SOLUTION OF PERIODIC

POLYALPHABETIC SYSTEMS

Section I

Systems Using Standard Cipher Alphabets

Approaches to Solution

9-1.

When standard alphabets are used with monoalphabetic systems, three approaches

are possible. The simplest occurs when text can be immediately identified. Identifica-

tion of only two or three letters in a standard unilateral alphabet is sufficient to

reconstruct and confirm the entire alphabet. The other two methods, where text is not

readily identifiable, are to match frequency patterns to the normal A through Z pat-

tern and to generate all possible solutions. All three of these methods also apply to

standard alphabet periodic polyalphabetics.

by Probable Word Method

9-2. Solution

When the alphabets in a periodic system are known or suspected to be standard, the

identification of one plaintext word is usually enough to recover the whole system. The

period must be identified first, as explained in the previous chapter, either by analysis

of repeat intervals or by the phi test. Then when a word is recognized from repeats or

stereotypes, the alphabets can be written and tried throughout the cryptogram. If they

produce good plaintext throughout, the problem is solved.

9-1

Factor analysis does not show us a clearcut period length, but if we select the four

letter repeat as the most likely causal repeat, 7 appears to be the correct period. If we

also try STOP as the four letter repeat, it gives us the following text and alphabets.

From the partial plaintext that this produces, STOP is clearly correct. Such words as

RECONNAISSANCE, HEAVY, and REINFORCED are apparent, any one of which

will complete the solution. For another type of probable word approach, applicable to

periodics or aperiodic, see paragraph 10-3c on crib dragging.

9-2

Solution by Frequency Matching

9-3.

With monoalphabetic systems using standard alphabets, the solution was very easy

whenever a message was long enough to give a recognizable pattern. The characteristic

pattern of highs and lows of a standard sequence cannot be easily concealed. The same

technique applies to polyalphabetic systems, although messages necessarily must be

longer to produce a recognizable pattern for each separate alphabet.

a. Factor analysisshows common factors of three and six for all repeat intervals.

Based, on this, a frequency count for six alphabets is produced, as listed in

Figure 9-1. If the period were actually three, the first and fourth, the second and

fifth, and the third and sixth frequency counts would be similar. This is clearly not

the case, so the period is confirmed as six.

9-3

b. The easiest patterns to match are generally those with the highest ICs. The first,

second, and fifth alphabets have the highest ICs, and all can be matched fairly

easily. In the first, plaintext A equals ciphertext B. In the second, plaintext A

equals ciphertext A, and in the fifth, plaintext A equals ciphertext O. Other

alphabets can be matched, too, but using these as an example, the partially

reconstructed text is shown below.

c. The letter combinations produced by the three recovered alphabets are consistent

with good plaintext. Expanded plaintext can be recognized in many places. The

first word is ENEMY for example. Filling in added plaintext is a surer and quicker

means of completing the solution at this point than trying to match more alphabets.

Here is the complete solution.

9-4

Solution by the Generatrix Method

9-4.

With standard alphabets or any known alphabets, the method of completing the plain

component can be used. This method, when applied to periodic systems, is commonly

called the generatrix method. The advantage of this method over frequency matching

is that it will work even with fairly short cryptograms. Just as with a monoalphabetic

system (see paragraph 4-11), the first step is a trial decryption at any alphabet align-

ment, followed by listing the plain component sequence vertically underneath each

letter of the trial decryption. Whenever the plain and cipher sequences are identical

and in the same direction, no trial decryption is necessary. The key difference with

periodic systems is that the process must be applied to the letters of each alphabet

separately. Plaintext will not be immediately obvious when you look at the generated

lines of letters from only a single alphabet, so selection must be initially based on letter

frequencies and probabilities rather than recognizable text. The process is illustrated

with the following cryptogram enciphered with direct standard alphabets.

a. The cryptogram has a period of five, which can be confirmed either through

periodic-phi tests or factor analysis of all the repeats, including two letter repeats,

which are not underlined.

b. The most obvious step to try is to substitute STOP for the four letter repeat. It does

not produce plaintext elsewhere, however. More powerful methods of solution are

required.

c. The cryptogram can be readily solved by the generatrix method. The first step is to

separate the letters produced by each alphabet. The letters from each of the five

alphabets are listed separately below. Notice that if you read all the first letters, it

produces the first group of the cryptogram. The second letters produce the second

group and so on.

9-5

d. No trial decryption is required, because the same sequence is expected for both the

plain and cipher components. Therefore, the next step is to complete the plain com-

ponent sequence for each letter grouping. This is illustrated in Figure 9-2.

9-6

e. To aid in selection of the most likely generated letter sequences, numeric

probability data has been added to each line of the listing. The numbers listed

below each letter are assigned on the basis of logarithmic weights of the letter

probabilities. To the right of each group of logarithmic weights is the sum of the

weights for that group. Using this kind of weighting lets us determine the relative

probabilities of each line by adding the weights for each letter. The weights in

Figure 9-2 have been added according to the log weights shown in Table 9-1.

f. The listing in Figure 9-2 was computer generated. When this work must be done

manually, it is easier to generate the sequences without the probability data. Then

scan the generated rows for each alphabet to visually select those with the most high

frequency letters. Finally, if necessary, the probability data can be added only for

the selected rows.

g. Only rarely will the correct rows consist entirely of those with the highest totals.

Normally, you will have to try different combinations of the high probability rows

until you find the correct match. The best place to start is with those rows that

stand out the most from others in the same alphabet groups. In the illustrated

problem shown below, alphabets four and five provide the most likely starting

point. In each case, the sum of the log weights for one row are well above any others.

These are listed below, superimposed above each other with room for the other three

alphabets to be added.

1:

2:

3:

4: MRELTNEARHTT 97

5: YENESTIVETN 88

h. As the rows are superimposed, the plaintext will appear vertically. The next step is

to see which high probability rows from other alphabets will fit well with the

starting pair. Trying both of the two highest probability rows for alphabet three

produces the next two possibilities.

9-7

i. Reading the plaintext vertically, the grouping on the right is better than the one on

the left. The DTS sequence in the left grouping is unlikely, and all the letter com-

binations on the right are acceptable. Furthermore, the EMY combination at the

beginning of the right grouping suggests ENEMY. The letter sequences for the first

two alphabets which begin with E and N respectively are both high probability

sequences. The complete solution is shown below.

â€œENEMY HAS RETAKEN HILL EIGHT SEVEN THREE IN HEAVY

FIREFIGHT LAST NIGHTâ€

Section II

Systems Using Mixed Alphabets

With Known Sequences

Approaches to Solution

9-5.

When mixed sequences are used in periodic systems, a variety of different techniques

can be used to solve them. When the plain and cipher sequences are known, the same

techniques used with standard alphabets can be used, adapted to the known

sequences. When one or both of the sequences are unknown, new techniques must be

used. Each situation is a little different. The major paragraphs of this section deal with

each situation: both sequences are known, the ciphertext sequence is known, or the

plaintext sequence is known. Techniques for solving periodics when neither sequence is

known are covered in the next section.

9-8

Solving Periodics With Known Mixed

9-6.

Sequences

Exactly the same techniques that were used with standard alphabets can be used with

any known mixed sequences.

a. Successful assumption of plaintext allows you to directly reconstruct the cipher

alphabets, as before.

b. The generatrix method works, making sure that a trial decryption is first performed

with the sequences set at any alignment. All possible letter combinations are then

generated by completing the plain component sequence, as before. The key points

to remember are to perform the trial decryption and to use the plain component as

the generatrix sequence, not a standard sequence.

c. Frequency matching also works, but there are some differences in its application.

Frequency counts must be arranged in the cipher sequence order, not in standard

order. The pattern that the frequency counts are matched to must be adjusted to

the order of the known plain component. Rearrange the patterns of peaks and

troughs to fit the plain component. For example, shown below is the pattern for a

standard plain sequence and the pattern that results if a keyword mixed sequence

based on POLYALPHABETIC is used as the plain component.

The new pattern resulting from the mixed plaintext sequence is just as easy to

match frequency counts to as the more familiar standard pattern. If it should prove

difficult to match by eye alone, there is also a statistical test, called the chi test,

which can be used to aid the matching process. Paragraph 9-7 demonstrates the use

of the chi test.

9-7. Solving Periodics With Known Cipher

Sequences

The technique of frequency matching can be used any time the cipher sequence is

known, whether or not the plain sequence is also known. When the plain sequence is

known, the frequency patterns of the cipher sequences are best matched to the ex-

pected plain pattern as explained in paragraph 9-6. When the plain sequence is un-

known, the frequency patterns of the cipher sequences can be matched to each other.

In either case, the key is that the known cipher sequence allows the frequency count to

be arranged in the order of the original cipher sequence. The following problem

9-9

demonstrates frequency matching with a known cipher component sequence. The

cipher component sequence in the problem in Figure 9-3 is a keyword mixed sequence

.

based on NORWAY.

a. Examination of the frequency patterns in Figure 9-3 shows that they do not match

the usual standard sequence-pattern. This means that the plain component

sequence was not a standard sequence.

b. If the cipher sequences can be correctly matched against each other, the crypto-

gram can then be reduced to monoalphabetic terms and solved easily.

c. Figure 9-4 is a portion of a computer listing that matches the frequency count of the

cipher letters of the first alphabet with the frequency count of second alphabet

letters at every possible alignment. The alignments are evaluated by the chi test. In

the chi test, each pair of frequencies for an alignment is multiplied. The products of

all the pairs are totaled to produce the chi value for that alignment. Figure 9-5

shows the computation carried out for the first alignment. The chi test is also called

the cross-product test.

9-10

9-11

d. Figure 9-6 shows the highest chi values for each match of the first alphabet with the

other four alphabets. For all matches except the fourth alphabet, the chi values

were clearly the highest. Two matches are shown for the fourth alphabet, because

the difference between the two values is not significant. Either match could be the

correct one.

9-12

e. To resolve which of the two matches with the fourth alphabet is correct, the highest

chi values for matches between the second and fourth and the third and fourth

alphabets have also been determined. These are shown in Figure 9-7.

f. The matches of alphabet four with alphabets two and three clarify which of the

matches with the first alphabet was correct. This becomes apparent when we set up

the other four alphabets.

g. The match of N of the first alphabet with P of the fourth alphabetic correct. The

second alphabet and third alphabet matches confirm this.

9-13

h. The next step in the solution is to reduce the cryptogram to monoalphabetic terms

using the matches just determined. An A through Z sequence is arbitrarily used for

the plain component, and the message is decrypted just as if it were the original.

i. Reduced to monoalphabetic terms, many more repeats in the text that were sup-

pressed by the multiple alphabets now appear. The solution is completed the same

as any other monoalphabetic system.

Solving Periodics With Known Plaintext

9-8.

Sequences by Direct Symmetry

When the plaintext sequence is known, but not the ciphertext sequence, a solution

technique known as direct symmetry is possible. Direct symmetry depends on the

probable word method for the initial entry into the cryptogram. It makes use of the

fact that the columns can be reconstructed in their original order as recoveries are

made. Consider the next example, which uses a standard plaintext sequence.

9-14

a. The period is five. The 14 letter repeat is probably RECONNAISSANCE.

b. With recovered letters filled in, we can see that the beginning phrase is the

stereotype, RECONNAISSANCE PATROL REPORTS.

9-15

c. With a known plain component, the columns are in their original order. This means

that the partially reconstructed cipher sequences are also in the right order. Each

cipher sequence is the same sequence, and whatever one row reveals about the spac-

ing of letters can be transferred to other rows as well. For example, in the second

row, X follows immediately after W. X can then be placed after W in row three.

Similarly, all common letters can be placed by carefully counting the intervals and

placing the same letters at the same intervals in each row. Here is what the matrix

looks like after all such values are placed.

d. Filling all the new values into the text reveals many more possibilities. Completion

of the solution is routine from this point.

e. The direct symmetry technique can also be used as an alternate method when the

cipher sequence is the known sequence. The matrix can be inverted, placing the

cipher sequence on the top of the matrix and the plaintext equivalents inside in

separate rows for each alphabet. Each row will be the plaintext sequence in the

correct order. Horizontal intervals recovered in one row can then be duplicated in

each sequence just as was demonstrated above for cipher sequence recovery. Unlike

the technique of frequency matching, it depends on successful plaintext assump-

tions, however. It is not as powerful a method of solution, but if plaintext can be

readily identified, it may be the quickest way to solve a cryptogram.

9-16

Section III

Solving Periodics With Unknown Sequences

Solving Periodics by Indirect Symmetry

9-9.

When neither the plaintext nor the ciphertext sequence is known, the matrix cannot be

initially recovered with sequences in the correct order. Frequency matching cannot be

used, either. However, some of the interval relationships are preserved even when the

columns are not placed in the correct order, and these interval relationships can be

exploited to aid in matrix recovery.

a. To illustrate how interval relationships are preserved, consider the following two

matrices. The first is the matrix in its original form. The second is the same matrix,

rearranged with the plain component in A through Z order. This is the form in

which you will normally recover a matrix with unknown sequences until enough is

known to rearrange the columns in the correct order.

b. The key principle to understand when working with ananalystâ€™s matrix, like the

second one above, is that every pair of columns and every pair of rows represents an

interval in the original matrix. To illustrate this, look at the plaintext A column

and the plaintext G column in the bottom matrix. The letters D and R appear in

the first cipher sequence. If you count the distance between the D and R in the

original (top) matrix, you see that the interval is nine. Similarly, the interval for the

other pairs in the two columns, R and X, U and P, and M and S, are also nine. For

any two columns that you compare, the horizontal interval between the letters in

each alphabet will be the same. The interval will not always be nine, of course. It

depends on which two columns you are comparing. The point is that between any

pairs in the same row in the same two columns, the interval will be the same.

c. Next compare the letters in the first cipher sequence and the second in the bottom

matrix. In the first column, the letters D and R appear, which we already noted are

nine letters apart horizontally in the original matrix. The letters R and X appear in

9-17

another column in the first and second sequences, as do U and P, and M and S. The

first and second cipher sequences are an interval of nine apart. Whichever pair of

letters you look at in the first and second cipher sequences, they are nine apart in

the original cipher sequence. Each pair of cipher sequences represents a different

interval. For example, the interval between the first and third cipher sequence is

eleven. The interval between the first and fourth is seven. The interval between the

second and third is two, and so on.

d. There are a number of ways in which we can use an understanding of these interval

relationships to help solve a polyalphabetic cryptogram. The use of interval

relationships where sequences are unknown and columns are out of order is called

indirect symmetry. This contrasts with the earlier situation with known sequences

and columns in the correct order, where we used direct symmetry to aid in the

solution.

e. To put indirect symmetry to use, consider the following example. Initial recoveries

in a polyalphabetic system have produced the following information.

f. In comparing the plaintext A and E columns, we see that the letters R and T and the

letters M and F are the same interval apart. We do not know what the interval is,

but we know it is the same in each case.

g. The same interval appears when we compare the first and third cipher sequences,

where R and T appear in the first column. Since we know the interval will be the

same for any pair of letters between the first and third sequences, and we know M

and F have the same interval as R and T, we can add the letter F in the plaintext I

column in the third sequence under the letter M.

h. Any time we can establish an interval relationship for two pairs in a rectangular

pattern as above, and can find three of the four letters, also in a rectangular pattern

elsewhere, we can add the fourth letter to complete the pattern. The pairs must be

read in the same direction in each case. Notice that we cannot add F in the plain-

text G column in the first sequence. The interval from the first to the third sequence

is not the same as the interval from the third to the first.

i. Matching pairs are usually found by reading horizontally in one case, and vertically

with one letter in common in the second case, as in the above example. Matching

relationships may be found anywhere in matrix, however, and are not restricted to

9-18

cases with one letter in common. You can find most such matching pairs by examin-

ing every column in which you have recovered at least three letters. For each letter

in the column, look for a match with letters on the same row that are the same as one

of the other letters in the column. When you find such letters, check for every possi-

ble complete rectangular relationship, and see if you can find the same relationship

with one letter missing elsewhere. Often the addition of one or two letters is all you

need to recognize more plaintext in the cryptogram and complete a solution.

j. If you have reason to believe that the plaintext sequence is the same as the cipher

sequences, you can use the plaintext sequence in establishing interval relationships,

too. All the techniques that apply to the ciphertext sequences apply to the plaintext

sequence as well, when it is the same sequence.

Extended Application of Indirect Symmetry

9-10.

Indirect symmetry can be used in other ways, too. For example, when enough letters

have been recovered, you can list all the pairs of letters between each pair of sequences,

and develop partial decimated chains of letters for each, as was explained in paragraph

4-8 with monoalphabetic substitution. These partial chains from different alphabet

combinations can then be combined together geometrically to recover the original

sequence. This technique is illustrated in the following indirect symmetry problem.

a. Through recognition of the stereotyped beginnings and the use of many numbers,

the text shown has been recovered, and the recovered values filled into the matrix.

9-19

More values can be filled into the text, but we will first concentrate on the applica-

tion of indirect symmetry.

b. To recover additional values through indirect symmetry, examine each column

with more than two recovered letters in it. Beginning with the fifth column, take

each letter in turn, and scan the same row as the selected letter for letters that are

the same as those in the column. The first letter, Z, has no letters in common in its

row with the letters M, B, P, and N.

c. For the second letter, M, the common letter Z does appear in its row. Having found

a common letter, examine each rectangular relationship that exists between the two

columns. We first see that Z and W have the same interval as M and Z. Links with

this common letter will not add any more values, however.

d. The next rectangular relationship shows that P and L have the same interval as M

and Z. Reading M and Z vertically, we look for P or L on the same rows as the M

and Z to complete the relationship. We find neither P in the second row nor L in the

first row. If either occurred, we could fill in the other. The letters can be written in a

column off to the side for future use.

e. Having observed all relationships from the column with the common letter Z, we

look for another column with a common letter on the M row. B and P do not occur

except in our added column. The letter N does occur in the second row, however.

Examining relationships in the N column, we see that Z and J have the same inter-

val as M and N reading horizontally. With that established, we read M and N ver-

tically and look for Z in the second row or J in the last row. This time we find Z in

the second row. We can add J in the last row in the same column with Z to complete

the rectangular relationship.

f. Continuing this process, all the letters shown in bold print can be added to the

matrix without making any new plaintext recoveries.

g. It would be easy at this point to return to plaintext recovery to complete the solu-

tion, but another technique can be used to recover the original cipher sequences and

rebuild the matrix. This technique involves listing all links that result by matching

each cipher sequence with every other cipher sequence. Sequence 1 is matched with

9-20

sequences 2, 3, 4, and 5, in turn. Then sequence 2 is matched with 3, 4, and 5;

sequence 3 is matched with 4 and 5; and sequence 4 is matched with 5. If the plain-

text sequence were the same as the ciphertext sequence, it would only have been

necessary to match the plaintext with each cipher sequence to get all combinations.

When all links have been plotted and combined into partial chains wherever possi-

ble, the results are shown below.

h. Each set of partial chains represents a decimation of the original sequence.

Sometimes, you will be fortunate at this point to find that one of the partial chains

directly represents the original sequence (decimation one). When this happens, the

original sequence is the obvious starting point. It does not occur in this example, so

the best technique is usually to select a set with one of the longer chains as a

starting point and relate all other sequence combinations to it. Notice that the

chains produced by sequences 1-2 and by sequences 2-3 are obviously produced by

the same interval, since many of the partial chains are identical. They make a good

starting point for this problem. Begin by listing each chain fragment on paper,

horizontally. Write the separate chains in different rows so they will not run into

each other.

i. The next step is to relate other chains to the existing plot. By examining the inter-

vals or patterns that letters from other chains have in relation to the starting chains,

they can be added by following the same rule. For example, the 1-3 combination can

9-21

be added by observing that it will fit the starting chains by skipping every other

letter. This will also enable linking the fifth fragment, AS, with the fourth. After

adding all the 1-3 chains, the plot looks like this example.

j. Next, search for another combination that can be added to the plot. The 3-4 com-

bination links by counting backwards every fifth letter, as shown by the V and C of

the NZIVC chain. This ties all the chain fragments together into one longer chain.

When all combinations are added, each by their own rule, it results in almost com-

plete recovery.

k. This technique is known as linear chaining. Sometimes you will be unable to com-

bine the fragments into one long chain. When all intervals are even, you will always

end with two separate 13-letter chains, which may be combined by trial and error or

by figuring out the structure of the original matrix. A second technique, called

geometric chaining, which could have been applied here also, is explained in

paragraph 9-11.

l. Continuing, the chain above must be a decimation of the original sequence. Since V,

W, and X are spaced consistently nine apart, trying a decimation of 9 produces the

next sequence.

m. With G missing from alphabetical progression, the sequence is keyword mixed,

based on GAMES. We can now return to the polyalphabetic matrix and rearrange

the columns using the GAMES sequence on each cipher row.

9-22

n. The unused letters can be determined by returning to the plaintext and deciphering

the rest of the message. The plaintext sequence turns out to be a simple transposi-

tion mixed sequence based on OLYMPIC. The repeating key is KOREA.

o. The approach shown to solving this problem is not necessarily the way in which you

would solve it in actual practice. It would probably be more effective to return to

the plaintext earlier than was done in this example. This approach was selected to

show the variety of indirect symmetry techniques that can be used, not necessarily

because it would yield the quickest solution.

9-11. Solution of Isologs

Whenever isologs are encountered between periodic messages with different period

lengths, it is possible to recover the original cipher sequences without any initial plain-

text recovery. The cryptograms can then be reduced to monoalphabetic terms and

quickly solved. Two different techniques may be used, depending on whether the same

alphabets or different alphabets are used in the isologs.

a. When isologous cryptograms use the same alphabets with different repeating keys,

the cipher sequences can be recovered by the indirect symmetry process. Take the

following two messages, for example.

9-23

(1) To solve the isologs, the two messages are first superimposed with the alphabets

numbered for each.

(2) With periods of 3 and 4, there are 12 different ways in which the alphabets of the

first are matched to the alphabets of the second. These begin with the first

alphabet of message 1 matched with the first alphabet of message 2 and con-

tinue through alphabet 3 matched with alphabet 4. After these 12 matches, the

cycle of matches starts over again. For other periods, the number of different

alphabet matches is the least common multiple of the two period lengths. The

least common multiple of 6 and 4 is 12. The least common multiple of 6 and 9 is

18. For periods of 8 and 9, 72 different alphabet matches are required.

(3) Analysis continues by plotting the links for each alphabet pair. For example,

the first link is A1=D1, the second link is O2=C2, and the third link is P3=F3.

The next example shows all links plotted and combined into partial chains.

9-24

(4) The 1-3 plot shows that the same alphabets were used in both these positions.

(5) The partial chains can be combined into one long chain by a process of

geometric chaining. Geometric chaining will often produce results when linear

chaining is not effective. Geometric chaining is plotted horizontally and ver-

tically, instead of in one straight line. Relationships between alphabet matches

can be discovered more readily with this method.

(6) Geometric chaining begins, as with linear chaining, by selecting one alphabet

match to plot horizontally. We can select the 1-1 match for its 5-letter chain as a

starting point. Next, select a second alphabet match to intersect it plotted ver-

tically. For our example, we will use the 2-2 match, producing the following in-

itial plot.

(7) To this initial plot, we add as many other fragments from the 1-1 and 2-2

matches as we can at this time. We can also set up plots separated from these

for each one that cannot be linked to it.

9-25

(8) The next step is to find another alphabet match that can easily be added to the

plot. For example, the 1-2 match proceeds in the diagram along a lower left to

upper right diagonal, as shown by the NSC and XJ fragments. All the 1-2 frag-

ments can be added by the same diagonal rule. This ties in the separate plots

from above, also.

(9) Each additional alphabet combination can be added to the plot now. In many

cases, you may see different possibilities for rules. For example, the 3-4 match

can be seen to proceed by an up 3, left 1 rule, as shown by the TO link. A simpler

equivalent is to plot by the upper left to lower right diagonal, as shown by the

PK link. The simplest way to describe the 3-3 match is up 1, right 2, as shown

by the TK or BY links. This is similar to a knightâ€™s move in chess. When all

matches are plotted, they produce this diagram.

(10) The rows can easily be extended into one 26-letter chain at this point, but if

alphabetic progression can be spotted by any other rule, it can be used instead.

For example, starting with the V in the upper left part of the diagram, VWXY

appears by a descending knightâ€™s move. Continuing from the Y that repeats

near the left side, the sequence can be extended further. The complete

sequence appears below.

9-26

(11) Using the new recovered sequence and the relationships between the alphabets

of messages 1 and 2, the matrices for both messages can be set up. Using the

first cipher sequence for message 1, all the cipher sequences for message 2 can

be lined up with it using the links already plotted. Here is how the message 2

alphabets line up with alphabet one. The first 1-1, 1-2, 1-3, and 1-4 links from

the isologs are shown in bold print to demonstrate how they were lined up.

(12) Similarly, the alphabets in the first matrix can be completed by plotting the

relationships between the second message and the first. The solution then

becomes a matter of reducing them to monoalphabetic terms.

(13) In cases where the two periods have a common factor, the sequences can still

be recovered, but they cannot be fully aligned. In this case, the chi test can be

used to match the sequences by frequencies, if necessary, once the sequences

are known.

b. A different technique must be used if different alphabets are used between the

isologs, not just different repeating keys. For example, consider the next two

messages.

9-27

(1) The sequences are different in the two messages, and they cannot be directly

chained together. If you listed the links resulting from the two messages using

the previous technique, they would lead nowhere and contradictions would

quickly develop. The cipher sequences of each must be kept separate.

(2) The method of recovering the cipher sequences when they are different is to set

up periodic matrices one over the other, as shown below. Message 1 and message

2 equivalents are then plotted in the correct sequence for each in the same

columns. Initially, this will result in more than 26 columns, but as incomplete

columns are combined with each other, the matrices will collapse to the correct

width. This method could be used with more than two isologs also, by superim-

posing as many matrices as there are isologous messages.

9-28

(3) The first three groups of each message are plotted above. Each time a previously

used letter appears in the same sequence, the two columns can be combined.

For example, in message 2, the Zs in the third sequence allow those two columns

to be combined, and similarly, the Xs in the fourth sequence can be combined.

In the next example, the complete messages are plotted and all possible

columns are combined.

(4) These matrices can easily be completed by direct symmetry, remembering that

the sequence in each matrix is different.

(5) Either cryptogram can now be reduced to monoalphabetic terms and solved, as

before.

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