Beale's Papers, Ciphers, and Key: The Order Created
But, evidence suggests that the actual order was this: Paper No. 2 was
prepared sometime before January 4th, 1822. To avoid leaving the paper in the clear, it was enciphered using a key based
on the Declaration of Independence, thus allowing the clear paper to be destroyed. Paper No. 3 was prepared subsequent to the long and detailed letter to
Mr. Morriss of January 4th, but prior to his very brief letter to Mr. Morriss of January 5th. But as Paper No. 2 had
already been enciphered, it was necessary to amend cipher B2 to include the phrase "number three herewith."
Paper No. 1 was created next. But Beale had not yet settled on a method of cipher for Papers No. 1 and No. 3. Thus,
to avoid leaving the papers in the clear while developing a more intricate plan for their protection, he appended Paper
No. 1 to the end of Paper No. 3 and enciphered them using a simple method (easy to encipher and easy to decipher).
This simple method of cipher is unknown, and thus part of the mystery confronting the cryptanalyst. The clear papers
were destroyed. Ultimately, Beale decided to encipher the remaining two papers using a homophonic cipher that may also have
been a book cipher, but based on a different keytext. His homophonic cipher may also have had some extra "twist"
to make it more intricate and secure. But when it came time to encipher Papers No. 1 and No. 3 with his homophonic
cipher, the plan changed. Rather than decipher the papers, only to immediately reencipher them, he
instead decided to directly encipher the already enciphered papers. Thus, the papers were enciphered twice, using two different
methods. During the final step of encipherment, a form of double encipherment was used to create five monotonically increasing letters strings in cipher B1. However,
the strings were most likely created by allowing the double encipherment process, in some cases, to assign cipher numbers
to the homophonic enciphering key. This would provide enough degrees of freedom for the double encipherment
process to succeed. The key to his ciphers was created last.
Here is a list of accumulated evidence pertinent to the issue addressed here:
A computer test provided
strong statistical evidence that B1 and B3 were created with the same cipher and key. The distribution of cipher numbers in
B1 and B3 suggests that B3 was enciphered first and B1 was enciphered second as a continuation of cipher B3.
In his letter of January 4th, Beale refers to the members in his party, saying "
I thought, at first, to give you their names in this letter, but reflecting that someone may read the letter, and thus be
enabled to impose upon you by personating some member of the party, [I] have decided the present plan is best." His short, yet, "to the point" letter of January 5th, with a description of the contents of Paper
No. 3, was no doubt prompted as a consequence of actually preparing Paper No. 3 on that very day.
Cipher B2 contains a mixture of cipher numbers that occur once (freq=1) and some that occur multiple
times (freq>1). For the most part, the freq=1 cipher numbers are randomly distributed over the entire cipher. However,
there is an unusually long run of 95 freq>1 cipher numbers that overlaps the words "number three herewith"
in the deciphered text. This suggests that B1 might have been amended to include new words "number
three herewith" and that the process involved replacing the letters in this phrase using cipher numbers
already present in the cipher text. If so, then any freq=1 cipher number replaced by a cipher number associated with the
new words would be or become a freq>1 cipher number. This could explain the unusually long run of 95
freq>1 cipher numbers.
A computer program was written to create 10,000 plain texts that simulate Beale's
Papers No. 3 and No. 1. No. 1 was appended to the end of No. 3 and each plain text was enciphered with a homophonic cipher.
Different types of homophonic ciphers were tried. In each case, a count was made of the number of repeated 2-grams
(pairs of cipher numbers that repeat) in each cipher text. The number of repeats for all 10,000 cipher texts was plotted as
a distribution. The number of repeated cipher text 2-grams in B3B1 (B1 appended to the end of B3) was also computed,
and this number was compared with the plotted distribution for each trial in which 10,000 cipher texts were produced. In each
case, the number of repeated 2-grams in B3B1 fell outside and below the distribution for the 10,000 cipher texts. In
other words, there are too few 2-grams in B3B1 for these two ciphers to be the result of enciphering a plain text with a homophonic
enciphering key. If the ciphers are real, it means that the plain text had to first undergo a step, e.g., an additional
enciphering step, that caused the resulting intermediate text to have a somewhat "flattened" distribution,
which would account for the paucity of repeated 2-grams in B3B1.
A computer simulation program
was written to test the hypothesis that Beale's longest monotonically increasing letter string could be created using a process
of double encipherment. The assumption was made that double encipherment was performed as part of the step
of enciphering an intermediate text with a homophonic enciphering key to produce the final cipher text. The program succeeded
in producing monotonically increasing letter strings of length 21 in roughly one in 22 attempts. It succeeded in producing
monotonically increasing letter strings of length 21 and of the form A...P (similar to Beale's longest letter string ABCDEFGHIIJKLMMNOOPPP)
or better, namely A...P or A...Q or A...R or A...S or A...T or A...U, in roughly one in 100 attempts. However, the results
are disappointing and would have to be characterized as failure.
Double encipherment could be made to
work if nulls could be employed. Nulls decode as blanks or "nothing." But as a computer test has shown
that B1 and B3 were created with the same cipher and key, nulls won't work due to the fact that the cipher numbers are, in general, spread across both ciphers (B3 and B1). This means that cipher
numbers in B1 denoting nulls would, in many cases, also occur in B3. This would reduce the effective length
of the plain text representing Paper No. 3, which is already so short that any reduction to its length would seem to
invalidate B3 as a credible cipher.
But, there is a way that the letter strings could have
been created with double encipherment, although it is not double encipherment in the usual sense. The strings were created
first, even before the homophonic enciphering key was created. Instead of starting with an initialized enciphering
key and selecting cipher numbers that can be decoded as desired plain text letters using two different keys, cipher numbers are
assigned to letters in the enciphering key that can be decoded to desired plain text letters using two different keys. In
effect, this provides an extra degree of freedom that permits double encipherment to succeed. As more and more numbers are
assigned to letters in the key, it is possible in some cases to perform double encipherment in the usual sense, i.e., selecting
cipher numbers already assigned to letters in the enciphering key rather than assigning additional cipher numbers to
letters in the enciphering key.