The Game Tile Cipher
(See also the Slide Rule Cipher and The Enigma Machine)

Here's a nifty little cipher based on shuffling around word game tiles.  From your favorite board game that has letter tiles, select one each of all 26 letters and arrange them as shown in figure 1.

Figure 1. The initial arrangement.

To encode a cleartext letter, find that letter in the set of tiles and read down to the next letter below it.  If the letter is on the bottom row wrap around to the top row.  For example, cleartext 'D' will be enciphered as 'M' and cleartext 'Y' will be enciphered as 'G'.  Skip over the empty square when doing this operation.

Now comes the fun part.  You and your correspondent agree on some set of sliding operations to perform between enciphering each letter.  For example, lets use the series of operations:

1. Encipher one letter.

2. Slide one tile up and slide one complete row into the empty spot.

3. Encipher one letter.

4. Slide one tile up and slide one complete row into the empty spot.

5. Encipher one letter.

6. Slide one tile down and slide one complete row into the empty spot.

7. Encipher one letter.

8. Slide on tile down and slide one complete row into the empty spot.

9. Go to step 1 and repeat.

That way the alphabet gets mixed up between each letter.  The above procedure returns to the starting point after 52 letters, but procedures could be chosen that would give longer period lengths.

Here's what the tile set looks like after each of those operations:

Figure 2. Original Alphabet.

Figure 3. After step 2.

Figure 4. After step 4.

Figure 5. After step 6.

Figure 6. After step 8 and ready to start over.

Using these operations the cleartext word HELLO becomes QOSRU.

The best way to use this type of cipher would be to perform some complex operation between each letter that would more thoroughly scramble the alphabet.  Such an operation would have to one that could be repeated many times without returning to the original tile configuration.  However, rather than performing a different operation between each letter as we did in the above example, it would be better to use the same operation over and over so that this single multi-move operation could be memorized and then repeated from memory after each letter without having to keep track of where we are in the sequence as with the above example.

A More Secure Game Tile Cipher

Here's another method of shuffling game tiles around to make a more secure polyalphabetic cipher.  Arrange two sets of letter tiles, preferably in different colors, as shown in figure 7.

Figure 7. The two-color alphabet setup.

Now after each letter is encrypted perform these easy to remember operations.  On the top row (the plaintext row), take the rightmost letter of each color group and slide it in front of the rest of that color group.  On the bottom row (the ciphertext row), take the leftmost letter of each color group and slide it to the end of that color group.  Figure 8 shows the tiles above after one shuffling operation.

Figure 8. The same alphabet after one shuffling operation.

The first color group will return to its original setup after three operations.  The second color group will take 5 operations to return to the starting point.  The third color group takes 7 steps and the longest color group takes 11 steps.  Altogether it takes 3 x 5 x 7 x 11 = 1155 steps before the entire alphabet returns to its original configuration.  The bottom alphabet also returns to its original configuration in 1155 steps so that is to total period length of this polyalphabetic cipher.

If the bottom row were made up of three color groups of length 2, 11 and 13 then the period of the bottom row would be 286 and the combined period would be 30,030 (286 x 1155 divided by their common factor 11).  Thus with just two rows of letter tiles you can easily create a polyalphabetic cipher that won't repeat the cipher alphabet for over 30,000 letters.  For low volume traffic, such as passing notes behind teacher's back, this should do quite nicely.

Roy's Playing Card Shuffle Adapted

On March 8, 2004 a poster named "Roy" wrote to the usenet newsgroup sci.crypt with a method of shuffling playing cards systematically to get a key stream for a stream cipher.  I thought the same method of shuffling might be applied to letter tiles to create the next alphabet in a polyalphabetic cipher.  Here's my adaptation of Roy's Shuffle to letter tiles.

Begin with the two alphabet rows as shown in the section above.  The top row will be the plaintext row and will not be shuffled.  The bottom row will be the cipher text row and will be shuffled one or more times between each letter of the message.

Figure 9. The starting alphabet for Roy's Shuffle.

Now each letter is assigned an arbitrary numerical value, but we will only use the value 1 through 9 assigned (for example) as follows:

      A R S = 3
      B Q T = 4
      C P U = 5
      D O V = 6
      E N W = 7
      F M X = 8
      G L Y = 9
      H K Z = 1
      I J   = 2

We could, of course, distribute the number values in a different way, or use the numbers 1 through 13 with two letters assigned to each number, or even use the values 1 through 26.  The enemy trying to read you rmessage (or your little brother trying to read your diary), not knowing which method you used to assign the numeric values will have one more obstacle to overcome.

The shuffling is done as follows:

• Look at the leftmost letter in the bottom, or ciphertext row. (A to begin with)

• Find it's numeric value in the table. (A is 3)

• Take that many tiles off the left end of the ciphertext row.

• Move those tiles to the end, reversing their order as you move them.

After the first shuffling operation the three moved and reversed tiles are found at the end of the row like this:

Figure 10. After one shuffle that moved three tiles.

Now D is at the head of the row.  D's numeric value in the table is 6 so we remove the leftmost 6 tiles, reverse them and put them on the end like this:

Figure 11. The tiles after two shuffles.

Using this numbering scheme and this starting arrangement the alphabet must be shuffled in excess of 2.1 billion times times before returning to the original letter arrangement.  The exact number of shuffles may be considerably larger than that, but that's how far I got before I gave up and stopped the computer run.  For comparison purposes, a row of 18 tiles had to be shuffled 800,194 times before returning to the starting arrangement and increasing that by 2 tiles to a row of 20 tiles boosted the number of shuffles required to 129,258,576.   Looking at the curve as the row size increases I would take a rough at stab and say that it will take around 2 to 3 trillion (2,000,000,000,000 - 3,000,000,000,000) shuffles to get the 26 tiles back into position.  (This is based on the rate at which the log of the number of shuffles grows as the number of tiles grows.)

This shuffling method also has the interesting property that it can be reversed by doing the same shuffle, but from the opposite end.  For example, here is the arrangement we find after 10,000 shuffles:

      XBSOFULKVHQEPCNTWIZDRGJMYA

By treating the rightmost letter, A as the first letter, we look its value up in the table (3) and remove 3 letters from the end of the alphabet and move them to the beginning, reversing them as we go.  Thus we know that after 9,999 shuffles it would have been:

      AYMXBSOFULKVHQEPCNTWIZDRGJ

Being able to reverse the process requires knowing the numerical value assigned to each letter.  If that is kept secret then the enemy will not be able to reverse the shuffling process, and will not be able to undo your cipher even if you are careless enough to leave your letter tiles laying out in the last arrangement you used.

Here, just for fun, is the plaintext MISSISSIPPI enciphered using one shuffle before each letter: PRISEMVWTOK.

To Complicate Matters A Bit

If you really want to confuse the enemy, shuffle the tiles more than once between each letter.  In fact you could agree on some set of numbers numbers like 213857, then shuffle the alphabet twice for the first letter, then once for the next letter,  three times for the third letter, eight times for the fourth letter, and so on, repeating the key number sequence over and over.  As an aid to keeping track of where you are in the sequence, arrange a short deck of playing cards: 2A3857 (or whatever your secret key is).  After each letter is enciphered turn over the next card and shuffle the letter row that many times before the next plaintext letter is enciphered.  When you turn up the last card, flip the discard pile back face down and go through it again until the message is completely encoded.

With this method even you will not be able to reverse the shuffling process unless you remember exactly where you left off in the sequence of key numbers.

And of course you can start with a keyword alphabet like: PATHFINDERBCGJKLMOQSUVWXYZ which makes the job of cracking it a bit more difficult.

And Just To Be Devious

Why not alternate using the top row as the plaintext alphabet with using the bottom row as the plaintext alphabet and the top row as the ciphertext alphabet.

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