Usage patterns in the Beale Ciphers…

Even though Beale Ciphers B1, B2, and B3 each consist of similar-looking strings of numbers, it’s far from obvious that they have been generated in the same way (i.e. that they all result from using the same cipher system).

Usage Patterns

We can quickly map the usage of the first 1000 index values (I remain a bit suspicious of higher numbers), with the following bitwise key:
* ‘.’ => unused
* ‘1’ => used in B1
* ‘2’ => used in B2
* ‘3’ => used in B1 + B2
* ‘4’ => used in B3
* ‘5’ => used in B1 + B3
* ‘6’ => used in B2 + B3
* ‘7’ => used in B1 + B2 + B3

77773777777757777776775777777376777777777777676777
7766576377737777755577754577317777755555.75777575.
77516774525774757757774525554135276747531.6.143147
42.3..4227.5.4..4..1454.4564.45.4124..1.7.171624.7
515.65574175554565525..51454.7111745.42421.44752.2
52...4...414154...46..247....1..132522.34..1.46.4.
73457..3.444474644475.11.5.5........4..11.46.1.4..
4.2..1...31..55.1..32..44....444.....2.....2..4..2
1.1.22.12.144..541.2..1..1....1..4.1..43......1...
.........55.11.2....1..1..........33........4....1
1...2.....2..................1.......1.3..1....3..
...2..2..........1......2.....33.................2
1131..3...41.1..54.2..11......4...........24..2...
...1.........1.21.......41...1.1.4................
...............14..........1.......1..............
.......1.....................1....................
......2...21.14.41554.11....1..........1....4.....
.....4.1..11.4.......1..1..............5....141...
...........1....411.14.4......1..4.5..............
44.......111..1.........5.........1..............1

From this, we can see that even though the numbers that are used in all three ciphers are biased towards low numbers (e.g. look at all the ‘7’ values at the top), B1 numbers (and to a slightly lesser extent B3 numbers) appear throughout the number range. Furthermore, apart from the numbers near the top, there seems to be no systematic relationship between the usage map of any two pair of ciphers (not even B1 and B3).

And yet we have quite strong evidence that the same enciphering tables derived from the DoI were used for both B2 (which has been solved) and B1 (which remains unsolved). I think this alone is strong evidence that for all their underlying “causal similarities” (for want of a better phrase), B1 and B2 were not generated by the same ciphering system.

Note also that the map shows runs of adjacent indices that appear in only one of the three ciphertexts (e.g. “4444” in B3) or that appear in both B1 and B3 but not in B2 (e.g. “55555”). However, these look broadly within the range of normal randomness, so I doubt these are highlighting anything unusual.

Jarlve’s Incremental Series

In a comment here a few days ago, Jarlve observed that all three Beale ciphers have stretches of numbers that were numerically ordered to a degree that was somewhat unusual. And furthermore:

Testing the significance of these incremental series versus randomizations, then B1 = 4.61 sigma, B2 = 2.72 sigma and B3 = 9.86 sigma.

If we map B1’s “incrementality” (i.e. where ‘.’ => decrement, and ‘*’ => increment), we can indeed see a six-long increment sequence about 60% of the way through, plus a couple of five-long increment sequences. What is just as striking is that the long decrement sequence in B1 is four-long (twice), which points to some kind of subtle asymmetry.

B1:

*.*...**.*.*.*..**.**.***.**.*.**.**.*.**..***..**
.*..**.*.***...*.**..**..**.*..*.*.*.***.*..*..***
..*.*.*.***...***.*.**.***.*..*..**.****.*.**.*.**
..*.*..*..*.**.*.***....***.**..***.**.*.*..*.**.*
*.**.**..**.*.*.*..***..***.*.*.*****.**.*.**.*...
***.*..****.*.**.***..*..****..****.*.*.***..***..
*.******.*..**.***.*..*..**...**.*..**.*.*.*.*.**.
*..*.*..*.**.****..*.*..**..*.*.*.*..****..*..**..
**.*..**..***.*.***..**.*..**.**.****.****.*..*.*.
.***....*.***..*..*****.*..*.**..*.*.**...***.*.*.
**.***...*.****..**

Compare this with B2, which has a six-long decrement sequence (about 30% of the way through), and a pair of five-long increment sequences.

B2:

..*.*..***..**..*.*.*.**.**..*..*..***....*.*.**..
*.*..**.**.*.*.**..***.**.*..***.**.*.*.*.**.*.*.*
*.*.**...*...**..**.**.***.*.**..**..***.*..*.**.*
.***.*.*.*.*..*..**..*.***.*.***..**..***.*..**..*
.*.**......*.**.*....*..*.*.***.***..**.*..***..**
.*.*.*..**..*.*.*...*.***.*.**.*.*.**.*.*.**.*.*.*
.***.***.*..**..*.****...*..*.*.**..***.**.*.**.**
.*.*.**.*.*.*..*.*....***.*..*..*..**..**...**.*..
*.*.**.****..**.*..*.**.***...*.*...**.*..*..*.*.*
*.***.**..**.**.*.**.*...*.**..*.*.****.*****..**.
***..*.*..**.*.**.*.*.**..***..**...***.*....*.*..
*.**...***.*.****..*.*.*.*.*..**..**.**..*.*.*.*..
**.**...*..*.*.*.*..***.*.****..*.**..*.*.*..*.*.*
*.**.*..***..*..**...**.*.*..**....*.**.*..*****.*
***.*.*..**.**.*.***.**.*..**.*...**..*..*.****.*.
***..***.***

But all of this in B1 and B2 is almost as nothing to B3’s extremely unbalanced set of increment series, firstly in a patch in the middle (two seven-long increments and two six-long increments) and then in a long patch at the end (where the positive increment sequences are 9, 9, 7, 6, 7, 9, and 6 long). By way of contrast, the longest decrement sequences in B3 are a single 6-long set, and a single 5-long set).

B3:

.*.*..*.**.*.**..***..*.*..*.***..*.*.*..*.*.**.*.
**.*.*.*.*..*..**.***..*.**.****.***.**.*.*.**.***
*.*.*.*...*...**.*.*.**.**..*.**..*.***.*.***..*..
*.***.**.**..*****...*.*.*..****..*....***..*.****
.*****..***...*.*.****.*******..**.*.*.*..**..**.*
***.**...**..*..**.*.*.***.**.**.****.****.******.
***..****.**...*.*.**..*.***.*******..*...**.*****
*.**..*.*.**.*.***.*.**..*.*.**..*****..***.**..*.
***.**.****.****.*.***.*...**...**.*......*.*..**.
**..*.*****.****.*****.*.*.*.*.**.*.*....*.*.*****
****.*********..**.**.*******..**.*..******.*.****
..*.**.****.*****.....*.**..*******.****..*.*.***.
.*********.******

Putting All This Together

I think Jarlve’s incrementing series perhaps offer a quite different dimension to what Jim Gillogly (perhaps better known for breaking parts of the Kryptos ciphers) mused in his “Dissenting Opinion” on the Beale Ciphers, where he opined:

I visualize the encryptor selecting numbers more or less at random, but occasionally growing bored and picking entries from the numbered Declaration of Independence in front of him, in several cases choosing numbers with an alphabetic sequence.

Whereas this loosely seems to fit B1 (where mysterious alphabet-like strings do indeed appear, but which require the cipher table used in B2 to have been used in a different manner), the immediate problem is that it doesn’t really capture what happens in B3 (where no mysterious alphabet-like strings appear if you apply B3’s index values to the DoI) at all. There, (thanks to Jarlve) we can say that the same encryptor seems to have instead chosen numbers with a strong bias towards incrementing numeric series.

But why would that be?