We will now consider the class of substitution ciphers where a number
of alphabets are used, the number and choice of alphabets depending
on a key word or equivalent and being used periodically throughout
the message.
In this class belong the methods of Vigenere, Porta, Beaufort, St. Cyr,
and many others. These methods date back several hundred years,
but variations of them are constantly appearing as new ciphers. The
Larrabee cipher, used for communication between government departments,
is the Vigenere cipher of the 17th Century. The cipher disk method
is practically the Vigenere cipher with reversed alphabets.
In using these ciphers, there is provided a number of different cipher
alphabets, usually twenty-six, and each cipher alphabet is identified
by a different letter or number. A key word or phrase (or key number)
is agreed upon by the correspondents. The message to be enciphered is
written in lines containing a number of letters which is a multiple
of the number of letters of the key. The key is written as the first
line. Then each column under a letter of the key is enciphered by the
cipher alphabet pertaining to that letter of the key. For example,
let us take the message, "All radio messages must hereafter be put
in cipher," with the key Grant, using the Vigenere cipher alphabets
given below. Each of these alphabets is identified by the first or left
hand letter which represents A of the text. We thus will use in turn
the alphabets beginning with G, with R, with A, with N, and with T.
G R A N T G R A N T
A L L R A D I O M E
S S A G E S M U S T
H E R E A F T E R B
E P U T I N C I P H
E R
Using the alphabet indicated by G, we get
G J
Y Y
N L
K T
K
Continuing for the other alphabets, we get
G C L E T J Z O Z X
Y J A T X Y D U F M
N V R R T L K E E U
K G U G B T T I C A
K I
This method of arranging the message into lines and columns and then
enciphering whole columns with each cipher alphabet is much shorter
than the method of handling each letter of the message separately. The
chance of error is also greatly reduced.
All these cipher methods can be operated by means of squares containing
the various alphabets, cipher disks or arrangements of fixed and
sliding alphabets. For example, this was the original cipher of
Vigenere:
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
B C D E F G H I J K L M N O P Q R S T U V W X Y Z A
C D E F G H I J K L M N O P Q R S T U V W X Y Z A B
D E F G H I J K L M N O P Q R S T U V W X Y Z A B C
E F G H I J K L M N O P Q R S T U V W X Y Z A B C D
F G H I J K L M N O P Q R S T U V W X Y Z A B C D E
G H I J K L M N O P Q R S T U V W X Y Z A B C D E F
H I J K L M N O P Q R S T U V W X Y Z A B C D E F G
I J K L M N O P Q R S T U V W X Y Z A B C D E F G H
J K L M N O P Q R S T U V W X Y Z A B C D E F G H I
K L M N O P Q R S T U V W X Y Z A B C D E F G H I J
L M N O P Q R S T U V W X Y Z A B C D E F G H I J K
M N O P Q R S T U V W X Y Z A B C D E F G H I J K L
N O P Q R S T U V W X Y Z A B C D E F G H I J K L M
O P Q R S T U V W X Y Z A B C D E F G H I J K L M N
P Q R S T U V W X Y Z A B C D E F G H I J K L M N O
Q R S T U V W X Y Z A B C D E F G H I J K L M N O P
R S T U V W X Y Z A B C D E F G H I J K L M N O P Q
S T U V W X Y Z A B C D E F G H I J K L M N O P Q R
T U V W X Y Z A B C D E F G H I J K L M N O P Q R S
U V W X Y Z A B C D E F G H I J K L M N O P Q R S T
V W X Y Z A B C D E F G H I J K L M N O P Q R S T U
W X Y Z A B C D E F G H I J K L M N O P Q R S T U V
X Y Z A B C D E F G H I J K L M N O P Q R S T U V W
Y Z A B C D E F G H I J K L M N O P Q R S T U V W X
Z A B C D E F G H I J K L M N O P Q R S T U V W X Y
The first horizontal alphabet is the alphabet of the plain text. Each
substitution alphabet is designated by the letter at the left of a
horizontal line. For example, if the key word is BAD, the second,
first and fourth alphabets are used in turn and the word WILL is
enciphered XIOM.
The Larrabee cipher is merely a slightly different arrangement of
the Vigenere cipher and is printed on a card in this form:
----------------------------------
A ABCDEFGHIJKLMNOPQRSTUVWXYZ
abcdefghijklmnopqrstuvwxyz
----------------------------------
B ABCDEFGHIJKLMNOPQRSTUVWXYZ
bcdefghijklmnopqrstuvwxyza
----------------------------------
C ABCDEFGHIJKLMNOPQRSTUVWXYZ
cdefghijklmnopqrstuvwxyzab
----------------------------------
etc.
----------------------------------
Y ABCDEFGHIJKLMNOPQRSTUVWXYZ
yzabcdefghijklmnopqrstuvwx
----------------------------------
Z ABCDEFGHIJKLMNOPQRSTUVWXYZ
zabcdefghijklmnopqrstuvwxy
----------------------------------
The large letters at the left are the letters of the key word. It
will be noted that these letters correspond to the first letters of
the cipher alphabets (in small letters) as in the Vigenere cipher.
A much simpler arrangement of the Vigenere cipher is the use of a
fixed and sliding alphabet. Either the fixed or sliding alphabet must
be double in order to get coincidence for every letter when A is set
to the letter of the key word.
Fixed Alphabet of Text
| |
|ABCDEFGHIJKLMNOPQRSTUVWXYZABCDEFGHIJKLMNOPQRSTUVWXYZ |
+-----------------------------------------------------+
| ABCDEFGHIJKLMNOPQRSTUVWXYZ |
| |
| Movable Alphabet of Cipher |
+-------------------------------------------+
As shown here, A of the fixed or text alphabet coincides with T
of the movable cipher alphabet. This is the setting where T is the
letter of the key word in use. The lower movable alphabet is moved
for each letter of the message and the A of the fixed alphabet is
made to coincide in turn with each letter of the key before the
corresponding letter of the text is enciphered. It is obviously
only a step from this arrangement to that of a cipher disk, where
the fixed alphabet, (a single one will now serve) is printed in a
circle and the movable alphabet, also in a circle, is on a separate
rotatable disk. Coincidence of any letter on the disk with A of the
fixed alphabet is obtained by rotating the disk.
The well known U. S. Army Cipher Disk has just such an arrangement of
the fixed alphabet but the alphabet of the disk is reversed. This has
several advantages in simplicity of operation but none in increasing
the indecipherability of the cipher prepared with it. The arrangement
of fixed and sliding alphabets which is equivalent to the U. S. Army
cipher disk is this:
Fixed Alphabet
| |
| ABCDEFGHIJKLMNOPQRSTUVWXYZABCDEFGHIJKLMNOPQRSTUVWXYZ |
+------------------------------------------------------+
| ZYXWVUTSRQPONMLKJIHGFEDCBA |
| |
| Movable Alphabet |
+------------------------------------------+
It will be noticed that with this arrangement of running the alphabets
in opposite directions, it becomes immaterial which alphabet is used
for the text and which for the cipher for if A = G then G = A. This
is not true of the Vigenere cipher.
It is perfectly feasible to substitute a card for the U. S. Army
cipher disk. It would have this form:
ABCDEFGHIJKLMNOPQRSTUVWXYZ
1 AZYXWVUTSRQPONMLKJIHGFEDCB
2 BAZYXWVUTSRQPONMLKJIHGFEDC
3 CBAZYXWVUTSRQPONMLKJIHGFED
etc.
25 YXWVUTSRQPONMLKJIHGFEDCBAZ
26 ZYXWVUTSRQPONMLKJIHGFEDCBA
The first horizontal line is the alphabet of the text. The other
twenty-six lines are the cipher alphabets each corresponding to the
letter of the key word which is at the left of the line.
One of the ciphers of Porta was prepared with a card of this kind:
-----------------------------
AB ABCDEFGHIJKLM
NOPQRSTUVWXYZ
-----------------------------
CD ABCDEFGHIJKLM
ZNOPQRSTUVWXY
-----------------------------
EF ABCDEFGHIJKLM
YZABCDEFGHIJK
-----------------------------
etc.
-----------------------------
WX ABCDEFGHIJKLM
PQRSTUVWXYZNO
-----------------------------
YZ ABCDEFGHIJKLM
OPQRSTUVWXYZN
-----------------------------
In this cipher the large letters at the left correspond to the letters
of the key and, in each alphabet, the lower letter is substituted for
the upper and vice versa. For example, with key BAD to encipher WILL
we would get JVXY. Note that with either B or A as the key letter,
the first alphabet would be used.
A combination of the Vigenere and Porta ciphers is this:
----------------------------------
A ABCDEFGHIJKLMNOPQRSTUVWXYZ
ABCDEFGHIJKLMNOPQRSTUVWXYZ
----------------------------------
BC ABCDEFGHIJKLMNOPQRSTUVWXYZ
BCDEFGHIJKLMNOPQRSTUVWXYZA
----------------------------------
DE ABCDEFGHIJKLMNOPQRSTUVWXYZ
CDEFGHIJKLMNOPQRSTUVWXYZAB
----------------------------------
etc.
----------------------------------
VW ABCDEFGHIJKLMNOPQRSTUVWXYZ
LMNOPQRSTUVWXYZABCDEFGHIJK
----------------------------------
XY ABCDEFGHIJKLMNOPQRSTUVWXYZ
MNOPQRSTUVWXYZABCDEFGHIJKL
----------------------------------
Z ABCDEFGHIJKLMNOPQRSTUVWXYZ
NOPQRSTUVWXYZABCDEFGHIJKLM
----------------------------------
Here again the large letters at the left correspond to the letters
of the key and, in each pair of alphabets, the upper one is that of
the plain text and the lower is that of the cipher.
This cipher can also be operated by a fixed and sliding alphabet.
Fixed Alphabet of Text:
| |
| ABCDEFGHIJKLMNOPQRSTUVWXYZABCDEFGHIJKLMNOPQRSTUVWXYZ |
+------------------------------------------------------+
| ABCDEFGHIJKLMNOPQRSTUVWXYZ |
| |
| Sliding Alphabet of Cipher |
| Index |
+------------------------------------------+
BDFHJLNPRTVX
A Z Letters of Key.
CEGIKMOQSUWY
The other ciphers mentioned are merely variations of these that have
been discussed. It is immaterial, in the following analysis, which
variety has been used. The analysis is really based on what can be
done with a cipher made up with a mixed cipher alphabet which may
be moved with reference to the fixed alphabet of the text, (See Case
7-b). Clearly this is a much more difficult proposition than dealing
with a cipher in which the cipher alphabets run in their regular
sequence, either backward or forward. In fact, in the analysis of Case
7, we may consider any cipher prepared by the method of Vigenere or
any of its variations as a special and simple case.
It was long ago discovered that, in any cipher of this class,
(1) two like groups of letters in the cipher are most probably the
result of two like groups of letters of the text enciphered by the
same alphabets and (2) the number of letters in one group plus the
number of letters to the beginning of the second group is a multiple
of the number of alphabets used. It is evident, of course, that we
may have similar groups in the cipher which are not the result of
enciphering similar groups of the text by the same alphabets but if
we take all recurring groups in a message and investigate the number
of intervening letters, we will find that the majority of such cases
will conform to these two principles.
Changing the key word and message to illustrate more clearly the
above points, the following is quoted from the Signal Book, 1914,
with reference to the use of the cipher disk in preparing a message
with a key word. [2]
"--This simple disk can be used with a cipher word or, preferably,
cipher words, known only to the correspondents.... Using the key word
'disk' to encipher the message 'Artillery commander will order all
guns withdrawn,' we will proceed as follows: Write out the message
to be enciphered and above it write the key word ... letter over
letter, thus:
| | | | | | | | | |
DISK|DISK|DISK|DISK|DISK|DISK|DISK|DISK|DISK|DISK|DIS
ARTI|LLER|YCOM|MAND|ERWI|LLOR|DERA|LLGU|NSWI|THDR|AWN
| | | | | | | | | |
DRZC|SXOT|FGEY|RIFH|ZRWC|SXET|AEBK|SXMQ|QQWC|XBPT|DMP
| | | | | | | | | |
"Now bring the 'a' of the upper disk under the first letter of the
key word on the lower disk, in this case 'D'. The first letter of
the message to be enciphered is 'A': 'd' is found to be the letter
connected with 'A', and it is put down as the first cipher letter. The
letter 'a' is then brought under 'I' which is the second letter of the
key word. 'R' is to be enciphered and 'r' is found to be the second
cipher letter.... Proceed in this manner until the last letter of the
key word is used and beginning again with the letter 'D', so continue
until all letters of the message have been enciphered. Divided into
groups of five letters, it will be as follows:
"DRZCS XOTFG EYRIF HZRWC SXETA EBKSX MQQQW CKBPT DMF."
So much for the Signal Book; now let us examine the above message
for pairs or similar groups and count the intervening letters to
demonstrate principles (1) and (2);
CSX--CSX 16 = 4×4
SX--SX 16 = 4×4
SX--SX 8 = 2×4
WC--WC 16 = 4×4
The key word might contain 2, 4 or 8 letters from the evidence but we
may eliminate 2 as unlikely and preparation of frequency tables of each
of the four alphabets would soon show that 4 is the correct number.
A later and more extensive example (Case 7-a) will show
pairs not separated by multiples of the number of alphabets
used, but the evidence in nearly every case will be practically
conclusive. Especially is this so if chance assists us by giving groups
of three or more letters like the group CSX in the above example. The
number of alphabets having been determined each alphabet is handled
by the methods of Case 6 already discussed.
Case 7-a.--The following message appeared in the "personal" column
of a London paper:
"M. B. Will deposit £27 14s 5d tomorrow,"
and the next day we find this one:
M.B. CT OSB UHGI TP IPEWF H CEWIL NSTTLE FJNVX XTYLS FWKKHI BJLSI SQ
VOI BKSM XMKUL SK NVPONPN GSW OL. IEAG NPSI HYJISFZ CYY NPUXQG TPRJA
VXMXI AP EHVPPR TH WPPNEL. UVZUA MMYVSF KNTS ZSZ UAJPQ DLMMJXL JR
RA PORTELOGJ CSULTWNI XMKUHW XGLN ELCPOWY OL. ULJTL BVJ TLBWTPZ XLD
K ZISZNK OSY DL RYJUAJSSGK. TLFNS UVD VV FQGCYL FJHVSI YJL NEXV PO
WTOL PYYYHSH GQBOH AGZTIQ EYFAX YPMP SQA CI XEYVXNPPAII UV TLFTWMC FU
WBWXGUHIWU. AIIWG HSI YJVTI BJV XMQN SFX DQB LRTY TZ QTXLNISVZ. GIFT
AII UQSJGJ OHZ XFOWFV BKAI CTWY DSWTLTTTPKFRHG IVX QCAFV TP DIIS JBF
ESF JSC MCCF HNGK ESBP DJPQ NLU CTW ROSB CSM.
The messages in question appeared in an English newspaper. It is
fair to presume then that the cipher is in English. This is checked
negatively by the fact that it contains the letter W which is not
used in any of the Latin languages and that the last fifteen words of
the message consist of from two to four letters each, an impossible
thing in German. It contains 108 groups which are probably words,
as there are 473 letters or an average of 4.4 letters per group,
while we normally expect an average of about 5 letters per group. The
vowels AEIOU number 90 and the letters JKQXZ number 78. It is thus
a substitution cipher (20% of 473=94.6).
Recurring words and similar groups are AIIWG, AII; BKSM, BKAI; CT,
CTWY, CTW; DLMMJXL, DL; ESF, ESBP; FJNVX, FJHVSI; NPSI, NPUXQG; OSB,
OSY, ROSB; OL, OL; PORTELOGJ, PO; SQ, SQA; TP, TP; TLBWTPZ, TLFNS,
TLFTWMC; UVZUA, UVD, UV; XMKUL, XMKUHW; YJL, YJVTI.
Frequency Table for 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 Y Z
15 13 15 8 13 20 16 16 30 21 13 27 14 19 15 26 13 9 33 30 17 12 19 20 19 11
This clearly eliminates Cases 4, 5 and 6.
Referring to the recurring words and groups above noted, we figure
the number of letters between each.
AII ... AII 45 = 3×3×5
BK ... BK 345 = 23×3×5
CT ... CT 403 No factors
CTW ... CTW 60 = 2×2×3×5
DL ... DL 75 = 3×5×5
ES ... ES 14 = 2×7
FJ ... FJ 187 No factors
NP ... NP 14 = 2×7
OL ... OL 120 = 2×2×2×3×5
OS ... OS 220 = 11×2×2×5
OSB ... OSB 465 = 31×3×5
PO ... PO 105 = 7×3×5
SQ ... SQ 250 = 2×5×5×5
TLF ... TLF 80 = 2×2×2×2×5
TP ... TP 405 = 3×3×3×3×5
UV ... UV 115 = 23×5
XMKU ... XMKU 120 = 2×2×2×3×5
UV ... UV 73 No factors
YJ ... YJ 85 = 17×5
The dominant factor is clearly 5, so we may consider that five
alphabets were used, indicating a keyword of five letters. Writing
the message in lines of five letters each and making a frequency
table for each of the five columns so formed, we find the following:
Frequency Tables
Colum 1 Column 2 Column 3 Column 4 Column 5
A 11 A 111111111 A 1 A 1 A 11
B B 111 B 111 B B 1111111
C 1111111 C 1 C 111 C 1111 C
D 11 D 11 D 1 D D 111
E 1111 E E 11 E 1111111 E
F 111 F F 111111111 F 111 F 11111
G 111111111 G G 111 G 11 G 11
H 111 H 11111 H 111 H 111 H 11
I 11 I 11 I 1111111 I 11111111111111111 I 11
J 11111 J 1 J 111111 J J 111111111
K 111111 K 11111 K K 1 K 1
L L 1111111111111111111 L 11 L 11111 L 1
M M M 1111111 M 1111 M 111
N 1111111 N 111 N 1111 N N 11111
O 11111 O O 111111111 O 1 O
P 1111111 P 1111111 P 11111111 P 1111 P
Q 11111 Q Q Q 11 Q 111111
R R 1 R 1 R 111111 R 1
S S 11111111 S 111111 S 111111111111 S 1111111
T 1111111 T 111 T 11111 T 1 T 11111111111111
U 1111111 U 111 U 111111 U U 1
V 11111 V V 11 V 11111 V
W 111 W 1111 W W 11111 W 1111111
X 11 X X 1111 X 11111111 X 111111
Y 1111 Y 11111 Y Y 111 Y 1111111
Z Z 11111 Z 111 Z Z 111
In the table for Column 1, the letter G occurs 9 times. Let us
consider it tentatively as E. Then if the cipher alphabet runs
regularly and in the direction of the regular alphabet, C (7 times)=A
and the cipher alphabet bears a close resemblance to the regular
frequency table. Note TUV (=RST) occurring respectively 7, 7, and 5
times and the non-occurrence of B, L, M, R, S, Z, (=Z, J, K, P, Q,
and X respectively.)
In the next table, L occurs 19 times and taking it for E with the
alphabet running in the same way, A=H. The first word of our message,
CT, thus becomes AM when deciphered with these two alphabets and the
first two letters of the key are C H.
Similarly in the third table we may take either F or O for E, but a
casual examination shows that the former is correct and A=B (even if
we were looking for a vowel for the next letter of the keyword).
In the fourth table, I is clearly E and A=E. The fifth table shows
T=14 and J=9. If we take T=E we find that we would have many letters
which should not occur. On the other hand, if we take J=E then
T=O and in view of the many E's already accounted for in the other
columns, this may be all right. It checks as correct if we apply the
last three alphabets to the second word of our message, OSB, which
deciphers NOW. Using these alphabets to decipher the whole message,
we find it to read:
"M. B. Am now safe on board a barge moored below Tower Bridge
where no one will think of looking for me. Have good friends
but little money owing to action of police. Trust, little girl,
you still believe in my innocence although things seem against
me. There are reasons why I should not be questioned. Shall try to
embark before the mast in some outward bound vessel. Crews will
not be scrutinized so sharply as passengers. There are those who
will let you know my movements. Fear the police may tamper with
your correspondence but later on when hue and cry have died down
will let you know all."
The key to this message is CHBEF which is not intelligible as a word
but if put into figures indicating that the 2d, 7th, 1st, 4th, and
5th letter beyond the corresponding letter of the message has been
used the key becomes 27145 and we may connect it with the "personal"
which appeared in the same paper the day before reading:
"M. B. Will deposit £27 14s 5d tomorrow."
Case 7-b.
Message
DDLRM ERGLM UJTLL CHERS LSOEE SMEJU
ZJIMU DAEES DUTDB GUGPN RCHOB EQEIE
OOACD EIOOG COLJL PDUVM IGIYX QQTOT
DJCPJ OISLY DUASI UPFNE AECOB OESHO
BETND QXUCY LUQOY EHYDU LXPEQ FIXZE
PDCNZ ENELQ MJTSQ ECFIE ARNDN ETSCF
IFQSE TDDNP UUZHQ CDTXQ IRMER GLXBE
IQRXJ FBSQD LDSVI XUMTB AEQEB YLECO
IYCUD QTPYS VOQBL ULYRO YHEFM OYMUY
ROYMU EQBLV UBREY GHYTQ CMUBR EQTOF
VSDDU DAFFS CEBSV TIOYE TCLQX DVNLQ
XYTSI MZULX BAXQR ECVTD ETGOB CCUYF
TTNXL UNEFS IVIJR ZHSBY LLTSI
On the preliminary determination, we have the following count of
letters out of a total of 385:
A 8 L 23 J 9
E 38 N 11 Q 22
I 19 R 14 V 9
O 21 S 20 X 13
U 24 T 21 Z 6
Total 110 Total 89 Total 59
28% 23% 15%
Every letter except K and W occurs at least six times. We may say then
that it is a substitution cipher, Spanish text, and certainly not Case
4, 5 or 6. We will now analyze it for recurring pairs or groups to
determine, if it be Case 7, how many alphabets were used. The following
is a complete list of such recurring groups and pairs with the number
of letters intervening and the factors thereof. In work of this kind,
the groups of three or more letters are always much more valuable
than single pairs. For example, the groups, HOBE, OYMU, RMERGL and
UBRE show, without question, that six alphabets were used. It is not
necessary, as a rule, to make a complete list like the following:
AE 74=2×37 IE 110=2×5×11 RE 50=2×5×5
AE 120=2×2×2×3×5 IM 302=2×151 RMERGL 198=2×3×3×11
BE 88=2×2×2×11 IO 250=2×5×5×5 SC 132=2×2×3×11
CD 132=2×2×3×11 IX 78=2×3×13 SD 262=2×131
CFI 12=2×2×3 LY 158=2×79 SI 230=2×5×23
CH 36=2×2×3×3 JT 150=2×3×5×5 SI 34=2×17
CO 42=2×3×7 LL 367 No factors SI 264=2×2×2×3×11
CO 126=2×3×3×7 LQ 164=2×2×41 SI 12=2×2×3
CU 114=2×3×19 LQX 6=2×3 SL 78=2×3×13
DD 186=2×3×31 LU 124=2×2×31 SQ 54=2×3×3×3
DD 116=2×2×29 LU 110=2×5×11 SV 27=3×3×3
DE 285=5×57 LU 234=2×3×3×13 SV 63=3×3×7
DL 218=2×109 LX 66=2×3×11 SV 90=2×3×3×5
DN 14=2×7 LXB 132=2×2×3×11 TD 47 No factors
DQ 120=2×2×2×3×5 LY 158=2×79 TD 165=3×5×11
DU 36=2×2×3×3 ME 22=2×11 TD 96=2×2×2×2×2×3
DU 24=2×2×2×3 MU 24=2×2×2×3 TN 239 No factors
DU 38=2×19 MU 240=2×2×2×2×3×5 TS 14=2×7
DU 165=3×5×11 MU 18=2×3×3 TS 156=2×2×3×13
EA 30=2×3×5 ND 47 No factors TSI 50=2×5×5
EB 78=2×3×13 NE 48=2×2×2×2×3 UBRE 12=2×2×3
EC 180=2×2×3×3×5 NE 18=2×3×3 UD 60=2×2×3×5
ECO 126=2×3×3×7 NE 192=2×2×2×2×2×2×3 UDA 270=2×3×3×3×5
EES 14=2×7 OB 6=2×3 UL 114=2×3×19
EF 105=3×5×7 OB 234=2×3×3×13 ULX 198=2×3×3×11
EI 8=2×2×2 OE 93=3×31 UY 89 No factors
EI 152=2×2×2×19 OI 144=2×2×2×2×3×3 UZ 162=2×3×3×3×3
EQ 88=2×2×2×11 OO 7 No factors VI 148=2×2×37
EQ 264=2×2×2×3×11 OY 6=2×3 VT 33=3×11
EQ 44=2×2×11 OY 46=2×23 XQ 114=2×3×19
EQE 176=2×2×2×2×11 OYMU 6=2×3 XQ 144=2×2×2×2×3×3
ER 12=2×2×3 PD 75=3×5×5 XU 99=3×3×11
ES 78=2×3×13 QBL 24=2×2×2×3 YE 184=2×2×2×23
ET 135=3×3×5 QC 95=5×19 YL 106=2×53
ET 9=3×3 QE 108=2×2×3×3×3 YL 144=2×2×2×2×3×3
ET 54=2×3×3×3 QE 68=2×2×17 YM 6=2×3
ET 31 No factors QR 132=2×2×3×11 YRO 12=2×2×3
HE 245=5×7×7 QTO 210=2×3×5×7 ZE 6=2×3
HOBE 66=2×3×11 QX 198=2×3×3×11 ZH 183=3×61
Out of one hundred and one recurring pairs we have fifty with the
factors 2×3=6; out of twelve recurring triplets, nine have these
factors; and the four recurring groups of four or more letters all have
these factors. The percentages are respectively 49.5%, 75% and 100%
and we may be certain from this that six alphabets were used. But,
before the six frequency tables are made up, there is one more point
to be considered; why are there so many recurring groups which do not
have six as a factor? The answer is that one or more of the alphabets
is repeated in each cycle; that is, a key word of the form HAVANA has
been used. If this were the key word, the second, fourth and sixth
alphabets would be the same. We will see later that in this example
the second and sixth alphabets are the same and this introduces the
great number of recurring groups without the factor 6.
We will now proceed to make a frequency table for each alphabet. As
the message is written in thirty columns, we take the first, seventh,
thirteenth, etc., as constituting the first alphabet; the second,
eighth, fourteenth, etc., as constituting the second alphabet and
so on. The prefix and suffix letter is noted for each occurrence
of each letter. The importance of this will be appreciated when the
form of the frequency tables is examined. None bears any resemblance
to the normal frequency table except that each is evidently a mixed
up alphabet. The numbers after "Prefix" and "Suffix" refer to the
alphabet to which these belong, for convenience in future reference.
Frequency Tables
First Alphabet
Letter Prefix (6) Suffix (2)
A 111 3 DUD ESF
B 11111111 8 OOOFEEOS EOESYSCY
C 0
D 1111 4 TYT DJUD
E 1 1 E S
F 0
G 0
H 0
I 11111111 8 EOFFEOVS OSEFQYJ
J 0
L 1 1 O J
M 1 1 F O
N 1111 4 FEDU EEEE
O 1 1 E O
P 11 2 GE ND
Q 1111 4 UEOE OFBB
R 1111111 7 EEEYYBB GSGOOEE
S 1 1 D V
T 11111111 8 JUJMQYVF LDSBPQDT
U 0
V 11 2 UF MS
X 111111 6 YQTQQA QUQDYQ
Y 1 1 O E
Z 111 3 UUM JHU
Second Alphabet
Letter Prefix (1) Suffix (3)
A 0
B 111 3 TQQ ALL
C 1 1 B C
D 111111 6 DTPDXT LBCNVE
E 11111111111 1 1 ABNBNINRRYN EOATLATYQTF
F 111 3 QIA IQF
G 11 2 RR LL
H 1 1 Z O
I 0
J 1111 4 ZLDI ILCR
L 1 1 T L
M 1 1 V I
N 1 1 P R
O 1111111 7 OIBQRMR AOEYYYY
P 1 1 T Y
Q 11111 5 XXITX OIRCR
R 0
S 11111111 8 REIATBVB LMLIQQDV
T 1 1 T N
U 111 3 XDZ CLL
V 1 1 S I
X 0
Y 1111 4 BIXB LCTL
Z 0
Third Alphabet
Letter Prefix (2) Suffix (4)
A 1111 4 OEEB CERE
B 1 1 D G
C 111111 6 JUDYQC PYNUMU
D 1 1 S D
E 111 3 EOD SST
F 11 2 FE SS
G 0
H 0
I 111111 6 JMSFQV MGUXRX
J 0
L 11111111111111 14 DGLSJSUEGYBBUY RMCSPYXQXEUVXL
M 1 1 S E
N 11 2 DT PX
O 1 1 O G
P 0
Q 1111111 7 EQSFHSE ETESCDT
R 1111 4 NQQJ CXEZ
S 0
T 1111 4 EEEY NSCS
U 0
V 11 2 SD TN
X 0
Y 111111 6 OPOOOE ESHMMG
Z 0
Fourth Alphabet
Letter Prefix (3) Suffix (5)
A 0
B 0
C 11111 5 LRAQT HHDDL
D 11 2 QD LU
E 11111111 8 MQAYQALR JICHCQCC
F 0
G 1111 4 BOIY UCIH
H 1 1 Y E
I 0
J 0
L 1 1 L T
M 11111 5 LIYYC UUUUU
N 111 3 TCV DZL
O 0
P 111 3 LCN DJU
Q 1 1 L M
R 111 3 LAI MNM
S 111111111 9 LEETQYFTF ODHCEVCII
T 1111 4 QQVE OOIG
U 1111 4 ICLC PDLY
V 1 1 L U
X 1111111 7 LILRILN PZBUBL
Y 11 2 LC DLJ
Z 1 1 R H
Fifth Alphabet
Letter Prefix (4) Suffix (6)
A 0
B 11 2 XX EA
C 1111111 7 GDDSDSD OOFFOEV
D 111111 6 SCPYNCU UEUUQTQ
E 11 2 SH TP
F 0
G 1 1 U T
H 111111 6 CCSDGZ EOOYYS
I 11111 5 EGTSS EYOMV
J 111 3 EPX UOP
L 111111 6 YDUCNX UDYQQU
M 111 3 RQR EJE
N 1 1 R D
O 111 3 STT ETF
P 11 2 UX FE
Q 1 1 E E
R 0
S 0
T 1 1 L S
U 1111111111 10 MMGPXMMVMD JDGUMYEBBD
V 1 1 S O
X 0
Y 1 1 U F
Z 11 2 XN EE
Sixth Alphabet
Letter Prefix (5) Suffix (1)
A 1 1 B X
B 11 2 UU RR
C 0
D 1111 4 UNLU ANSA
E 1111111111111 13 MHOIDPZZMBQUC RREOIQPNRIBQB
F 1111111 7 PCCJEOC NIIBMVT
G 1 1 U P
H 0
I 0
J 11 2 UM TT
L 0
M 11 2 UI TZ
N 0
O 111111111 9 HCJCHCVIG BLIBBIQYB
P 0
Q 1111 4 DDLL XTXX
R 0
S 11 2 HT BI
T 111 3 OED DDX
U 1111111 7 JDDDLUL ZTVAQZN
V 11 2 CI TI
X 1 0 I
Y 11111 5 IHLUH XDRRT
Z 0
We will now set down some of the determinations which can be made at
once from these frequency tables. Clearly several mixed alphabets have
been used. As was to be expected from the analysis of the recurring
groups, we note that the frequency tables for alphabets 2 and 6 are
of so nearly the same general form that certainly these two alphabets
are one and the same. If a Spanish word has been used as a key word,
this means that A is probably represented by a vowel in these two
alphabets and probably equals A or O, because these two letters are
such common finals in Spanish.
1st Alphabet. Probable vowels T, X; probable common consonants, B, I,
N, R. We conclude this because of the frequency of occurrence of T and
X and the variety of their prefixes and suffixes. On the other hand, B,
I, N, and R have for prefixes and suffixes, in a majority of cases, E,
F, O and S which are the probable vowels in the 2d and 6th alphabets.
2d and 6th Alphabets.--Probable vowels E, F, O, S; probable common
consonants, D, J, Q, U, Y.
3d Alphabet.--Probable vowels C, I, L; probable common consonants A,
Q, T, Y.
4th Alphabet.--Probable vowels, E, G, S, T; probable common consonants,
C, M, N, P, U, X.
5th Alphabet.--Probable vowels, D, L, U; probable common consonants,
C, H, I.
Now this cipher may have been made up from five distinct alphabets
with letters chosen at random but it is much more likely to have
been prepared with a cipher disk or equivalent, having the regular
alphabet on the fixed disk and the mixed alphabet on the movable
disk. An equivalent form of apparatus (not using the mixed alphabet
in question) is one like this:
| Fixed Alphabet of Text |
| ABCDEFGHIJLMNOPQRSTUVXYZABCDEFGHIJLMNOPQRSTUVXYZ |
+------------------------------------------------------+
| PCJVRQZBAODFSUTMXIYHLGEN |
| Movable Alphabet of Cipher |
+-------------------------------------------+
Here A of the plain text is enciphered by S and the other letters come
as they will. If we move the cipher alphabet one space to the left,
A will be enciphered by U and the whole sequence of the alphabet will
be changed.
We will therefore use some such form as the above and see if we can
insert our letters, as they are determined, in such a way as to have
each of the cipher slips identical. We may start thus:
ABCDEFGHIJLMNOPQRSTUVXYZABCDEFGHIJLMNOPQRSTUVXYZ
1st Alphabet t x
2d ol qei ms d c u
3d ol qei ms d c u
4th ol qei ms d c u
5th d c u ol qei ms
6th ol qei ms d c u
In the 1st alphabet, T and X are placed as A and E respectively on the
basis of frequency. In the 2d and 6th alphabets, O and E are placed as
A and E respectively on the basis of frequency. In the 4th alphabet,
E and S are placed as A and E, and in the 5th, D, U and L are placed
as A, E and O for the same reason. We now have an excess of E's and
a deficiency of A's, which will be corrected if, in the 3d alphabet,
we place L, I and C as A, E and O respectively. As a check, this
gives us TOLEDO as the key word.
In the second alphabet, O is four letters to the left of E; we may
place O four letters to the left of E in the fourth and it comes under
V. Note that in the fourth frequency table O (=V) does not occur. In
the same way in the fourth alphabet, S is four letters to the right
of E; placing it in the same position with respect to E in the second
and sixth, we have S under I. We have already noted that S probably
represents a vowel in these two alphabets. In this way, we may add
D and U to the third alphabet from their position in the fifth with
respect to L and we may add I and O to the fifth from their position
in the third with respect to L. In every case we check results from
the frequency tables and find nothing unlikely in the results.
Now in the second and sixth, let us try Q, D and U as D, N and R
respectively. We may add these letters to the third, fourth and
fifth alphabets by the method of observing the number of letters to
the right or left of some letter already fixed. We now add L to the
second, third, fourth and sixth from its position with reference to
D and U in the fifth. M is probably D in the fourth and we may add
it to each of the alphabets, except the first, in the same way. The
table is now complete as shown.
Let us try these letters on the first line of the message and see if
some other letters will be self-evident.
Alphabet 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6
Message D D L R M E R G L M U J T L L C H E R S L S O E E S M E J U
Deciphered _ N A _ U E _ _ A D E _ A B A L _ E _ I A E N E _ I G A _ R
Referring to our frequency tables as a check on suppositions, we find
everything agrees well enough if we assume the first line to read:
UNAFUERZA DE CABALLERIA ENEMIGA
We will now put the newly found letters in the table. The letters
previously found are in capitals and the new letters in small
letters. The addition of D (=U) to the first alphabet permits us
to add all the letters of the other alphabets to the first by the
methods already discussed. Each of the other letters may then be
added to every alphabet by these methods:
ABCDEFGHIJLMNOPQRSTUVXYZABCDEFGHIJLMNOPQRSTUVXYZ
1st T xhgoljqei msr d c u
2d t xhgOLjQEI MSr D C U
3d t xhgOLjQEI MSr D C U
4th t xhgOLjQEI MSr D C U
5th t xhgOLjQEI MSr D C U
6th t xhgOLjQEI MSr D C U
One alphabet checks another in this way and we find everything to
fit so far. We will decipher a few words more of the cipher message
by the above alphabets and see if we can determine some new letters.
Alphabet 5612345612345612345612345612345612345612345612345612345612
Message JUZJIMUDAEESDUTDBGUGPNRCHOBEQEIEOOACDEIOOGCOLJLPDUVMIGIYXQ
Deciphered PR_CEDEN_EDEARAU__UEZ_ILLA_ECASEHA_LAENAZUCAICA_AR_HEUS_ED
Again referring to the frequency tables the first word is
evidently PROCEDENTE. We have also HALLA and MARCHEUSTED. The
letter B may be determined from another cipher group, JFBSQDLD
(56123456) =POSICION. The letter N may be determined from BETNDQXUC
(123456123) =SERRADERO. The letters F and Y may be determined from
JCPJOISLYDUASIUPF (23456123456123456)= COMPANIA PARTIENDO. The
completed alphabets, arranged as before, are:
ABCDEFGHIJLMNOPQRSTUVXYZABCDEFGHIJLMNOPQRSTUVXYZ
1st TYVNXHGOLJQEIZMSRBADFCPU
2d TYVNXHGOLJQEIZMSRBADFCPU
3d TYVNXHGOLJQEIZMSRBADFCPU
4th TYVNXHGOLJQEIZMSRBADFCPU
5th TYVNXHGOLJQEIZMSRBADFCPU
6th TYVNXHGOLJQEIZMSRBADFCPU
The key word is TOLEDO and the completely deciphered message is:
"Una fuerza de caballeria enemiga procedente de Aranjuez y
Villaseca se halla en Azucaica. Marche usted con su compania
partiendo de la casa de la serradero por las alturas de lo este
y norte de Azucaica con el fin de reconocer su numero y clase
de fuerzas y en disposicion que se halla. (Q) Esta acantonada
(Q) Se hallan otras tropas detras de ella (Q). El resultado del
reconocimiento necesito saberlo dentro de tres horas y media
cuando mas. Pongo a sus ordenes un ciclista (X) Fin."
Special Solution for Case 7
When a short message is enciphered with a long key word, the methods
of analysis already discussed may fail; first, because there will
be no recurring pairs to indicate the number of alphabets used and,
second, because there will be so few letters in each alphabet that
the methods of Case 6 will not be easily applied.
However, if we know or correctly assume one word, preferably a fairly
long one, in the cipher text, a solution is very simple. For example,
the following message is believed to refer to reënforcements and to
contain that word.
YANZV ZNLPP KQFXI JBPWA
NRUQP EPLOM CCWHM I
Let us assume that REINFORCEMENTS is the first word and that it is
represented by the cipher group YANZVZNLPPKQFX. We may put the test in
this tabular form, using a cipher disk and a Larrabee cipher card to
determine the value of A for each letter under these two systems. Any
other alphabets suspected may be tried out at the same time.
If
Y A N Z V Z N L P P K Q F X
equals
R E I N F O R C E M E N T S
then, with cipher disk, A equals
P E R M A N E N T B O D Y P
and, in Vigenere cipher, A equals
H W F M Q L W J L D G D M F
It is evident that the guess as to the appearance of the word
REINFORCEMENTS was correct, that it is the first word of the message,
that the cipher disk was used in preparing the cipher and that the
key words are PERMANENT BODY.
This is, of course, an especially favorable case and we will take
one less favorable to show how this method can be applied.
Two Mexican chieftains, A and B, have been communicating with the
following cipher alphabet:
Plain text ABCDEFGHIJLMNOPQRSTUVXYZ
Cipher PCJVRQZBAODFSUTMXIYHLGEN
This alphabet has been determined from many radio messages from
A, the superior, to B, his sub-ordinate, who has a force of about
2,000 men near the border. A uses the form ORDENO QUE instead of the
more familiar MANDO QUE in all his messages giving orders to B. The
following message is received from A by B's radio station (and other
listening stations) and about an hour later there is a good deal of
noise and movement as if B's force were breaking camp.
IIHAH YDXRP EGQGV JJEEE HOBGV
GJCAG XAESA VVXLE IILHM PSQAG
BDGAV GSQAZ
This is a substitution cipher, but it is not Case 6 using the usual
alphabet of the communications from A to B and, in fact, is not Case
6 at all. The recurring pairs and triplets point to a key word of
ten letters and this would give us but six letters per alphabet if
it is Case 7.
The preparations for a move lead us to believe that A has given an
order to B and he has, in that case, probably used the expression
ORDENO QUE in the message. We will try the first nine letters of
the message as in the other example, first preparing a cipher disk
or equivalent sliding arrangement having on it the alphabet usually
used between these chieftains or A-B cipher.
| Fixed Cipher Alphabet |
| PCJVRQZBAODFSUTMXIYHLGENPCJVRQZBAODFSUTMXIYHLGEN |
+------------------------------------------------------+
| ABCDEFGHIJLMNOPQRSTUVXYZ |
| Sliding Plain Text Alphabet |
+------------------------------------------+
A-B Cipher
If I equals O then A equals R
I R C
H D X
A E R
H N B
Y O Q
Clearly there is nothing here and the assumed words, if they occur, are
in the middle of the message. We may jump to the combination PEGQGV at
once since the preceding letters do not make ORDENO QUE. We try this
without result and proceed to EGQGVJ, GQGVJJ, QGVJJE, GVJJEE, VJJEEE,
JJEEEH, JEEEHO, EEEHOB, EEHOBG, EHOBGV, HOBGVG, OBGVGJ, BGVGJC, GVGJCA,
VGJCAG, all without result. This work requires less time than might be
imagined and is the kind of work which can be divided among a number
of operators. Now let us come to the next combination GJCAGX. We add
the next three letters, AES, against QUE.
If
G J C A G X A E S
equals
O R D E N O Q U E
Then, in the A-B cipher, A equals
A D E R O V I V A
The key is found; VIVA_ADERO and a trial of M in the blank space
shows correct results. This checks with our theory that a ten letter
key word was used and deciphering the message we have:
PARA EL ATAQUE CONTRA TORREON ORDENO QUE SUS TROPAS MARCHEN ESTE
NOCHE X.
The reason for breaking camp is now evident.
This method may be used, with some labor, on short words like THE, AND,
etc. Parts of the key will appear whenever an assumed word is found
in the message and the whole key may be assembled if enough of the
parts are available. Even if only part of the key may be so recovered,
it will always lead to the ultimate solution of the cipher by trial
of the partially recovered key on the message letter by letter.
As an example of recovery of a key by use of short common words, let
us refer to the message of Case 7-a. There are twenty-four groups of
three letters each in this message and we will try them against THE,
ARE and YOU, assuming that the Vigenere cipher is used.
1 2 3 4 5 6 7 8 9 10 11 12
If OSB VOI GSW CYY ZSZ BVJ XLD OSY UVD YJL SQA HSI
equals THE THE THE THE THE THE THE THE THE THE THE THE
or ARE ARE ARE ARE ARE ARE ARE ARE ARE ARE ARE ARE
or YOU YOU YOU YOU YOU YOU YOU YOU YOU YOU YOU YOU
then A equals VLX CHE NLS JRU GLV IOF EEZ VLU BOZ FCH ZJW OLE
or OBX VXE GBS CHU ZBV BEF XUZ OBU UEZ YSH SZW HBE
or QEH XAO IEC EKE BEF DHP ZXJ QEE WHJ AVR UCG JEO
13 14 15 16 17 18 19 20 21 22 23 24
If BJV SFX DQB AII OHZ IVX JBF ESF JSC NLU CTW CSM
equals THE THE THE THE THE THE THE THE THE THE THE THE
or ARE ARE ARE ARE ARE ARE ARE ARE ARE ARE ARE ARE
or YOU YOU YOU YOU YOU YOU YOU YOU YOU YOU YOU YOU
then A equals ICR ZYT KJX HBE VAV POT QUB LLB QLY UEQ JMS JLI
or BSR SOT DZX ARE OQV IET JKB EBB JBY NUQ CCS CBI
or DVB URD FCH YUO QTF KHD LNL GEL LEI PXA EFC EES
In column 5, we have, for YOU, the key BEF; column 6 gives the same key
for ARE; column 10 gives the key FCH for THE and column 15 gives the
same key for YOU; column 12 gives the key HBE for ARE and column 16
gives the same key for THE; column 23 gives the key EFC for YOU. The
only possible key for the message is a five-letter one made up of the
letters BEFCH or EFCHB or FCHBE or CHBEF or HBEFC. If the key in this
case were a word, we would have no difficulty in determining it; as it
is, there is no real difficulty in the matter as we may now divide the
message into blocks of five letters and note that ZSZ (=YOU) form the
3d, 4th and 5th letters of a group. The corresponding key letters, BEF,
are then the 3d, 4th and 5th letters of the key which must be CHBEF.
This special solution for Case 7 depends so largely on the intuition
of the operator in choice of a word that it is not, in general,
advisable to use it unless the message is very short and the regular
methods of analysis have been tried unsuccessfully. It is, however, a
wonderfully short cut in difficult cases where the other methods fail.