problem! Can we devise a computer program that accepts Latin Squares and outputs either a yes or a no? The answer is yes, and the program will also run in polynomial time (where n = grid size, not the domain name length). Steve’s Off the Grid system operates on no more than 6 letters of a domain name, so the set of all possible plain texts is a finite size of 26^6 (multiply again by 52 if we need to consider the possibility of starting on a different row or column). For a given input grid, the testing program simply tries encrypting all possible domain names then looks for duplicate passwords. Although this is clearly a P time problem, a naive implementation will still take a long time to execute on a 26×26 grid, so clearly this cannot be decided “off the grid”.

There are several double pendulum videos on YouTube that demonstrate its chaotic motion, however only a few illustrate one key property of chaotic systems — very tiny differences in starting conditions can yield very large variations in the eventual path that the double pendulum follows. This is not true of a single pendulum. To illustrate this, I created a release mechanism to ensure that the double pendulum would always start in the same configuration and filmed multiple releases. Overlaying four trial runs in the same video reveals the very divergent paths followed by the same double pendulum in spite of starting from almost identical initial conditions. Very minor, near imperceptible deviations in the repeatability of the release mechanism were enough to cause this.

A few months ago I had based my custom CD on the Kubuntu 8.10 desktop CD which worked fine on the notebook PC I use for managing finances. Now that Ubuntu 9.04 has been out for a month I thought now would be a good opportunity to upgrade, or so I thought. As it turned out, WiFi support on Kubuntu 9.04 is seriously broken. This I wasn’t expecting since in the previous version wifi was quite usable. Newer version should be an improvement, no? So much for the adage “if it ain’t broke don’t fix it”. It seems that the Kubuntu team saw it fit to replace the knetworkmanager applet with the newer and much nicer looking plasmoid network manager, but ended up somehow breaking wifi. My attempts to work around the problem using wicd met with little success. I simply could not get Kubuntu to connect to my encrypted wireless access point.

So by late evening my frustrations motivated a switch to Xubuntu 9.04. I’m now blogging this using my freshly burned live CD-RW via the wifi connection on my notebook. The custom modifications included installing Keepassx, Truecrypt 6.2, python-tk, and my own password hashing script (I’ll blog on this later). Also removed a bunch of packages I shouldn’t really need to use for online banking. The CD image came to less than 550MB. I could probably shrink this further if I had more time to do through the list of installed packages in more detail. With Kubuntu I always had about 50-100MB extra worth of additional software. I should have been using the leaner Xubuntu from day one especially given the security requirements. A minimal desktop implies lower complexity and fewer places for exploitable security holes. It seems much more responsive also.

I let YouTube supply the audio track for this one.

The audio clip was from Magnatune Records for this one.

The panorama was created using stitch assist mode of the Canon Powershot A570 IS and Canon’s included photo stitching software. From this a scrolling panorama video clip was created using a Python Image Library script, mjpegtools, and mencoder.

Encoding the puzzle pieces

First I would have to encode each piece as a vector of 4 integers to represent what shape was on each of the four sides. I chose the numbers such that protrusions were positive and recesses were negative, and that pieces would fit together if matching sides summed to zero.

Finding the perfect fit

The puzzle pieces were thus reduced to a list of 4-vectors:

-4 -1 1 4
-3 -4 4 3
-1 -3 2 2
-3 -2 4 2
-4 -3 4 2
-1 -1 4 3
-4 -3 1 4
-1 -4 3 1
-3 -2 3 1
-4 -2 1 1
-4 -1 4 4
-2 -4 4 3
-2 -1 4 2
-1 -4 2 1
-1 -2 2 3
-4 -3 2 2

Solving the puzzle was now a matter of scanning through all arrangements and orientations of the pieces. The search space can be heavily pruned by discarding solutions whenever mismatches are found. I’ve made use of recursion to handle the backtracking for when no piece can be found that builds upon a trial configuration. Coding in Python took less than half an hour.

#!/usr/bin/env python

# load piece data

f = open("puzzle_pieces.txt")
pieces = []
while 1:
    line = f.readline()
    if line=="": break
    a = map(int,line.split())
    if len(a)<4: break
    pieces.append(a)
f.close()

#create spaces for 4x4 tiles
tiles = [None]*16

#keep track of which tiles are already used
used = [0]*len(pieces)

#does piece j0 orientation k0 fit the tile t1 above
def fit_top(j0,k0,t1):
    j1,k1 = t1
    return pieces[j1][(k1+3)%4] == -pieces[j0][(k0+1)%4]

#does piece j0 orientation k0 fit the tile t1 on the left
def fit_left(j0,k0,t1):
    j1,k1 = t1
    return pieces[j1][(k1+2)%4] == -pieces[j0][k0%4]

#recursively fit pieces and backtrack upon failure
def fit_pieces(i,used):
    for j in range(len(pieces)):
        if used[j]: continue
        used[j] = 1
        #try every orientation
        for k in range(4):
            if (i<4 or fit_top(j,k,tiles[i-4])) \
                   and (i%4==0 or fit_left(j,k,tiles[i-1])):
                tiles[i] = j,k
                #piece fits; advance to next tile
                if i==15 or fit_pieces(i+1,used):
                    return 1 #success
        #make tile available for future use
        used[j] = 0
    #failed - must backtrack
    return 0

#solve
if fit_pieces(0,used):
    #output solution
    n = 0
    for j,k in tiles:
        print n,":",
        for i in range(4):
            print pieces[j][(i+k)%4],
        print
        n = n + 1
        if n%4==0: print

Execution only took a few seconds on an old PC to yield the following solution:

0 : -4 -1 1 4
1 : -1 -1 4 3
2 : -4 -1 4 4
3 : -4 -3 2 2

4 : -3 -4 4 3
5 : -4 -3 1 4
6 : -1 -4 3 1
7 : -3 -2 4 2

8 : 1 -3 -2 3
9 : 2 -4 -3 4
10 : 3 -1 -2 2
11 : 2 -2 -1 4

12 : -1 -3 2 2
13 : -2 -4 4 3
14 : -4 -2 1 1
15 : -1 -4 2 1

which was confirmed by practical means:

I wonder how long it would’ve taken me without the use of computers.

The user only has to remember one master password which should never to be entered on any website. It is only to be used to unlock an encrypted database that contains all the login details for each bank account. There are several plugins that improve web browser integration, minimise cut-n-paste, and automate online account logins.

Keepass currently only runs on WIndows. Luckily for Linux and Mac OS X users there is the functionally equivalent, KeepassX, which can also open Keepass database files.