PoIygraphlc Substitution Systems



Section I
Characteristics of Polygraphic

Types of Polygraphic Systems
As first explained in Part One, polygraphic cipher systems are those in which the
plaintext units are consistently more than one letter long. The most common type is
digraphic substitution, which replaces two letters of plaintext with two letters of
ciphertext. There are also such systems as trigraphic and tetragraphic substitution.
The larger types are rare, and awkward to use in military applications, so they are not
included in this manual.

Digraphic System Characteristics
The simplest type of digraphic substitution, if not the simplest type to construct, uses
a 26 by 26 matrix with plaintext values as coordinates to two-letter ciphertext values
within the table. A sample of a digraphic substitution matrix is shown in Table 6-1.

a. As the example shows, with any digraphic system, repeated plaintext digraphs can
cause a ciphertext repeat. Repeated single letters do not cause ciphertext repeats.
Digraphic systems suppress individual letter frequencies, but show normal fre-
quency patterns for pairs of letters. Since there are 676 possible digraphs in the
English language, many more groups of text are needed for digraphic frequencies to
be very useful as a direct aid to analysis.

b. Repeated plaintext words and phrases cause ciphertext repeats only when they
begin in the same odd or even position. If both occurrences of a plaintext repeat
begin in the odd position or both begin in the even position, the ciphertext repeats.
If one occurrence is in an odd position and one is in an even position, they will
produce different ciphertext. As a result, nearly half of all plaintext repeats are
suppressed. This is shown in these three alternate examples, all enciphered from
Table 6-1.

c. In the first example, all three ZEROs produce a repeat when they all begin in the
even position. In the second example, they all begin in the odd position, and only
the portions of the three ZEROS that appear as complete digraphs (the ERs)
produce a repeat. In the third example, the two ZEROs that begin in the even posi-
tion produce repeats, but the first ZERO, which begins in the odd position, does
d. The suppression of individual letter frequencies and a significant portion of plain-
text repeats means that digraphic systems are considerably more secure than
unilateral systems and most multiliterals.

Four-Square System
Large table digraphics are awkward systems for military usage. In their place, there
are several much more convenient small matrix digraphic systems available with
about the same degree of security. The first of these is the four-square.

a. The four-square consists of four 5 by 5 matrices in a square. The two plaintext
letters and the two ciphertext letters of each encipherment each use a different

square. The squares marked p1 and p2 usually, but not always, contain standard
sequences. The two squares marked c 1 and C2 can include any mixed sequence.

b. Encipherment or decipherment follows a rectangular pattern. Whether enciphering
or deciphering, the letters of the digraphs are located in the appropriately labeled
squares. These letters form diagonally opposite corners of a rectangle. The
equivalents, plaintext or ciphertext, are the remaining corners of the same
rectangle. For example, plaintext MO determines the rectangle outlined in the
square below. Plaintext M determines the upper row and the left column of the rec-
tangle. Plaintext O determines the bottom row and the right column of the rec-
tangle. The ciphertext equivalent, KF, is then found in the remaining corners in the
appropriately labeled squares.

c. For a second example, to encipher RT, R is located in the pl square, and T is
located in the p2 square. The ciphertext equivalent of RT is found in the remaining
corners of the rectangle prescribed by RT. The first ciphertext letter, S, is found in
the cl square in the plaintext T column and the plaintext R row. The second cipher-
text letter, N, is found in the C2 square at the intersection of the plaintext R column
and the T row. Tracing the letters from pl to p2 to cl to C2 is shown below.

d. Decipherment is handled in exactly the same way, except that the ciphertext
letters in the cl and C2 squares determine the rectangle by which the plaintext let-
ters are found.

Vertical Two-Square
The two types of two-squares are simpler than the four-square system. The first is the
vertical two-square, which uses two 5 by 5 matrices one on top of the other. Normally
both squares contain mixed sequences.

a. The rectangular rule used with the four-square is used with the two-square, also.
Whenever the letters to be enciphered are in the same column, however, the letters
become their own equivalents. The encipherment of ON and TE in the example
illustrates this.

b. The case where the plaintext letters remain unchanged in the ciphertext is called a
transparency. A weakness of this system is that in the long run, about 20 percent of
the digraphs in a cryptogram will be transparencies. This is enough to give away
more plaintext in many cases and enable a speedy solution.

Horizontal Two-Square
The second kind of two-square is the horizontal two-square, like the vertical, it uses
two 5 by 5 matrices.

a. The rectangular rule again applies. In the horizontal two-square, values on the
same row are replaced with the same letters in the reverse order. This is illustrated
by the encipherment of the plaintext letters be and ig in the example.

b. Digraphs in ciphertext which are the same as the plaintext in reverse, are called
reverse transparencies. Like the direct transparencies of the vertical two-square,
they occur in the long run in about 20 percent of the digraphs. They severely
weaken the security of the system.

Playfair Cipher
The Playfair cipher is the most common digraphic system. Playfair is always
capitalized, because it was named for a Lord Playfair of England. It is the simplest of
systems to construct, using only a 5 by 5 matrix, yet it is more secure than uniliterals
and most multiliterals. The rules of encipherment and decipherment are a little more
complex than the previous digraphic systems. Sizes other than 5 by 5 are occasionally

a. The first rule of encipherment and decipherment is the familiar rectangular rule.
This applies any time the two letters to be enciphered or deciphered are not in the
same row or column. The first four digraphs in the example follow this rule. One
additional step must be remembered. In tracing the encipherment or decipherment
in the matrix, always move vertically from the second letter to the third letter. For
example, to encipher TH, locate the T and the H and move vertically from the H to
the letter that is in the same column as the H and the same row as the T. Following
this rule, TH is enciphered as QB, not BQ. Similarly, to decipher CU, locate the C
and the U, move vertically from the U to find the first plaintext letter E and then
the second plaintext letter S.
b. When the two letters to be enciphered or deciphered are in the same row, follow the
rule, encipher right, decipher left. To encipher or decipher, pick the letter to the
right or left of each letter of the given digraph, as appropriate. In the example, the
plaintext letters R and D are in the same row. They are enciphered with the letters
immediately to the right of each letter, producing ciphertext AJ (or AI). If a letter
to be enciphered is at the right edge, as in the encipherment of HE, the next letter
to the right of the right edge is considered to be the letter in the same row at the far
left. The letter to the right of E is P. Similarly, if deciphering, the letter to the left of
the left edge is the letter at the far right in the same row. The letter to the left of F is
N. Each row is treated as if it were written in a circle with the first letter of a row
immediately following the last letter.

c. When the two letters to be enciphered or deciphered are in the same column, use the
rule encipher below, decipher above. To encipher EA in the example, the letters
below E and A are N and E respectively. To decipher ZU, the letters above Z and U
are U and N respectively. As with the rows, columns are treated as if they were writ-
ten in a circle. The letter after the bottom letter in a column is the top letter; the let-
ter before the top letter is the bottom letter.

d. The rules encipher right, decipher left and encipher below, decipher above produce
the acronyms ERDL and EBDA. For many analysts, it is convenient to memorize
these pronounceable acronyms to remember the rules.

e. The rectangular rule and the row and column rules take care of all possible cases
except double letters. In the Playfair system, there is no rule for enciphering or
deciphering a double letter in the same digraph. When double letters are encoun-
tered in plaintext in the same digraph, the cryptographer must break up the double
letters with a null letter, such as inserting an X between them. As a result, double
letters will never be encountered in the ciphertext, except in error. This is only true
of the Playfair system. Four-squares and two-squares can handle double letters
without any problem.

Section II
Identification of Polygraphic Substitution

General Digraphic Characteristics
Certain identifying characteristics are common to all digraphic systems. Other charac-
teristics appear only with specific systems.
a. Message lengths, repeats, and distances between repeats are likely to be even in
length in all digraphic systems because the basic unit is two-letters. Furthermore,
the systems which use 5 by 5 matrices will often only use 25 letters, omitting either
the I or the J in ciphertext. In some cases, these values will be used alternately just
to ensure use of all letters.

b. Digraphic systems are most often mistaken for biliteral with variant systems,
because both exhibit ciphertext which breaks into units of two and both can use
most letters. The key distinction to look for between biliterals and digraphics is the
complete absence of any positional limitation (paragraph 5-5b) in digraphic
c. Two-square systems stand out because of the director reverse transparencies. Scan
the text for the presence of good plaintext digraphs, either direct or reversed, to
identify two-square systems. Direct transparencies indicate vertical two-squares;
reversed transparencies indicate horizontal two-squares.
d. If no double letters are present in a digraphic, it is probably a Playfair system.
e. Monographic frequency counts for digraphic systems are not as flat as random text
and not as rough as plaintext or unilateral systems. They generally fall in between
the two. The monographic phi test can be used to confirm this, if necessary.

Digraphic Frequency Counts
There are several types of frequency counts you can take for working with digraphic

a. The most common way to take a digraphic count is to break the text into digraphs
and count those digraphs. For example, given text ABCDE FGHIJ . . . , you would
normally break it as AB, CD, EF, GH, IJ, . . . . There are two other ways to take a
digraphic count, however. If you are unsure whether there may be indicator groups
or null letters at the beginning, you may not know where to begin breaking the text
into digraphs. As a comparison, you can skip the first character and begin
separating the text into digraphs beginning with the second character. This will
produce a completely different set of digraphs than the usual method: A, BC, DE,
FG, HI, J . . . . The third way to produce a digraphic count is to combine the two
methods to count all possible digraphs. In this case, you would count AB, BC, CD,
DE, EF, FG, GH, HI, IJ, . . . . Unless you have a reason to want an alternate method,
stick to the first method.
b. There are two ways to record your count on paper. One is to make a 26 by 26 square
on graph paper, and mark the digraphs in the appropriate cells. The other way,
useful with short cryptograms, is to write the letters A through Z horizontally, and
mark the digraphs by putting the second letter of each digraph under the first letter
of the digraph in the A through Z sequence. Then by scanning the columns under
each letter for repeated letters, you can readily spot repeated digraphs. This
method takes much less space than a 26 by 26 square and gives you the same infor-

Digraphic Coincidence Tests
The phi test and phi index of coincidence can be calculated for digraphic frequency
counts as well as monographic.

a. The digraphic phi test is calculated in essentially the same way as the monographic
test. In the monographic phi test, 1 out of 26 comparisons in random text was expec-
ted to be a coincidence for a probability of 0.0385. In the digraphic phi test, 1 out of
676 comparisons is expected to be a coincidence for a probability of 0.0015. The

probability of a coincidence in plaintext is 0.0069 instead of 0.0667. Thus, the
formulas for the digraphic phi test are”

b. As discussed in the first part of this chapter, digraphic ciphertext frequencies will
occur with the same numbers as plaintext frequencies when digraphic systems are
used. If the digraphic φ o is close to φ p but the monographic φ o is low, the system is
likely to be a digraphic system. If you are using the index of coincidence form of the
test, the expected 2 ∆ IC is 4.6. The results are much more variable than the
monographic test, because of the large number of different elements counted, but it
can still be used as a guide. As with any statistical test, the results should not be
used by themselves, but used along with all other available information.

Examples of System Identification
Three messages in unknown systems follow to show the process that leads to system
identification. Repeats are underlined, monographic and digraphic frequency counts
are shown, and monographic and digraphic ICs are calculated for each. The three
messages were all sent by the same headquarters to subordinate elements, and all con-
tained a common message serial number in their header.

a. Message texts and data.
Message 1:

Message 2:

Message 3:

b. Different analysts might approach the identification of the systems used in these
messages in different ways, but here is one example of how the systems can be iden-

(1) Although the messages all carry the same message serial number, which is
usually a sign of isologs, the messages are all different lengths. If they are
isologs, they are not enciphered in the same system.

(2) A comparison of monographic frequency counts confirms that they are in dif-
ferent systems. The highs and lows in each frequency count are too different for
any possibility of repeated use of the identical system.

(3) The ICs give a different picture in each. Message 1 has monographic and
digraphic ICs consistent with plaintext or a unilateral system. The digraphic IC
of 3.41 is slightly below the expected 4.6, but it is within acceptable limits.
Message 2 shows a low monographic IC of 1.26, but the digraphic IC of 5.38 is
also well within plaintext limits. This is typical of digraphic systems. Message 3
is quite high in both monographic and digraphic ICs.

(4) Messages 1 and 2 use nearly all letters. Message 3, which is twice as long as
message 1, uses only 14 different letters. The high ICs and the limited letter
usage are consistent with a biliteral with variants system. A close inspection of
the digraphic frequency count will show rows and columns with very similar
patterns, suggesting external variants that can be combined. Different letters
are used in the row position than those used in the column position. This
positional limitation confirms the identification of a biliteral with variants
(5) Message 1 has the most repeated text, which is consistent with a unilateral
system. Message 2 has only a few repeats and message 3 has only short and
fragmentary repeats. In message 3, the fragmented repeat on lines 7 and 10 are
in the identical relative position in message 2 as the ZTVK repeat in lines 2 and
5 of message 1. This similarity strongly confirms that the two messages are
(6) The identifications of the systems in messages 1 and 3 are clear at this point,
but message 2 still needs to be•clarified. The underlined repeats in message 2 are
in the same relative position as in message 1, if you adjust for the slightly
increased length of the message. Only some of the repeats from message 1
appear in message 2, however. This is consistent with a digraphic system, which
will only show repeats that begin in the same even or odd position.

(7) In message 2, a check of the long diagonal from the AA position to the ZZ posi-
tion of the digraphic frequency count shows that the only double letter that
appeared was the filler XX at the end of the message. The Playfair is the only

digraphic system which will not show double letters. Finally, because the
Playfair cannot encipher double letters, all double letters that occur in digraphs
must be broken up by the insertion of null letters. This characteristic explains
how it can be an isolog, but appear slightly longer. The three messages are all
clearly isologs, and the systems are confidently identified, lacking only the final
solution for full confirmation. Solution techniques for each of the major
digraphic system types are explained in the next chapter.