EXAMINATION OF TRANSPOSITION CIPHERS
After having decided that a cipher belongs to the transposition class,
it remains to decide on the variety of cipher used. As, by definition,
a transposition cipher consists wholly of characters of the original
message, rearranged according to some law, we may, in general, say that
such a cipher offers fewer difficulties in solution than a substitution
cipher. A transposition cipher is like a picture puzzle; the parts are
all there and the solution merely involves their correct arrangement.
Case 1.--Geometrical ciphers. This case includes all ciphers in which
a certain number of the characters are chosen so that they will form
a square or rectangle of predetermined dimensions; and then these
characters are arranged according to a geometrical design.
Taking the message:
A B C D E F G H I J K L M N O P Q R S T U V W X
of twenty-four letters and assuming a rectangle of six letters
horizontally, and four letters vertically, we may have:
(a) Simple Horizontal:
A B C D E F F E D C B A S T U V W X X W V U T S
G H I J K L L K J I H G M N O P Q R R Q P O N M
M N O P Q R R Q P O N M G H I J K L L K J I H G
S T U V W X X W V U T S A B C D E F F E D C B A
(b) Simple Vertical:
A E I M Q U D H L P T X U Q M I E A X T P L H D
B F J N R V C G K O S W V R N J F B W S O K G C
C G K O S W B F J N R V W S O K G C V R N J F B
D H L P T X A E I M Q U X T P L H D U Q M I E A
(c) Alternate Horizontal:
A B C D E F F E D C B A X W V U T S S T U V W X
L K J I H G G H I J K L M N O P Q R R Q P O N M
M N O P Q R R Q P O N M L K J I H G G H I J K L
X W V U T S S T U V W X A B C D E F F E D C B A
(d) Alternate Vertical:
A H I P Q X D E L M T U X Q P I H A U T M L E D
B G J O R W C F K N S V W R O J G B V S N K F C
C F K N S V B G J O R W V S N K F C W R O J G B
D E L M T U A H I P Q X U T M L E D X Q P I H A
(e) Simple Diagonal:
A B D G K O G K O S V X O K G D B A X V S O K G
C E H L P S D H L P T W S P L H E C W T P L H D
F I M Q T V B E I M Q U V T Q M I F U Q M I E B
J N R U W X A C F J N R X W U R N J R N J F C A
A C F J N R J N R U W X R N J F C A X W U R N J
B E I M Q U F I M Q T V U Q M I E B V T Q M I F
D H L P T W C E H L P S W T P L H D S P L H E C
G K O S V X A B D G K O X V S O K G O K G D B A
(f) Alternate Diagonal:
A B F G N O G N O U V X O N G F B A X V U O N G
C E H M P U F H M P T W U P M H E C W T P M H F
D I L Q T V B E I L Q S V T Q L I D S Q L I E B
J K R S W X A C D J K R X W S R K J R K J D C A
A C D J K R J K R S W X R K J D C A X W S R K J
B E I L Q S D I L Q T V S Q L I E B V T Q L I D
F H M P T W C E H M P U W T P M H F U P M H E C
G N O U V X A B F G N O X V U O N G O N G F B A
(g) Spiral, clockwise:
A B C D E F L M N O P A I J K L M N D E F G H I
P Q R S T G K V W X Q B H U V W X O C R S T U J
O X W V U H J U T S R C G T S R Q P B Q X W V K
N M L K J I I H G F E D F E D C B A A P O N M L
(h) Spiral, counter clockwise:
A P O N M L N M L K J I I H G F E D F E D C B A
B Q X W V K O X W V U H J U T S R C G T S R Q P
C R S T U J P Q R S T G K V W X Q B H U V W X O
D E F G H I A B C D E F L M N O P A I J K L M N
It is simply a matter of inspection to read a message in a cipher
of this type, once the dimensions of the rectangles have been
determined. We place the whole or a portion of the message in such
rectangles and read horizontally, vertically and diagonally forward
and backward. Parts of words will at once be apparent and the whole
message is soon deciphered. Two examples will show the process.
Message
ILVGIOIAEITSRNMANHMNG
This message contains eight vowels or 38% out of twenty-one letters,
and the letters LNRST occur 7 times or 33%, the letters XQJKZ not
appearing. It is therefore a transposition cipher. Twenty-one letters
immediately suggest seven columns of three letters each or three
columns of seven letters each. Trying the former we have:
I L V G I O I
A E I T S R N
M A N H M N G
and reading down each column in succession (Case 1-b) reveals the
message to be "I am leaving this morning."
Message
M S I B R O R S E E V U E E M C O R E R E L I D E T O E P Q
E N R E R N S E R Y E C O L L E R E U S P L U R C E L O A J
A E H U H P F A S O N N O A A E P I U A P P E A C U Q A R U
O P O E I I R R M I A F D A A R Q U B O Z A E G E R S F S X
There are 120 letters in this message with 57 vowels or 47% vowels,
and the letters LNRST occur 31 times or 26% of the whole.
Non-occurrence of K and W and vowel proportion leads us to the
assumption that it is a transposition cipher of a Spanish text. The
factors of 120 are 5 × 3 × 2 × 2 × 2. We may then have one rectangle
of 4 × 30 or one of 5 × 24 or two of 5 × 12, or three of 5 × 8, or
four of 5 × 6, or five of 3 × 8, or ten of 3 × 4, or twenty of 3 ×
2. The message being in a rectangle of 4 × 30, we can inspect it as it
stands and this is clearly not the arrangement if it be a geometrical
transposition cipher at all. It is best however to try the largest
possible rectangles first so we will put it in the form 5 × 24, thus:
M S I B R O R S E E V U E E M C O R E R E L I D
E T O E P Q E N R E R N S E R Y E C O L L E R E
U S P L U R C E L O A J A E H U H P F A S O N N
O A A E P I U A P P E A C U Q A R U O P O E I I
R R M I A F D A A R Q U B O Z A E G E R S F S X
Here an inspection shows this to be Case 1-f, alternate diagonal,
and the text to be "ME SITUO SOBRE PARRAL PORQUE ME PRESENCIA FUE
REVELADA POR U"; here the sense breaks but note that U is the twelfth
letter of the line and continue as if the rectangle were 5 × 12 and
we have "NA PAREJA QU." Now inspect the second rectangle of 5 × 12
in the same way and the sense continues "E SE ME ACERCO Y HUBO QUE
RECHAZAR POR EL FUEGO ALLI ESRERO ORDENES FINISX".
The practical way of examining a cipher of this type is to have several
men prepare rectangles of different dimensions, using the letters of
the cipher in the order received. The rectangles can be inspected very
rapidly when once prepared. Note that the dimensions of any rectangle
will rarely be such as to contain more than fifty letters, on account
of the necessity of filling up a rectangle with nulls if the number
of letters of the message is just a little greater than a multiple of
the rectangle. Also large rectangles give, for all but the diagonal
method, whole words in a line or column and these are easily noted.
The following ciphers come under Case 1:
Case 1-i.--The rail fence cipher, useful as an operators' cipher but
permits of no variation and is therefore read almost as easily as
straight text when the method is known. The message:
HOSTILE CAVALRY HAS RETIRED
is written:
O T L C V L Y A R T R D
H S I E A A R H S E I E
and is sent:
OTLCV LYART RDHSI EAARH SEIEX
Case 1-j.
Message
S S O H S T P F O R I E E A E
T Q N E T F A I X E G L F D R
A U L R N O S R X L H A T R O
To solve this cipher, read down the columns in this order 8, 1, 15,
2, 14, 3, 13, 4, 12, etc. A variation is to arrange the cipher so
the columns are read upwards. Another is to arrange the ciphers so
the columns are read alternately upward and downward. The factors of
the number of letters in this case give the shape of the rectangle
as usual.
It will be seen that there are a great number of possible transposition
ciphers that come under Case 1 but practically all of them are
useless from a military standpoint because they do not depend on a
key which can be readily and frequently changed. However such ciphers
constantly crop up in cipher examination, being used for special
communication between parties who consider the regular military
ciphers too complicated. Thus some of these expedients have been used.
Reversed Writing.--(Special case of Case 1-a).
LEAVING TONIGHT is enciphered THGINOT GNIVAEL or it may be reversed
by words, thus GNIVAEL THGINOT or by groups of five letters, thus
IVAEL NOTGN XTHGI.
Vertical Writing.--(Special case of Case 1-b). Same message is
enciphered,
LT
EO
AN
VI and is sent, LTEOA NVIIG NHGTX.
IG
NH
GT
Case 2.--This case includes all transposition ciphers in which lines
and columns of the text are rearranged according to some key word or
key number. There are many varieties of this case but their solution
usually is arrived at through the methods suggested for Case 1, that
is, arrangement into appropriate rectangles and examination of lines
and columns for words or syllables. Rearrangement of columns or lines
follows until the solution is completed.
Case 2-a.
Message
HIIGF TNGHI NTCVN IEIOT CYIFY LHAEA ESNBA EEEEN
RWGBN YDELR OAESG RNEBO VNLDA ICAOA LCNDT IRGVA
CDOIE SEREC DVPEI AFIFL RINEH ETT
There are 108 letters in this message and examination shows it to be
a transposition cipher, English text. The number of letters, 108,
immediately suggests a rectangle of 12 × 9 or 9 × 12 letters. Put
into this form we have:
Vowels Vowels
H I I G F T N G H I N T 3 H I I G F T N G H 2
C V N I E I O T C Y I F 5 I N T C V N I E I 4
Y L H A E A E S N B A E 6 O T C Y I F Y L H 2
E E E N R W G B N Y D E 4 A E A E S N B A E 6
L R O A E S G R N E B O 5 E E E N R W G B N 3
V N L D A I C A O A L C 5 Y D E L R O A E S 4
N D T I R G V A C D O I 4 G R N E B O V N L 2
E S E R E C D V P E I A 6 D A I C A O A L C 5
F I F L R I N E H E T T 4 N D T I R G V A C 2
D O I E S E R E C 5
D V P E I A F I F 4
L R I N E H E T T 3
The vowel count of the lines shows the first arrangement to be the
more likely. We will now number the columns and try pairing off certain
ones which in no line would give impossible combinations of letters.
1 2 3 4 5 6 7 8 9 10 11 12
H I I G F T N G H I N T
C V N I E I O T C Y I F
Y L H A E A E S N B A E
E E E N R W G B N Y D E
L R O A E S G R N E B O
V N L D A I C A O A L C
N D T I R G V A C D O I
E S E R E C D V P E I A
F I F L R I N E H E T T
These combinations appear among others:
1 6 2 4 5 2
H T I G F I
C I V I E V
Y A L A E L
E W E N R E
L S R A E R
V I N D A N
N G D I R D
E C S R E S
F I I L R I
The word FIGHT stares at us from the first line; let us arrange the
columns thus:
5 2 4 1 6 3
F I G H T I
E V I C I N
E L A Y A H
R E N E W E
E R A L S O
A N D V I L
R D I N G T
E S R E C E
R I L F I F
We have the words FIGHTI(NG), VICIN(ITY), RENEWE(D), ANDVIL(LA),
RDINGT(O), RECE(IVED). With this to go on, we must choose column 11
as the next one and then in order, columns 8, 10, 7, 12, 9. But note
that the order 11, 8, 10, 7, 12, 9, is the same as the order 5, 2, 4,
1, 6, 3. The message was written in twelve columns and the columns
have been transposed in that order. We may, although it is entirely
unnecessary, speculate on the key word used. It was probably
M E X I C O
4 2 6 3 1 5
meaning that the 4th column of the plain text was transferred in
enciphering so it became our 1st, the 2d column remained the 2d;
the 6th column became our 3d, etc.
Actually, this cipher was solved because the word VILLA was suspected
and all the necessary letters were found in line six of the arrangement
in twelve columns. The order 1, 6, 3, 11, 8 was tried and gave
this result.
1 6 3 11 8
H T I N G
C I N I T
Y A H A S
E W E D B
L S O B R
V I L L A
N G T O A
E C E I V
F I F T E
The remainder of the solution followed the lines already laid down
and, naturally, offered no difficulties, in view of the large number
of connected syllables available.
Case 2-b.
Message
SLCOF WEETN EBRDO ORVYM FFEDI
NMTEC ROIAR PERHO ESETS RFBHL
TENAH OPTAU SOMTL RTETT ASCBH
NIODC RENEN AAPRD LACYE ECIIE
SGUFN
This is a transposition cipher, English text, and contains 105
letters. The factors of 105 are 5 × 3 × 7 so that we must investigate
the following rectangles; 5 × 21, 15 × 7, three of 5 × 7, five of 3 ×
7 and seven of 5 × 3.
21 × 5 Vowels
S L C O F W E E T N E B R D O O R V Y M F 6
F E D I N M T E C R O I A R P E R H O E S 9
E T S R F B H L T E N A H O P T A U S O M 7
T L R T E T T A S C B H N I O D C R E N E 6
N A A P R D L A C Y E E C I I E S G U F N 9
Vowels 1 2 1 2 1 0 1 4 0 1 3 3 1 3 3 3 1 1 3 2 1
The vowel count of the columns of the rectangle 5 × 21 is very
satisfactory. Let us consider it as three blocks of 5 × 7 each,
since we must do this ultimately, and make a vowel count of columns
for these blocks.
5 × 21 Vowels
S L C O F 1
W E E T N 2
E B R D O 2
O R V Y M 1
F F E D I 2
N M T E C 1
R O I A R 3
P E R H O 2
E S E T S 2
R F B H L 0
T E N A H 2
O P T A U 3
S O M T L 1
R T E T T 1
A S C B H 1
N I O D C 2
R E N E N 2
A A P R D 2
L A C Y E 2
E C I I E 4
S G U F N 1
Vowels 7 9 8 7 6
Column
1 2 3 4 5
Vowels, 1st block 2 2 3 2 2
Vowels, 2d block 2 3 2 2 2
Vowels, 3d block 3 4 3 2 2
This is also excellent, so we will try three blocks 5 × 7 and see
if rearrangement of horizontal lines will give results reading the
columns vertically.
1 S L C O F P E R H O A S C B H
2 W E E T N E S E T S N I O D C
3 E B R D O R F B H L R E N E N
4 O R V Y M T E N A H A A P R D
5 F F E D I O P T A U L A C Y E
6 N M T E C S O M T L E C I I E
7 R O I A R R T E T T S G U F N
Among other combinations are:
3 E B R D O R F B H L R E N E N
2 W E E T N E S E T S N I O D C
1 S L C O F P E R H O A S C B H
5 F F E D I O P T A U L A C Y E
7 R O I A R R T E T T S G U F N
The addition of line 6 above line 3 and line 4 below line 7 will
complete this cipher. The successive columns should be read downward.
Case 2-c. In this case, both lines and columns are rearranged by
means of a key word or key words. The method of solution is the same
as Case 2-a and 2-b except that the lines must be rearranged after
the columns have been correctly arranged, or in some cases, vice
versa. This cipher is not infrequently met with because it seems
to offer safety by use of two key words and by the great but only
apparent complexity of the method.
Message
WVGAE EGENL TFTOH TEIEF RBTSE
INENG ONWRM GXIXN GOITN ROMRO
ESPAL HNEAC UDNNH DERME
This is a transposition cipher, English text and the number of letters,
70, leads us to try rectangles of 10 × 7 and 7 × 10.
Vowels Vowels
W V G A E E G E N L 4 W V G A E E G 3
T F T O H T E I E F 3 E N L T F T O 2
R B T S E I N E N G 3 H T E I E F R 3
O N W R M G X I X N 2 B T S E I N E 3
G O I T N R O M R O 4 N G O N W R M 1
E S P A L H N E A C 4 G X I X N G O 2
U D N N H D E R M E 3 I T N R O M R 2
O E S P A L H 3
N E A C U D N 3
N H D E R M E 2
The first form looks the more likely from the vowel count. We proceed
to number the columns and lines and try rearrangement of columns so
as to obtain possible letter combinations from every line.
1 2 3 4 5 6 7 8 9 10
1 W V G A E E G E N L
2 T F T O H T E I E F
3 R B T S E I N E N G
4 O N W R M G X I X N
5 G O I T N R O M R O
6 E S P A L H N E A C
7 U D N N H D E R M E
Among other combinations we have these:
3 5 1 4 2 8 10 6 9 7
1 G E W A V E L E N G
2 T H T O F I F T E E
3 T E R S B E G I N N
4 W M O R N I N G X X
5 I N G T O M O R R O
6 P L E A S E C H A N
7 N H U N D R E D M E
A very casual inspection of the lines shows that they should be
rearranged in order 6, 1, 2, 7, 3, 5, 4, as follows:
3 5 1 4 2 8 10 6 9 7
6 P L E A S E C H A N
1 G E W A V E L E N G
2 T H T O F I F T E E
7 N H U N D R E D M E
3 T E R S B E G I N N
5 I N G T O M O R R O
4 W M O R N I N G X X
Although of no particular importance, it may be stated that the
column key in this case was GRAND and the line key was CENTRAL,
both used as in enciphering Case 2-a.
Case 3. Route ciphers. In this case, whole words of the message
are transposed according to some of the methods of Case 1 or 2 or
their equivalents. The route cipher is little used at present. Its
development and use during the Civil War was caused by the inability
of the telegraphers of that day to handle regular cipher matter
correctly and rapidly. It was, even in those days, frankly only
a delaying cipher and, to be of any value, had to be filled with
meaningless words to conceal the message proper. An example from
the Signal Book will suffice to show the general character of route
ciphers. To one familiar with monoliteral transposition ciphers,
even the best of route ciphers offers but little difficulty.
"To encipher the message 'MOVE DAYLIGHT. ENEMY APPROACHING FROM
NORTH. PRISONERS SAY STRENGTH ONE HUNDRED THOUSAND. MEET HIM AS
PLANNED.' arrange as follows:
MOVE STRENGTH PLANNED SAY
DAYLIGHT ONE AS PRISONERS
ENEMY HUNDRED HIM NORTH
APPROACHING THOUSAND MEET FROM
Here the route is down the first column, up the fourth, down the
second and up the third."
This cipher was often complicated by the introduction of nulls for
every fifth word. Thus the above message might be sent:
MOVE STRENGTH PLANNED SAY NEVER DAYLIGHT ONE AS PRISONERS LEAVING
ENEMY HUNDRED HIM NORTH UNCHANGED APPROACHING THOUSAND MEET
FROM COME.
The words in italics are nulls and not a part of the message and the
receiver eliminates them before arranging his message in columns to
get the sense of it.
As an additional complication, it was customary for each correspondent
to have a dictionary or code in which the names of all prominent
generals and places and many of the prominent verbs,--as to march,
to sail, to encamp, to attack, to retreat,--were represented by
other words.
A route cipher using the code words of the War Department code might
have some advantages over the method of enciphering code messages as
prescribed in that Code.
General Remarks on Transposition Ciphers
It is the consensus of opinion of experts that the transposition
cipher is not the best one for military purposes. It does not
fulfill the first, second, and third of Kerckhoffs' requirements as
to indecipherability, safety when apparatus and method fall into the
hands of the enemy, and dependability on a readily changeable key word.
However, transposition ciphers are often encountered. They are
favorites with those who find the substitution ciphers too difficult
and too tedious to handle and who believe that their transposition
methods are either absolutely indecipherable or sufficiently so
for the purpose of concealing the text of a message for the time
being. They seem to be particularly popular with secret agents and
spies, presumably because special apparatus is rarely necessary in
enciphering and deciphering.
Although the number of transposition methods is legion, they can
practically all be considered under one of the three cases already
discussed. It is surprising how often transposition ciphers prepared
by complicated rules, will, on analysis, be seen to be very simple.
To be successful in solving transposition ciphers, one should
constantly practice reading backward and up and down columns, so that
the common combinations of letters are as quickly identified when seen
thus as when encountered in straight text. Combinations like EHT,
LLIW, ROF, DNA, etc., should be appreciated immediately as common
words written backward.
A study of the table of frequency of digraphs or pairs is also
excellent practice and such a table should be at hand when a
transposition cipher is under consideration. It assists greatly if
Case 2 be encountered and is of considerable use in solving Case 1.
The solution of route ciphers is necessarily one of try and fit,
with the knowledge that such ciphers usually are read up and down
columns. It is not believed that route ciphers will often be met with
at the present day.