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Transposition Ciphers are ciphers in which the plaintext message is rearranged by some means agree upon by the sender and receiver.
Note: Transposition ciphers differ from substitution cipher we studied in Chapter 2. In substitution ciphers, the letters are changed by creating a new alphabet (the cipher alphabet) and assigning new letters. In transposition ciphers, no new alphabet is created – the letters of the plaintext are just rearranged is some fashion.
Examples of Transposition Ciphers
1. Scytale Cipher – p. 4 of textbook.
2. ADFGVX – German WWI cipher .
3. Modern Block Ciphers – DES , AES cipher.
For columnar transposition ciphers, users agree upon some prescribed number of columns, and then the actual plaintext letters (with spaces and punctuation removed) are used to form an array of letters, similar to the array used in keyword columnar substitution ciphers, with this number of columns. The ciphertext is obtained by taking the columns of the resulting array in some specified order, and placing the letters in these columns in a row. We will demonstrate two methods for doing this.
Note: In columnar transposition ciphers, the plaintext is always in rows. The ciphertext is always in columns.
3.1.1
Simple Columnar Transpositions
In simple columnar transposition ciphers, the plaintext message is written horizontally in rows for a fixed and agreed upon number of columns. The ciphertext is then described letter by letter from the columns proceeding from left to right.
Note: Sometimes (not always) a sender and recipient will pad the message to make it a multiple of the number of letters in the keyword, giving an equal number letters in each column.

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Example 1: Encipher
THE JOKER SAID THAT IT WAS ALL PART OF
THE PLAN
using a simple 5 column transposition cipher.
Solution:

3
Example 2: Decipher
█
TOTBA AUJAA KMHKO ANTAU FKEEE LTTYR SRLHJ RDMHO ETEII that was enciphered using a simple 4 column transposition.
Solution:

4
Note: In general, given a simple columnar transposition with n total letters and c columns, we use the division algorithm to divide n by c to compute n
 qc
 r . In
█ tableau form, this looks like: q
Quotient q
# columns c c n # letters n
 qc r Remainder r
Then, the first r columns contain q

1 letters each for a total of
The remaining c
 r columns have q letters each giving r ( q

1 )
( c
 r ) q
letters.
total letters.
Example 3: Suppose a simple columnar transposition is made up of 50 total letter distributed over 9 columns. Determine the number of letters in each column that make up the transposition.
Solution:
█

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One problem with simple columnar transposition ciphers is that ciphertexts are always obtained by taking the columns of the array in order starting from the left. This can make a cipher more vulnerable to cryptanalysis. Keyword columnar transposition ciphers can help to alleviate this problem.
3.1.2 Keyword Columnar Transposition Ciphers
To increase security, we would like to “mix” the columns. The method we use involves choosing a keyword and using its alphabetical order of its letters to choose the columns of the ciphertext.
NOTE !! In a keyword columnar transposition ciphers, the keyword is NOT a part of the ciphertext. This differs from keyword columnar substitution ciphers (studied in Section
2.3), where the keyword is included in the cipher alphabet.
Example 4: Use the keyword
BARNEY
to encipher the message
ANDY
GRIFFITHS DEPUTY WAS BARNEY FIFE
using a keyword columnar transposition.
Solution:
█

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NOTE !! In a keyword columnar transposition, if one letter is repeated in the keyword, we order the repeated ciphertext columns from left to right. We do not remove the repeated keyword letters as we did with keyword columnar substitution ciphers in Chapter 1.
Example 5: Use the keyword
MATHEMATICS
and padding the message with X’s to assure equal column lengths, encipher the message
ALGEBRA AND TRIG CAN
BE FUN
using a keyword columnar transposition.
Solution:
█

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Example 6: Suppose we receive the message
ADDSH BGSAR OLGNN VCAII SFWDI AOTRN LSAUF RLLWL OENWE HIC that was enciphered using a keyword columnar transposition with keyword
GILLIGAN
. Decipher this message.
Solution: Since this message has 48 total letters and the keyword has 8 letters, each column under each keyword letter in the columnar transposition process will have
48
8

6 total letters. Using the alphabetical order of the keyword letters (keeping in mind that under the repeated letters G, I and L the columns are ordered from left to right), we can by placing the numbered sequence of 6 letters from the ciphertext:
    
(1) (2)
   
(3) (4)
 
(5)
 
( 6 )
   
(7) (8) under the corresponding matching keyword letter column number (the alphabetical ordering) to get the following array:
(2) (4) (6) (7) (5) (3) (1) (8)
G I L L I G A N
G I L L I G A N
S I S L A N D W
A S A W O N D E
R F U L T V S H
O W F O R C H I
L D R E N A B C
Hence the plaintext message is:
GILLIGANS ISLAND WAS A WONDERFUL
TV SHOW FOR CHILDREN
(note that the ABC was padded to the message in the original encipherment to ensure that the column lengths were equal).
█
Some Suggested Textbook Exercises for Practice for Section 3.1 p. 38: # 1-8

Some Suggested Textbook Exercises for Practice for Section 3.2 p. 42: # 1-6
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Transposition ciphers in general are fairly easy to break, especially in the case of simple transposition ciphers. With a little preliminary information about the plaintext, keyword columnar transpositions can be broken fairly quickly as well. We explore techniques of cryptanalysis for transposition ciphers in this section.
3.3.1 Cryptanalysis of Simple Transposition Ciphers
To try to break a simple transposition cipher, we try various column numbers for the columnar transposition until we get a message that makes sense.
Example 7: Suppose we want to decipher the message
TSINN RNRPT SBOAO ICEKN SOABE that we know was enciphered with a simple transposition cipher with no information about how many columns that were used.
Solution:
9

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█

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3.3.2 Cryptanalysis of Keyword Columnar Transposition Ciphers
1. Because keyword columnar transposition ciphers do not necessarily take the columns of the cipher array in order, cryptanalysis can be more difficult than it is for simple columnar transposition ciphers. If the columns are of the same length, it may be possible to divide the ciphertext up into equal blocks and rearrange the resulting columns until a resulting message results.
2. To simplify the process in all cases, if we know some of the original plaintext (call a crib) beforehand, we can decipher the message. The next example illustrates this method.

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Example 8: Suppose the message
AHLCC MSOAO NMSSS MTSSI AASDI
NRVLF WANTO ETTIA IOERI HLEYL AECVL W
was enciphered using a keyword columnar transposition and we know that the words
THE FAMILY
is a part of the plaintext. Decipher this message.
Solution: In the deciphering process, we will assume that the keyword that was used to encipher the message in the keyword columnar transposition is shorter than the known word (crib) given in the plaintext. Noting that the known word
   
9 letters is 9 letters long, we first assume that the keyword used is one less than this, that is, we assume that it is 8 letters long. If his is so, then the keyword columnar transposition will have 8 columns and the crib will appear in the columns in the form similar to
T H E F A M I L
Y
If the crib appeared in this fashion, then the digraph “TY” would appear in the ciphertext.
Since it does not, we will assume the keyword used in the columnar transposition has one less letter, that is, we assume that it is 7 letters long. Then the keyword columnar transposition will have 7 columns and the crib appears as
T H E F A M I
L Y which says that the digraphs TL and HY occur in the ciphertext. Since this does not occur, we assume the keyword used was 6 letters long. Hence, the crib appears as
T H E F A M
I L Y
One can see that the digraphs TI, HL, and EY all occur in the ciphertext. This says that the keyword is likely 6 characters long and hence 6 columns were used to create the ciphertext in the keyword columnar transposition. If we divide the total number of ciphertext letters ( n = 56) by this number of columns ( c = 6), we see by the division algorithm that
56

9

6

2 .
Hence, the quotient is q = 9 and the remainder is r = 2. Thus, in the columnar transposition, there are r = 2 columns with q + 1 = 10 characters and c – r = 6 – 2 = 4 columns with q = 9 characters. We now align the ciphertext into groups of 9 letters, which are numbered below:

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AHLCCMSOA ONMSSSMTS SIAASDINR VLFWANTOE TTIAIOERI
(1) (2) (3) (4) (5)
HLEYLAECV LW
(6) (7)
Next, we attempt to spell out the crib while lining up the digraphs TI, HL, and EY that occur. Doing this gives
(5) (1) (6) (4) (3) (2)
H V S O
T A L L I N
T H E F A M
I L Y W A S
A C L A S S
I C A N D S
O M E T I M
E S C O N T
R O V E R S
I A
When reading across the rows of this array from the top, the letters begin to form sensible
English starting with the columns labeled 1 and 6. Thus, it is likely that these columns are the first two in the original cipher array, and would therefore be the two columns that contain 10 letters instead of nine. So we will split the ciphertext into blocks again, using
10 letters in the blocks labeled 1 and 6, and nine letters in the rest.
AHLCCMSOAO NMSSSMTSS IAASDINRV LFWANTOET TIAIOERIH
(1) (2) (3) (4) (5)
LEYLAECVLW
(6)
(1) (6) (4) (3) (2) (5)
A L L I N T
H E F A M I
L Y W A S A
C L A S S I
C A N D S O
M E T I M E
S C O N T R
O V E R S I
A L T V S H
O W
Hence, the message is
ALL IN THE FAMILY WAS A CLASSIC AND
SOMETIMES CONTROVERSIAL TV SHOW
. █

Some Suggested Textbook Exercises for Practice for Section 3.3 p. 47: # 1-3
Some Suggested Textbook Exercises for Practice for Section 3.4 p. 52: # 1-2
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Toward the end of World War I, Germany began using a new type of cipher that combined features of both substitution and transposition ciphers. These new ciphers, called ADFGX ciphers, are named for the only five letters that can appear in ciphertexts.
These five letters were chosen because they sound very different from one another in
Morse code, thus minimizing transmission errors.
Fact: ADFGX ciphers were created by German Signals Officer Colonel Fritz Nebel, and first used by the German military in March, 1918.
ADFGX Cipher Encipherment Steps
1.
Label the rows and columns of an array with the letters A, D, F, G, and X. Then, using the alphabet, create a 5 × 5 array of letters agreed upon by the users and place them within the rows and columns labeled by A, D, F, G, and X . Note that since there are 26 total alphabet letters, the letters I and J are considered to be the same letter in ADFGX ciphers. Then, each plaintext letter is replaced with the pair of row and column (in that order) letter labels of the position that the plaintext letter occupies in the array. This yields a preliminary ciphertext that is twice as long as the plaintext and contains only the letters A, D, F, G, and X.
2. Next, use an agreed upon keyword to create a keyword columnar transposition cipher applied to the preliminary ciphertext, using one or more keywords agreed upon by the users. This yields the final ciphertext.
3. For decryption, the encryption steps must be undone in the opposite order. That is, the transposition cipher must be undone (i.e., decrypted normally) first, and the substitution cipher undone second.

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Example 9: Use the keywords
GO TO JAIL
with a ADFGX cipher with the array
A D F G X
A P G C E N
D B Q O Z R
F S L A F T
G M D V I W
X K U Y X H to encipher the message
I AM NOT A CROOK JUST YET.
Solution:
█

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Example 10: Suppose the ciphertext message
DAGXA DXAAX GFGGD XXGDG
AFFDX FFAFF DA
was enciphered with a ADFGX cipher using the keyword
SHAWSHANK with the array
A D F G X
A P G C E N
D B Q O Z R
F S L A F T
G M D V I W
X K U Y X H
Decipher this message.
Solution:
█

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The ADFGVX Cipher
In June, 1918, the Germans increased the size of the array of letters for ADFGX ciphers to 6 × 6 to create what was called the ADFGVX cipher. The major changes from the
ADFGX cipher was the following:
1.
All 26 letters of the alphabet could be included in the plaintexts (the letters I and J could be distinguished).
2.
The digits 0–9 could be included in the plaintext and hence in the array.
3.
It required that the additional letter, V, be used in ciphertexts.
Example 11: Use the keywords
BASKETBALL with a ADFGVX cipher with the array
A D F G V X
A 8 P 3 D 1 N
D L T 4 O A H
F 7 K B C 5 Z
G J U 6 W G M
V X S V I R 2
X 9 E Y 0 F Q to encipher the message
HEAT WIN SERIES 4 GAMES TO 1.
Solution:
█

Cryptanalysis of ADFGVX Cipher
The ADFGX and ADFGVX ciphers were cryptanalyzed and broken by French Army
Lieutenant Georges Painvin,
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This success is widely considered a primary reason why the French were able to stop
Germany’s ill-fated 1918 Spring Offensive.
Some Suggested Textbook Exercises for Practice for Section 3.5 p. 57: # 1-4
Some Suggested Textbook Exercises for Practice for Section 3.6 p. 61: # 1-7