Transposition Systems



Nature of Transposition
Transposition systems are fundamentally different from substitution systems. In sub-
stitution systems, plaintext values are replaced with other values. In transposition
systems, plaintext values are rearranged without otherwise changing them. All the
plaintext characters that were present before encipherment are still present after
encipherment. Only the order of the text changes.
a. Most transposition systems rearrange text by single letters. It is possible to
rearrange complete words or groups of letters rather than single letters, but these
approaches are not very secure and have little practical value. Larger groups than
single letters preserve too much recognizable plaintext.
b. Some transposition systems go through a single transposition process. These are
called single transposition. Others go through two distinctly separate transposition
processes. These are called double transposition.
c. Most transposition systems use a geometric process. Plaintext is written into a
geometric figure, most commonly a rectangle or square, and extracted from the
geometric figure by a different path than the way it was entered. When the
geometric figure is a rectangle or square, and the plaintext is entered by rows and
extracted by columns, it is called columnar transposition. When some route other
than rows and columns is used, it is called route transposition.

d. Another category of transposition is grille transposition. There are several types of
grilles, but each type uses a mask with cut out holes that is placed over the
worksheet. The mask may in turn be rotated or turned over to provide different pat-
terns when placed in different orientations. At each position, the holes lineup with
different spaces on the worksheet. After writing plaintext into the holes, the mask is
removed and the ciphertext extracted by rows or columns. In some variations, the
plaintext may be written in rows or columns and the ciphertext extracted using the
grille. These systems may be difficult to identify initially when first encountered,
but once the process is recognized, the systems are generally solvable.

e. Transposition systems are easy to identify. Their frequency counts will necessarily
look just like plaintext, since the same letters are still present. There should be no
repeats longer than two or three letters, except for the rare longer accidental repeat.
The monographic phi will be within plaintext limits, but a digraphic phi should be
lower, since repeated digraphs are broken up by transposition. Identifying which
type of transposition is used is much more difficult initially, and you may have to
try different possibilities until you find the particular method used or take advan-
tage of special situations which can occur.
f. Columnar transposition systems can be exploited when keys are reused with
messages of the same length. As will be explained in Chapter 13, the plaintext to
messages with reused keys can often be recovered without regard to the actual
method of encipherment. Once the plaintext is recovered, the method can be

Examples of Columnar Transposition
The most common type of transposition is columnar transposition. It is the easiest to
train and use consistently.

a. Simple Columnar Transposition. At its simplest, columnar transposition enters
the plaintext into a rectangle of a predetermined width and extracts ciphertext by
columns from left to right. For example, a simple columnar transposition with a
width of seven is shown below.

(1) The cryptographer receiving the above message knows only that a width of 7
was originally used. The cryptographer rebuilds the matrix by determining the
length of each column and writing the ciphertext back into the columns. With a
width of 7 and a length of 42, each column must have 6 letters. Inscribing the
ciphertext into columns from left to right recreates the original matrix, and the
plaintext can be read by rows.

(2) Not all messages will come out even on the bottom row. Here is the same
message with STOP omitted. The columns are not all the same length. In this
case, the matrix is called an incompletely filled matrix.

(3) The deciphering cryptographer must now perform the additional step of deter-
mining which columns will be longer than the others. With 38 letters and a
given width of 7, dividing 38 by 7 produces 5 with a remainder of 3. This means
that the basic column length is 5, but the first 3 columns are 1 letter longer.
Sometimes, cryptographers will avoid this additional step by padding message
texts so that the bottom row is always completely filled.

(4) The solution of these systems is extremely easy. The security depends on just
one number, the matrix width. All you have to do to solve a message enciphered
by simple columnar transposition is to try different matrix widths until you find
the right one. To try each width, you just do exactly what the deciphering
cryptographer does. Divide the total length by the trial width and the result and
remainder will tell you the basic column length and how many longer columns
there are.

(5) If you suspect that only completely filled matrices are being used, the solution is
easier. You only need to test widths that evenly divide into the message length
in that case. For example, with a length of 56, you would try widths of 7 and 8. If
neither of these worked, you would also try 4, 14, 2, and 28 to cover all
possibilities. It is better to try the possibilities closest to a perfect square before
you try very tall and very wide matrices.

b. Numerically-Keyed Columnar Transposition. Numerically-keyed transposition
systems are considerably more secure than simple columnar transposition. You
cannot exhaust all possibilities with just a few tries as you can with the simple
systems. The transposition process is similar to that used to produce transposition
mixed sequences.

(1) The numerical key is commonly based on a keyword or key phrase. Unlike
keywords used to produce mixed sequences, the keyword may have repeated
letters in it. To produce a numerical key from a keyword with repeated letters,
the repeated letters are numbered from left to right.

(2) As with simple columnar transposition, matrices may be completely filled or
incompletely filled. In either case, the plaintext is written horizontally and the
ciphertext is extracted by column in the order determined by the numerical key.
The following example shows an incompletely filled matrix.

(3) The decipherment process for the receiving cryptographer is more complicated
than with simple columnar transposition. The cryptographer must decide the
column lengths, as before. With the above message, the cryptographer divides
the length of the message by the length of the numerical key. In this case, 32
divided by 6 is 5 with a remainder of 2. The basic column length is 5 with two
longer columns at the left. The cryptographer then sets up a matrix with the key
at the top and marks the column lengths.

(4) The ciphertext is now entered by columns according to the numerical key to
produce the plaintext.

(5) The solution of numerically-keyed systems is more complex than for simple
columnar transposition. It is more than just trying all possibilities. The solution
of numerically-keyed columnar transposition is explained in Chapter 12.

Route Transposition
There are many other ways to transpose messages than columnar transposition using
squares and rectangles. The shape of the geometric figure used can be varied, and the
method of inscribing and extracting text can be varied. Columnar methods are the
most common in military usage, because they are the easiest to learn and use reliably,
but other methods may be encountered. Some of these common methods are shown

a. Route transposition using other geometric figures.
(1) The rail-fence cipher is inscribed by zigzag pattern and extracted by rows.

(2) The triangular pattern is inscribed by rows and extracted by columns.

b. The next examples show just some of the possibilities for route transposition using
squares or rectangles. Each example is based on REINFORCEMENTS ARRIVING
NOW to help you see how the route was entered. The route can be:

(1) Inscribed by spiral, out by columns.

(2) Inscribed by diagonals, out by alternating rows.

(3) In by outward spiral, out by alternating diagonals.

(4) In by L-pattern, out by spiral from lower right.

c. Completely filled squares or rectangles are more common with route transposition
than with columnar transposition. The reason is that it is often difficult for the
cryptographers to figure out how to handle an incompletely filled matrix. It is sim-
pler in practice to completely fill each matrix than to provide rules to cover every
incompletely filled situation.

d. The solution of route transposition is largely a matter of trial and error. When you
suspect route transposition, see if the message length is a perfect square or if the
matrix can be set up as a completely filled rectangle. Then try entering the cipher-
text by different routes, and look for visible plaintext by another route.