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CHAPTER 13

TRANSPOSITION SPECIAL SOLUTIONS



Special Exploitable Situations
13-1.
Military forces are rarely equipped to change cryptosystem keys with every message
transmitted. The logistics and management problems of providing enough different
keys and controlling their use are difficult to handle. Normally, keys will be reused for
a period before they are changed. With transposition systems, several special situa-
tions can arise when keys are reused that make a solution possible when the system
might otherwise resist successful analysis. One of these situations arises in columnar
transposition whenever similar beginnings and endings are used with the same width
matrix. The keys do not have to be the same in this case as long as the width is the
same. Another more general situation occurs whenever two or more different messages
of the same length occur using exactly the same keys. Each of these situations is
explained in the following paragraphs.

Similar Beginnings and Endings
13-2.
With columnar transposition, repeated message beginnings or endings can cause an
easily recognizable and exploitable situation. When the same width keys are used and
the beginnings are the same, the tops of the columns in each message will consist of the
same letters. When the length of the repeated beginning is several times as long as the
width of the matrix, these repeated letters are easy to spot.

a. The next two messages demonstrate the techniques that can be used when similar
beginnings are encountered. Repeated segments between the two messages are
underlined.




13-1
(1) There are eight repeated segments in each, which shows that the messages are
each eight columns wide. The repeated segments are not in the same order,
which shows that the two messages use different numerical keys.

(2) Message 1 has 95 letters. Dividing 8 into 95 gives 11 with a remainder of 7. This
means that all but one column must have 12 letters. The distance between
repeats shows that this is true. All segments have 12 letters except for the fifth
segment, which has 11 letters. The fifth segment, beginning IFA, must be the
right-hand column of the matrix.
(3) Message 2 has 92 letters. Four columns have 12 letters and four columns have
11 letters.
(4) All repeated segments contain three letters except for the ASOL segment. The
column beginning ASOL is probably the left-hand column.
(5) As a result of these observations, we can place the first and last columns in each
matrix, and we can separate the middle six columns into two groups of three,
based on the length of the columns in message 2.




(6) Completion of the solution from here is straightforward. Anagram each group of
three columns in each message, and the solution is complete. The similar begin-
ning is ALL REQUISITIONS FOR MEDICAL.
b. Messages with similar endings, such as a repeated signature block, show repeated
segments which represent the bottoms of columns instead of the top. The solution is
approached the same way, except that the text will not necessarily appear in the
same columns in both messages.




13-2
Messages With the Same Length and Keys
13-3.
Whenever two or more messages have the same length and are transposed with the
same keys, they can be solved together. The more messages you find that are the same
length and use the same keys, the easier they are to solve. This technique can be used
regardless of the type of transposition system.
a. Solving messages with the same length and keys is particularly effective with
columnar transposition. The next example shows how the solution can be
approached. The five messages all use the same keys. Their positions have been
numbered for easy reference and to aid in key recovery.




(1) The Q in message 2 in position 1 must certainly be followed by the U in
position 8.

(2) Position 1 must be at the top of a column in the original matrix, since columns
are extracted beginning at the top. Position 8 is also probably at the top of a
column. This applies not just to message 2, but to all five messages. The
position 1 column can be written next to position 8.




(3) Position 2 must be from the second row of the matrix. If position 8 is from the
top row, then position 9 must be from the second row, also. Similarly, positions



13-3
3 and 10 are from the third row. Positions 4 and 11 are from the fourth row. Posi-
tions 5 and 12 are probably from the fifth row, although these are short messages
and there might not be as many as five rows.




(4) Now the task is to find additional columns to add to the fragments already
started. For example, the QU in message 2 should be followed by a vowel, and
the most likely letter after JU in message 5 is N. There are three columns with
an N in message 5, and only one of these, position 19, has a vowel in message 2.
Therefore, we will add columns 19, 20, 21, 22, and 23 to our fragments.




(5) All of the fragments produce good plaintext except, possibly, the last one. QUA
will usually be followed by an R. Of the two columns with an R in message 2,
column 12 provides the best combinations.




13-4
(6) All of the matches give good plaintext, except the fifth set, which clearly does
not belong now. It is easy now to see words to build on, such as ARTILLERY,
QUARTERS or HEADQUARTERS, JUNCTION, SUPPORT, FIVE, and
others. All of these leads are added to the completely anagrammed messages.




(7) The final step in the solution is to recover the numerical keys. Looking at the
beginning, the pattern starts to repeat after seven letters, so the original matrix
was seven letters wide. The numerical key, derivable by observing the order in
which the columns were extracted, was 4275136.

b. The technique of solving messages of the same length and keys can be used with
any transposition system. It can be used as the basis for recovery of more difficult
transposition systems such as large grilles and double transposition. The cyclic pat-
tern of columnar transposition aided the solution of the example above. Given four
or more messages of the same length and keys, however, the complete messages can
often be anagrammed without the help of the cyclic pattern.




13-5



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