Several monotonic increasing letter strings occur in the recovered plain text when B1 is deciphered with the key to B2. In a monotonic increasing letter string, each letter is either one greater than the previous letter or equal to the previous letter. The observed deciphered strings and their corresponding locations in B1 are these:

Location       String 
37       AAAAB
111       ACBCDDE
Several monotonic increasing letter strings occur in the recovered plain text when B1 is deciphered with the key to B2. In a monotonic increasing letter string, each letter is either one greater than the previous letter or equal to the previous letter. The longest letter string ABCDEFGHIIJKLMMNOOPPP is just too long to be accidental. It could only have been created by someone who knew the key to B2.
The letter strings have the appearance of being created via a process of double encipherment. Thus, if the letter strings are genuine it means that they were created by Beale as part of the process of enciphering B1. Moreover, it means that the cipher numbers corresponding to the letter strings can be deciphered with the key to B1 and the key to B2 to produce two different meaningful decodings.
As a practical matter, because double encipherment introduces an extra degree of difficulty into the enciphering process, it is reasonable to expect that imperfect letter strings of the form ABCDEFGHIIJKLMMNOOPPP might be created with Beale's double encipherment method, but less likely that perfect letter strings of the form ABCDEFGHIJKLMNOPQRSTU would be created.  
The process of double encipherment can be described, although a few assumptions are first made. Assume that B1 makes use an encipherment method, consisting possibly of one or more encipherment operations, in which the final operation is encipherment with a homophonic cipher, e.g., a book cipher based on a keytext different from the Declaration of Independence. Other methods of homophonic encipherment are also possible. In any case, now suppose that one wishes to create a monotonically increasing letter string AAAB at location 37 in B1. Suppose also that the further that the four letters beginning at location 37 in the intermediate text are THMX.
The first step consists of enciphering the letter "T" with the key for B1 and enciphering the letter "A" with the key to B2. However, with double encipherment it means that we must find a cipher number that enciphers the letter "T" using the key to B1, which also enciphers the letter "A" using the key to B2. If the set of possible cipher numbers capable of enciphering letter "T" with the key to B1 is large and the set of possible cipher numbers capable of enciphering letter "A" with the key to B2 is large, then it is probable that one can find a cipher number in the first set of cipher numbers that also occurs in the second set of cipher numbers.
The more cipher numbers there are in each set of numbers, the greater the chance of finding a number that repeats in both sets. If one fails to find a cipher number that occurs in both sets, then the operation of double encipherment ends (one can go no further). In that case, the process would begin again with the plain text letter "H" at position 38 in Paper No. 1. However, if the step of double encipherment is successful, then an attempt is made to encipher letter "H" at position 38 in Papers No. 1 with the key to B1 and encipher letter "B" or letter "A" with the key to B2. In this case, the chance of success is increased because there are two choices, "B" or "A" that can be enciphered with the key to B2. However, the two choices have a natural precedence, namely, the next letter "B" is tried first and the previous letter "A" is tried second. In other words, an attempt is first made to encipher letter "B," and failing that, an attempt is made to encipher letter "A." The process continues in this manner. One cannot predict in advance the letters in a letter string that will repeat or the number of times a particular letter will repeat — the outcome is unpredictable.
"Notes on the Beale Ciphers" published anonymously on the Internet proposes a method for creating a double key which could be used by an encryptor to perform double encryption.
As mentioned previously, the strings AAAAB, AAABBCDEFF, AABBCCCCDDE, and ACBCDDE, were probably practice trials in which Beale experimented to see just how easy or difficult it would be to perform double encipherment.
On the basis of the four practice trials (AAAB, AAABBCDEFF, AABBCCCCDDE, and ACBCDDE), Beale probably realized that it would be difficult to construct a letter string that included the uncommon letters "J" and "K." To remedy the problem, he may have resorted to the following trick: For the fifth, and longest letter string, instead of beginning the letter string with letter "A," Beale may have begun the string with letters "JK."
There are six words in Beale's numbered Declaration of Independence that begin with letter "J" and three words that begin with the letter "K." They are

 Location  Word        Location  Word 
 120 just   305 king 
 567 justice  626 kept
 576 judiciary  994 known
 581 judges   
 665 jurisdiction   
 749 jury   

There are 6 × 3 = 18 ways in which these cipher numbers can be paired together, namely:    
(120, 305)  (567, 305)  (576, 305)  (581, 305)  (665, 305)  (749, 305)
(120, 626)  (567, 626)  (576, 626)  (581, 626)  (665, 626)  (749, 626)
(120, 994)  (576, 994)  (576, 994)  (581, 994)  (665, 994)  (749, 994)
Beale then consulted the key to B1 to determine the plain text letter pairs corresponding to each of the 18 pairs of cipher numbers. For sake of discussion, suppose that the 12 pairs of letters were these:
CA,  IA,  FA,  RA,  DA,  IA
CP,  IP,  FP,  RP,  DP,  IP
CT,  IT,  FT,  RT,  DT,  IT
Of course, the actual letter pairs that Beale would have found by deciphering the pairs of cipher numbers with the key to B1 are unknow to us. 
The fourth trial letter string ends at location 117 in B1. Thus, beginning with location 118, a check of the plain text in Paper No. 1 is made, looking for one of the twelve pairs of letters. For sake of argument, I shall assume that the first occurrence is letter pair "CA," found at location 198 in sample plain text "AND HERE YOU CAN SEE AN OLD." Of course, we don't kow what the actual plain text was.
Then, Beale worked backward from letter "J" at location 198 in B1, which I assume produced the string "ABCDEFGHII" and he worked forward from letter "K" at location 199 in the letter string, which I assume produced the string "LMMNOOPPP." To increase his chance of success, Beale could have also checked the plain text before location 198 in Paper No. 1 and after location 199 in Paper No. 1 to ensure that there was no uncommon letter that might interfere with the creation of the letter strings. The example plain text, actual cipher numbers, and corresponding monotonic increasing letter string are shown below:
Beginning Location: 188 
   A    N   D    H     E    R     E   Y    O   U    C    A   N   S    E    E    A    N    O   L   D 
147436 195 320   37 122 113   6 140   8 120 305  42  58 461  44 106 301   13 408 680 
   A   B    C   D   E   F   G   H   I  I   J   K   L  M   M   N   O   O   P   P   P

If Beale did use this trick to "jump start" his double encipherment operation, it may be assumed that the letter pair corresponding to cipher numbers (120, 305) in Beale's key to B1 happened to be a fairly common occurring 2-gram in standard English text, as cipher numbers 120 and 305 are the ones that actually do appear in B1. On the other hand, the appearance of numbers 120 and 305 in the letter string could be an indication that Beale did not need to use this "jump start" trick.

Because there is only one word in Beale's numbered Declaration of Independence that begins with letter "Q," Beale found that he was unable to continue the double encipherment operation beyond letter "P." He tried to continue by repeating letter "P" two additional times, hoping to encounter the needed letter in the plain text of Paper No. 1 that would allow letter "Q" in the letter string to be enciphered. But this failed. Beale either gave up, or found that he was unable to continue.

There are other "tricks" that Beale could have employed to facilitate the double encipherment process.

One such method would be to make use of nulls in the key to B1. Nulls represent "nothing" and can be thought of as spaces or blanks. An easy way to make use of nulls would be for the key to B1 to be based on a book cipher in which the individual letters in the keytext are consecutively numbered, including the spaces between words. In that case, the spaces would represent nulls. And, as there would be many nulls (approximately one for every four or five letters in the keytext) to choose from, there would be no difficulty in forcing the double encipherment process to succeed. Thus, if the usual double encipherment step happened to fail, a recovery could be easily made by inserting one or more nulls into the plaintext (Paper No. 1) at the appropriate point.

Another method would be to allow letters in the plain text to repeat in superincreasing fashion just as they might repeat in the letter string. If such a method were used, the enciphered plain text might look like AANNDHEEREEYOOUCCAANSEEANOLD instead of ANDHEREYOUCANSEEANOLD. Beale would have been satisfied that Mr. Morriss would have no difficulty reading such a recovered plain text and understanding its meaning. 

Yet another method works only if the key for cipher B1 is defined by Beale himself. Think of the key as a table with as many rows as there are letters in the alphabet. The key is populated by assigning cipher numbers or homophones to each letter in the key, by writing the homophones in the proper rows in the key. Thus, the assignment of homophones is a random or haphazard process. If a keytext is involved, the keytext itself determines the assignment of homophones to the letters in the key. In that case, the assignment is deterministic not random. For sake of discussion, suppose that Beale is the one who assigns homophones in the key. In such a case, double encipher works as follows: The process starts with an empty enciphering key. At first, letters in the letter string are produced as a consequence of assigning or adding needed homophones to the empty enciphering key. This extra degree of freedom can be enough to permit double encipherment to succeed. But, as more and more homophones are assigned to letters in the key, it becomes possible in some cases to perform double encipherment in the usual sense, without assigning new homophones to letters in the key.  In effect, the process of double encipherment consists of adding homophones to the enciphering key (whenever necessary) or making use of homophones already assigned to the enciphering key (whenever feasible). Once double encipherment has been completed, additional homophones are assigned to letters in the key, to complete the key, and the remaining plain text is enciphered in the usual sense.


Published Sept 12, 2011  

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Updated September 22, 2012  

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