MONOALPHABETIC MULTILITERAL

SUBSTITUTION SYSTEMS

Section I

Characteristics and Types

Characteristics of Multilateral Systems

5-1.

As explained in Chapter 3, monoalphabetic unilateral systems are those in which the

ciphertext unit is always one character long. Multilateral systems are those in which

the ciphertext unit is more than one character in length. The ciphertext characters

may be letters, numbers, or special characters.

a. Security of Multilateral Systems. By using more than one character of ciphertext

for each character of plaintext, encipherment is no longer limited to the same num-

ber of different cipher units as there are plaintext units. Although there is still only

one alphabet used in multilateral systems, the alphabet can have more than one

ciphertext value for each plaintext value. These variant ciphertext values provide

increased security. Additionally, the plaintext component of alphabets can be

expanded easily to include numbers, punctuation, and common syllables as well as

the basic 26 letters. When used, the variation in encipherment and the reduced

spelling of numbers, punctuation, and common syllables minimize the exact

weaknesses that we used in Chapter 4 to break into unilateral systems.

b. Advantages and Disadvantages. The increased security possible with variant

multilateral systems is the major advantage. The major disadvantage is that by

substituting more than one character of ciphertext for each plaintext value, the

length of messages and resulting transmission times are increased. A second disad-

vantage is that more training and discipline are required to take advantage of the

increased security. If training and discipline are inadequate, the security advan-

tages are lost easily.

5-0

Types of Multilateral Systems

5-2.

Multiliteral systems are further categorized by the type of substitution used. The

major types areâ€”

Biliteral systems, which replace each plaintext value with two letters of ciphertext.

Dinomic systems, which replace each plaintext value with two numbers of cipher-

text.

Trilateral and trinomic systems, which replace each plaintext value with three

letters or numbers of ciphertext.

Monome-dinome systems, which replace plaintext values with one number for some

values and two numbers for other values.

Biliteral with variants and dinomic with variants systems, which provide more than

one ciphertext value for each plaintext value.

Syllabary squares, which may be biliteral or dinomic, and which include syllables as

well as single characters as plaintext values.

Cryptography of Multilateral Systems

5-3.

The cryptography of each type of multilateral system, including some of the odd varia-

tions is illustrated in the following paragraphs. Most of these systems are coordinate

matrix systems in which the plaintext values are found inside a rectangular matrix and

the ciphertext values consist of the row and column coordinates of the matrix.

a. Simple Biliterals and Dinomics. The simplest multilateral systems use no varia-

tion. They typically use a small rectangular matrix large enough to contain the

letters of the alphabet and any other characters the system designer wants to use as

plaintext values.

(1) The plaintext values are the internals of the matrix. They may be entered

alphabetically, follow a systematic sequence, or they may be random. They

may be entered in rows, in columns, or by any other route.

(2) The row and column coordinates are the externals. Conventionally, the row

coordinates are placed at the left outside the matrix, and the column coor-

dinates are placed at the top. As with the internals, the coordinates may be

selected randomly or produced systematically.

(3) A ciphertext value is created by finding the plaintext value inside the matrix

and then combining the coordinate of the row with the coordinate of the column

for that plaintext value. Either can be placed first, although placing the row

coordinate before the column coordinate is more common.

5-1

(4) Five by five is a common size for a simple system (Figure 5-1). The 26 letters are

fitted into the 25 positions in the matrix by combining two letters. The usual

combinations are I and J or U and V. It is up to the deciphering cryptographer to

determine which of the two is the correct value. There are few, if any, words in

common usage in which good words can be formed using either letter of the I/J

or U/V combinations. Other common sizes are 6 by 6 (which gives room for the

10 digits), 4 by 7, and 3 by 10. Many other sizes are possible.

(5) Example A in Figure 5-1 is a simple 5 by 5 matrix with I and J in the same plain-

text cell of the square. The coordinates and the sequence within are in

alphabetic order.

5-2

(6) Example B is a simple 3 by 10 matrix with orderly coordinates and a keyword

mixed sequence inscribed within. The four extra cells are used for punctuation

marks.

(7) Example C is a 6 by 6 matrix with a spiral alphabetic sequence followed in the

spiral with the 10 digits. The coordinates in this case are related words.

(8) Example D is a 5 by 5 matrix with numeric coordinates. The plaintext sequence

is keyword mixed entered diagonally. In this case, there is deliberately no

repetition between the row and column coordinates. This allows the coordinates

to be read either in row-column order or in column-row order without any

ambiguity, as in the sample enciphered text. This is unusual, but you should be

alert to such possibilities.

b. Triliterals and Trinomics. Trilateral and trinomic systems are essentially the

same as biliteral and dinomic systems. The difference is that either the row coor-

dinates or the column coordinates consist of two characters instead of one, creating

a three-for-one substitution. Such systems offer no real advantage except to provide

a slightly different challenge to the cryptanalyst, and have the distinct disadvan-

tage of tripling the length of messages. They are easily recognized, and offer no

increase in security.

c. Monome-Dinomes. Monome-dinomes are coordinate matrix systems constructed

so that one row has no coordinate. The values from that row are enciphered with the

column coordinate only. This means that some ciphertext values are two characters

in length (dinomes) and others are only one (monomes). If the values used as row

5-3

coordinates are also used as column coordinates, no plaintext values are placed in

the monome row under those repeated column coordinates. The blanking of cells in

the monome row is shown in the example below.

Resulting message:

25720 67463 63485 69575 40000

(1) If the cells corresponding to the row coordinates in the monome row are not

blanked, the deciphering cryptographer will have difficulty. Decipherment

proceeds left to right, and when a 5 or a 6 is encountered in the matrix shown, it

will always be a row coordinate or combine with a preceding row coordinate. It

will never stand alone as a monome. If the 5 and 6 cells were not blanked, the

deciphering cryptographer could not tell if a 5 or 6 were a monome or the begin-

ning of a dinome. The cryptographer would have to rely on context to figure out

which was intended, and that could lead to errors.

(2) The additional examples of monome-dinomes shown below demonstrate the

various ways they can be constructed. The last example (top of page 5-5) is a

monome-dinome-trinome.

5-4

Resulting message:

31323 12331 3023271318 90000

d. Variant Systems. Variants in a multiliteral system allow plaintext characters to

be enciphered in more than one way. Variants can be external or internal.

(1) External variant systems have a choice of coordinates. Either row coordinates

or column coordinates or both can have variants. Examples A and B in

Figure 5-2 provide two ways to encipher every letter.

5-5

Example C provides four ways to encipher every letter. Example D was con-

structed to provide the most variants for the most common letters. The letters

E, T, and O can all be enciphered in eight different ways. R, N, and I can be en-

ciphered in six different ways. A, S, D, L, U, H, and M can be enciphered in

four different ways. Q, X, Z, and the comma can only be enciphered one way.

When any of the systems are conscientiously used, repeated words in the text

will not produce repeated ciphertext segments.

(2) Internal variant systems use larger matrices to provide variants inside the

matrix. Each common plaintext letter appears more than once. Here are two

examples of internal variant systems.

The first example above places the letters in the matrix according to their

expected frequency in plaintext. If their use is well balanced, all letters in the

square will be used with about the same frequency. The second square achieves

the same effect by using 10 words or phrases in the rows, which use all the

letters. The first letters of the column spell out an eleventh wordâ€”logarithms.

5-6

e. Syllabary Squares. Another type of internal variant system is the syllabary

square. This type includes common syllables as well as single letters. When these

are used, the same square may be used for a period, changing the coordinates more

frequently than the square itself.

The two sample encipherments of REINFORCEMENTS show that a syllabary

square suppresses repeats in ciphertext just as single letter variant systems do. It

also has the advantage of producing shorter text than single letter multilateral

systems.

f. Sum Checks. It is very easy for errors to occur when messages are transmitted and

received, whatever means of transmission are used. Because of this, some users

introduce an error detection feature into traffic known as sum checking.

(1) In its simplest form, a sum-check digit is added to every pair of digits in numeric

messages. The digit is produced by adding the pair of digits to produce the

5-7

third. If the result is larger than 9, only the second digit is used, dropping the

10â€™s digit, for example 8 plus 9 equals 7 instead of 17. This is also known as

modulo 10 arithmetic.

(2) Whenever the first two digits do not add up to the third, the receiving cryp-

tographer is alerted that an error has occurred. The cryptographer then tries to

figure out the correct digit from context or by assuming that two of the digits are

correct and determining what the third should be.

(3) There are many variations on the simple system of sum checking described

here. Sometimes the sum-check digit will be placed first or second in each

resulting group of three. Sometimes a sum check will be applied to a larger

group than two numbers. Sometimes a different rule of arithmetic will be used,

such as adding the sum-check digit so that the resulting three always add to the

same total. Sometimes a more complex system will be used that provides

enough information to resolve many errors as well as detect them, particularly

when computers are used in data and text transmissions.

(4) Computer produced sum checks can be used with any characters, not just num-

bers. Computer produced sum checks will normally be invisible to the user, as

they are automatically stripped out when a message is received. They may or

may not be invisible to the cryptanalyst. Recovery of computer produced sum

checks is well beyond the scope of this text, but you should be alert to their

existence.

Section II

Analysis of Simple Multilateral Systems

Techniques of Analysis

5-4.

The first steps in solving any multilateral system are to identify the system and

establish the coordinates. It makes little difference whether the system uses numbers

or letters for coordinates. The techniques are the same in either case. Once the system

is identified and the coordinates set up, a solution of the simpler systems is the same as

with unilateral systems. Variant systems require additional steps. Each type is con-

sidered in the following paragraphs.

5-8

Identification of Simple Biliteral and Dinomic

5-5.

Systems

Simple biliteral and dinomic systems are very easy to recognize and solve.

a. First, the two-for-one nature of the system will usually be apparent. The message

will be even in length. The majority of repeated segments will be even in length,

although when an adjacent row or column coordinate is the same, a repeat may

appear odd in length. The distance between repeats, counted from the first letter of

one to the first letter of the next, will be even in length.

b. Second, unless the identical letters or numbers are used for row and column coor-

dinates, there will be limitation by position. One set will appear in the row coor-

dinate position, and the other set will appear in the column coordinate position.

Even in the case where all coordinates are different and either the row or column

coordinate character may be placed first, each pair will be limited to one from one

set and one from the other. If you do not recognize it right away, charting contacts

will make it obvious.

c. For systems with letters as coordinates, not more than half the alphabet will be used

as coordinates. This severe limitation in letters used is the most obvious charac-

teristic, since only very short unilateral messages are ever that limited. A phi index

of coincidence will reflect that limitation, always appearing much higher than

expected for a unilateral system.

d. Dinomic systems, since they are limited to the 10 digits anyway, are not quite as

obvious. Simple systems should still show positional limitation, however.

Sample Solution of a Dinomic System

5-6.

The next problem shows the steps in solution of a sample dinomic system. These steps

apply equally to biliteral systems.

5-9

a. The most obvious thing about this cryptogram is that every pair of numbers begins

with 2, 4, 6, or 8. The final pair begins with 0, but since it appears nowhere else, it is

probably a filler. This suggests that we are dealing with a matrix with four rows.

b. Scanning the second digit of every pair, we see that there is some limitation in the

column position, also. All digits are used except 8. The matrix appears to have nine

columns, although it is possible that a column for 8 exists, but no values from it

were used. Four by nine is a reasonable size for a matrix.

c. Next, we check for repeats and underline them. We also prepare a dinomic

frequency count by setting up a 4 by 9 matrix and checking off each dinome that

appears.

d. The two longer repeats both include patterns of repeated values. Word patterns can

be constructed on repeated dinomes just as they were for repeated single letters.

The word patterns for the two longer repeats are shown below.

5-10

e. The word pattern lists in Appendix D show only one possibility for each pattern as

shown. The two are consistent with each other. Using these recoveries, we can set up

a matrix and place the values in it and the cryptogram.

f. The plaintext words ENEMY and AIRSTRIKE are now obvious. Placing the M

from ENEMY shows COMMANDING at the end of the message. Most of the

remaining plaintext letters are easily recovered.

5-11

g. The letters in the second row precede all the letters in the third row alphabetically.

This suggests an alphabetic structure, although the columns are clearly not in the

correct order. The first row probably contains a keyword. If we rearrange the

columns so the letters in the second and third rows fall in alphabetical order, we see

the next structure.

h. The plaintext letters area keyword mixed sequence based on INCOME TAX. After

placing the remaining letters, there are still 10 blank cells in the matrix. Seven of

them are used in the cryptogram, and they cluster together in segments of three or

four dinomes. They show the typical pattern of numbers. In particular, the four

5-12

plaintext values of groups 50 and 51 of the message indicate time, and 66 is

probably a 0. More likely than not, the remaining numbers fill the bottom row of

the matrix in numerical order, but these recoveries cannot be confirmed without

more information. If hill numbers could be compared to known numbers from an

enemy map sheet, we could accept the values with more confidence. At this point,

we are reasonably confident of the letter arrangement and the number 0, but the

remaining numbers are only a possibility. However, if this were a current real life

situation and the enemy referred to by the text is our own forces, we would certainly

consider reporting the likelihood of air strikes on our artillery positions.

Analysis of Monome-Dinome Systems

5-7.

The characteristics of biliteral and dinomic systems that stand out most are the

divisibility by two and the positional limitation that makes it easy to determine matrix

coordinates. By changing the length of the plaintext unit from character to character,

monome-dinome systems avoid both of these characteristics. In their place, however,

the frequency of the numbers (or occasionally, letters) used as row coordinates tends to

be higher than the other coordinates. Choosing the highest frequency numbers as row

coordinates gives a starting point to reconstruct a monome-dinome system. Consider

the next example.

a. Repeats are underlined and the number frequencies are shown in the example. A

dinomic system can be ruled out, because the repeats are an odd interval apart. The

distance between the repeats is 153 characters, counting from the first character of

one to the first character of the next. A three-for-one substitution is possible from

the position of the repeats, but no patterns or positional limitations appear when

divided into threes. The very high frequency of the numbers 0 and 9 in relation to

5-13

the other numbers suggests that the system is monome-dinome. The most likely

row coordinates are 0 and 9. Other row coordinates are possible, but at this point it

is best to start with the most likely candidates only.

b. Begin by breaking the message into monomes and dinomes using only the 0 and 9 as

row coordinates. Mark off the divisions in pencil, keeping in mind that some

changes may be required later. Start with the first character of the message and

work through in order to the end, marking off the monomes and dinomes. Whenever

the first character after a division is a 0 or 9, include it with the next character. If it

is any other character, leave it as a monome.

c. With the divisions in place, we can try a word pattern on the long repeat.

d. We next set up a monome-dinome matrix with row coordinates 0 and 9 and include

the recovered letters. Shown below is the partially recovered matrix and the crypto-

gram with all letters from RECONNAISSANCE placed in the plaintext and the

matrix.

5-14

e. These recoveries suggest additional plaintext, particularly the message beginning

AERIAL RECONNAISSANCE REPORTS ENEMY. Placing these new values

leads to additional recoveries.

5-15

f. Several things remain to be done to complete the solution. The columns can be

rearranged to recover a keyword in the top row and alphabetical progression in the

next two rows. Additionally, there are two unrecovered segments of text. Both of

them include a number of 5s, and the preceding text in each case suggests numbers.

The solution is that there is another row in the matrix with the 5 as its coordinate. It

was not used enough to select from frequency alone, but once enough text was

recovered, the structure can be seen. The added row includes the numbers. The

complete solution appears in the next example, with the recovery of specific num-

bers only tentative.

5-16

Application of Vowel-Consonant Relationships

5-8.

to Multiliterals

Vowel-consonant relationship solutions can be applied to multiliterals, too. As long as

you can determine the coordinates of the matrix, you can set up a dummy matrix with

any sequence of characters inside as a pseudoplain component. You then reduce the

cryptogram to unilateral terms by deciphering with the dummy matrix. Next, solve the

resulting unilateral cryptogram using any of the techniques learned with unilateral

systems, including the use of trilateral frequency counts and the vowel and consonant

lines.

5-17

Solution of Trilateral and Trinomic Systems

5-9.

Trilateral and trinomic systems are solved in exactly the same way as biliterals and

dinomics. The systems are identified by the tendency of messages to break into groups

of three instead of groups of two. With simple triliterals and trinomics, positional

limitation is even more evident than it is for biliterals and dinomics. Look for a limited

set of pairs of characters as either the first pair of characters or the last pair of charac-

ters in every three, Once these are found, set up your coordinates and solve as before.

Section III

Analysis of Variant Multilateral Systems

Identification of Variant Systems

5-10.

As with any coordinate system, analysis of variant multilateral systems begins with

determination of the coordinates. If the product of the row and column coordinates is

50 or more, the system is almost certainly a variant system of some kind.

Analysis of External Variant Systems -

5-10.

Frequency Matching

External variant systems are generally easier to solve than internal variant systems.

Frequency counts can usually be used to determine which coordinates combine with

each other on the same row or column, whenever the text is long enough to give a good

representative sample, as shown in the next problem.

5-18

a. The cryptogram used 10 different letters as row coordinates and 10 different letters

as column coordinates. Using these coordinates, a digraphic frequency count has

been completed as shown. For example, the letter I is paired with itself five times,

so the number 5 appears in the matrix at the point where the row and column of I

intersect.

b. Examining the frequency count, we can see that there are good frequency pattern

matches between certain rows and certain columns. For example, the I row and the

R row are nearly identical. Similarly, the A column and the I column are nearly

identical. Carrying this process further, we can match the row pairs, AU, DP, IR,

MN, and OS. The column pairs are AI, CN, GS, MO, and RU. At this point, we

have no idea in what order the coordinate pairs belong or which letter in each pair

comes first or if it even matters which letter comes first. We have enough informa-

tion, however, to reduce the cryptogram to unilateral terms.

c. To reduce the cryptogram to unilateral terms, we set up a matrix with the combined

coordinates and write any sequence of letters within it, for example, A through Y.

5-19

d. We see that repeats appear in the pseudotext that results from our trial decipher-

ment. The repeats that were suppressed by the variants are now visible with the

variants combined. The recovery of the plaintext is like any of the previous

problems. When we recover the plaintext and enter the recovered values in the

matrix in place of the trial sequence, we reach the solution shown below.

5-20

e. With the plaintext values filled into the matrix, we can see in what order the rows

and columns belong. Starting with the last row of the internals, we rearrange the

columns of the matrix in alphabetic order.

5-21

The first row of the internals should follow alphabetically after the third

rowâ€”scdef, gikln.

f. All that remains is to fill in the missing letters H, J, and Q in the plaintext sequence,

and to try to recognize how the coordinates were constructed. As mentioned earlier,

it is common practice to couple I with J or U with V when using a 5 by 5 matrix.

Since J did not appear in the plaintext, we may assume it occupies an alphabetical

position within the I block. The Q clearly belongs between the P and T, leaving the

H in the top row. The plaintext keyword is BRAHMS (the classical composer). With

that as a clue, the letters in the coordinates are shifted to their correct positions,

revealing the keywords PIANO, DRUMS, MUSIC, and ORGAN.

Analysis of Variants - Isologs

5-12.

Two or more encrypted messages with different encrypted text, but the same underly-

ing plaintext are called isologs. When isologs are encountered, your job is much easier.

Isologs are particularly useful in solving variant multilateral systems, either external or

internal.

a. Isologs can be recognized by one or more of these characteristicsâ€”

Identical message lengths.

Similar characteristics in the text, such as repeated segments or characters

occurring in the same position in each message.

5-22

External indications, such as identical times of file or identical message numbers

included in the header for each message. Normally, no two different messages

from the same sender receive the same file time or message number. When you

see the same time of file on the same date originating from the same unit, the

messages are likely to be isologs.

b. Two messages that showed the same time of file in the message header appear in

Figure 5-3.

c. Each message shows positional limitations. Message 1 has the letters

ADFGLNQRTX in the row coordinate position and BCHKLMPSVZ in the column

coordinate position. Message 2 has AEFGKLOQVZ in the row coordinate position

and BDHMPRSTWY in the column coordinate position. The two messages are not

encrypted in the same system, but they appear to be isologs.

d. The initial step in solving these isologs is to see what values equate to each other in

the two messages. Pick one of the most frequent digraphs in either message as a

starting point. For example, FH occurs four times in the first message. A frequency

count, while not strictly necessary, may be helpful in spotting the most common

values. The digraphs that occur in the same positions in message 2 as FH in

message 1 are OS, GW, GS, and another OS.

e. The next step is to find each of the digraphs in message 2 that equated to FH from

message 1. The letters OS, GW, and GS in message 2 and the digraphs in the same

position in message 1 are underlined in Figure 5-3.

5-23

f. We now see that RH, RP, FP, and FH in message 1 equate to GS, GW, and OS in

message 2. A check of the new values in message 1 adds the additional digraph OW

in message 2, completing the equations for that set. It appears that R and F are

variant row coordinates and P and H are variant column coordinates in message 1.

Similarly, the message 2 variants are G and O on the rows and W and S on the

columns.

g. Continue the process by picking additional repeated values. Complete the equa-

tions for each, working back and forth between the two messages, just as we did for

the initial digraph FH. Continue until all coordinates have been combined, or you

run out of digraphs to compare. You can set up a plot to keep track of the equations

as shown in the next example.

h. Other combinations could have been selected than the ones shown, but these are

sufficient to show all the variants in both matrices. From this point, either message

can be reduced to unilateral terms and solved. Then the recovered plaintext can be

applied to the other message to complete the recovery of the second matrix. Note

that if the same matrix was used in both messages, the similarity should be quickly

recognized and the solution accomplished more easily. The next paragraph shows

the simpler technique when the same matrix is used.

Solution Using Isologous Segments

5-13.

Segments of ciphertext which have the same underlying plaintext are known as

isologous segments. A technique similar to the one used in isolog solution can be used

any time repeated plaintext can be identified. This is likely to occur with repeated

beginnings and endings to messages or with long repeated words and phrases.

a. Recognizing repeated plaintext in variant systems requires painstaking inspection

of the ciphertext. Computer indexes of repeated plaintext, which show repeated

text on consecutive lines along with the preceding and following text makes repeats

5-24

easier to recognize. In any long plaintext repeat, some of the ciphertext digraphs or

dinomes are likely to repeat. Other ciphertext digraphs or dinomes are likely to

show common row or column coordinates. Pairs with neither row nor column coor-

dinates in common will generally be in the minority. Therefore, although a lot of

trial and error may be involved, the longer repeated plaintext segments can often be

identified. Consider the two message beginnings shown below.

b. The similarities of the text make it quite clear that the underlying plaintext is the

same in both cases, and the same matrix is used for both. Proceeding on the

assumption that the plaintext and matrix are the same, it is easy to match the

remaining values to determine the variants. For example, from the first dinome in

each message, 3 and 4 are column variants. From the second dinome in each

message, 8 and 9 are column variants. All the variants can be combined from this

short example, and the remainder of the solution is routine.

Analysis of Internal Variant Systems

5-14.

Internal variant systems are generally more difficult to solve than external variant

systems. With no coordinates to combine, frequency counts do not provide immediate

clues to variants. Similarly, isologous segments are harder to recognize. Some charac-

ters are likely to repeat in isologous segments with internal variant systems, but the

partial repeats caused by common row or column coordinates are much less likely to

occur. Still, given enough messages from a single system to produce repeats; given

operator carelessness in encryption; or given stereotyped traffic, these systems can

readily be solved, too. Once a plaintext entry is found, the remainder of a solution is

not difficult. When you find isologs or isologous segments, you can equate ciphertext

values just as was demonstrated in the internal variant examples. The only difference

is that you do not combine coordinates through this process, but instead find all cells

in the matrix that have the same plaintext value.

5-25

Analysis of Syllabary Squares

5-15.

Syllabary squares are closely related to small code charts, and the solution of both

types of systems is similar. The analysis of syllabary squares produces some distinct

differences.

a. Isologs or isologous segments are not necessarily the same length in each case. The

encipherment examples below are repeated from paragraph 5-3e.

b. Isologous segments can often still be recognized by the plaintext values which have

no variation. In the example, there is only one way to encipher the letters M and S.

When REINFORCEMENTS is enciphered, the ciphertext equivalents of M and S

will always be the same. Other values are likely to begin with the same row coor-

dinate, since syllables beginning with the same letter are likely to be on the same

row, such as the R and the RE. Still others will have a possible variation, but the

variation will not be used. The repeated CE syllable in both segments is an example

of this. As a result of all these considerations, isologous segments are often

recognizable and provide a point of entry to the system.

c. Solution of syllabary spelling will be further explained in Part Six, Analysis of Code

Systems.

5-26