(a) t . . . . i . i . a
(b) t h . n . i t i s a
(c) t h i n k i t i s a (Think it is a...)
Frequency Count on Alphabet 2, Only:
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
5 1 . 2 3 2 1 2 1 . . 3 2 3 5 . 1 . 1 . 2 . . 1 4 1
Presumably, during all of this time, we have been recording
substitutes in a key-frame. We have recovered fifteen of these in
alphabet 1, and also a number in alphabet 2. Those recovered from
alphabet 1 are shown at the top of Fig. 152. _O_ is the substitute
for _a_, _A_ is the substitute for _t_, _T_ is the substitute for
_s_, and _S_ is the substitute for _y_. Thus, if the cipher is
“Phillips,” then, in the original key-square, the two letters _A O_
were consecutive on one of the diagonals, the two letters _T A_ were
consecutive, the two letters _S T_, and the two letters _Y S_, so
that the complete diagonal must have been _Y S T A O_, and even
though the letter _o_ was not used at all in alphabet 1, we know that
its substitute must have been _Y_, since these diagonals may be
considered to begin with any one of the five letters. By beginning
at _c_-_H_, and following out another such chain, we may find another
complete diagonal, _C H K W D_; and, in addition, we may find parts
of diagonals. All of these are shown at (b).
Figure 152
An Alphabet No. 1 - Taken from a Key-Frame:
Plaintext: 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
SUBSTITUTES: O . H . B . . K R . W M X U . . . F T A . N D . S .
(a)
A T S Y Y
O A T S S
T
A
(b) O
*Y C E I L *V
S H B R M N
T K F *X U
A *W
O D
(c) (d) (e)
N D .│S│ S . N D . S I N D E . N D . S I N D E S
U C│.│T . T . U C B T R U C T . U C . T R U C B
.│H│L A H L A . . H L A F . L A . . H L A F . H
│.│K M O . K M O . . K M O . K M O . . K M O . .
│ │V W X Y . V W X Y . V W X Y V W X Y . V W X Y .
Whether or not we can go further than this, without consulting other
cipher alphabets, depends upon whether or not the original key-square
contained some of those alphabetical, or nearly alphabetical,
sequences which so often betray the poorly-mixed alphabet; usually,
these are most easily found toward the _X Y Z_ end of the normal
alphabet. In the given case (b), we are able to find the letters _V_,
_W_, _X_, and _Y_, each standing on a separate diagonal; thus, by
readjusting the beginning-points of their diagonals so as to place
these letters at the bottom, we are able to set together four of
these diagonals in the order shown at (c), leaving only the
part-diagonals _E B_ and _I R F_ still to be added. Their length will
show that _I R F_ belongs to the missing diagonal, but _E B_, by its
length, could belong to any one of three diagonals. Further
developments can be carried out with the rhomboid adjustment of (c),
or, if this is confusing, the conversion to a square can be made
immediately. The student may decide for himself which he prefers of
the two developments marked (d) and (e). Notice that this restoration
of the key-square can take place not only from a single alphabet, but
with only 15 substitutes known in that alphabet. But without the aid
of alphabetical sequences, we must, in the first place, have 20
substitutes, four for each diagonal, in order to recover the full
diagonals, after which, each one is entirely independent of the other
four, so that they cannot be adjusted and combined without consulting
one or more of the other alphabets. Here, the method varies a little,
according to just what we can recover, though a hasty glance at the
perfect case will serve to show the general path for all. To see this
as rapidly as possible, we will assume that we have recovered from
alphabet 1 the full five diagonals, and that, in alphabets 2, 3, and
4, we have discovered the substitutes for _e_, and also the originals
for which _E_ has been the substitute.
A careful consideration of the cipher itself will show that no letter
can have more than four different substitutes: the four letters in
the next column to its right which are not on the same line with
itself. Also, that no letter may act for more than four different
originals: the four letters in the next column to its left which are
not on the same line with itself. Any letter, in order to take all
four substitutes and act for all four originals in four successive
alphabets, must have started on the top line, which is the moving one.
Figure 153
Five Complete Diagonals Recovered From Any Alphabet No.1
Y C E I V
S H B R N
T K L F U
A W M P G
O D X Z Q
\ ↓
Y \ V │ │
S\I N│ E │ D E S I N
T\R│U C B │ U C B T R
A│F G H L │ F G H L A
│O\P Q K M│ O P Q K M
│ \Z W│X X Y Z V W
│ \ │D
↑
Now, considering Fig. 153: Our five recovered diagonals are imagined
to be those of a well-mixed square, so that we have no discoverable
sequences. It has been found that the letter _e_, in alphabets 1, 2,
3, and 4, has taken, successively, the substitutes _B_, _H_, _Q_, and
_Z_, and that the cipher-letter _E_ has served, successively, as the
substitute for _x_, _u_, _f_, and _o_. The four letters _B H Q Z_,
then, must all have stood in a single column _in exactly the order
named_; and the four letters _X U F O_ must have stood in another
column, two positions to the left of the first, but with a minor
difference in the order: some other letter (the one on the same line
as _E_) must have intervened between _X_ and _U_. The order in this
column, then, must have been _U F O X_. Our first step toward
combining the five diagonals is that of adjusting four of them so as
to set up the column _B H Q Z_. This automatically sets up four
letters of the other column _U F * X_ (_D_) — in the figure, the
_X D_ is present, but has not been adjusted to the _U F_ * — after
which, the fifth diagonal can be added to the others by placing the
_O_ of the column _U F O X_ (_D_). Now, since the letter _E_ has
taken, successively, all four of its substitutes and all four of its
originals, it must, in alphabet 1, have been standing on the top row.
Two parallel lines (if desired) can be ruled across the set-up to
show the top and bottom of the square, and two others (placed
anywhere, so long as they mark a width of five columns) may be ruled
to show the two sides. The outside letters, _Y_, _V_, and _D_, may
all be transferred to the opposite ends of their diagonals, after
which the rhomboid is easily adjusted to the form of a square.
It will be seen from what precedes that group-by-group encipherment
offers little, if anything, that is new, and no problems or theories
which the student could not figure out with what he has learned of
substitutions. Solving the actual cryptograms, of course, could
present some difficulties, according to the individual case.
Of the practice cryptograms to follow, No. 140 follows the plan of
the puzzlers, and should not prove very difficult in spite of its
brevity. No. 141 is also a “Phillips,” while No. 142 has been
enciphered with a mixed-alphabet slide applied to five-letter groups.
140. By NEMO. ("Phillips." Probable word: ASSOCIATION)
N D Q T F Q Z C N G B U Z H X N L U K Y F T E E W N R G U R M O X Q X
E Q Z L B G X H W F F N R P X P X V D D F I T G S E W R T I I T Z X E
R V W A R I S P E Y I G R Q C.
141. By PICCOLA. (This is the real McCoy - in 1938. But times change).
(a) K G E U H C K T S X P C K N C A D F X Q C B X T
(b) O U T U I U B F S B Q A P H N B Y Z X X L R U G
(c) O F U O S K H Q T K P W Q F E T B W W X P K B O K G H
(d) B L A M R P G X B W G C W K Q Z I A Q C U H Y R C
(e) G U C O S B B L P S B Q D K P G P K D S R C T B L I
(f) X R O S U I T T F G Y P C M C K F T F X O S R B H O A G M
(g) B O I B V B U K E E B D K B C O B Y W B T B M U H O O A B
(h) Y C U Y U T B I T F H S A N P H C W T.
142. By PICCOLA. (Direct examination - Snowball vs. Snowball).
(q) H F X L F M B L R N I N J W P Z Z G I S B B O Z X S F S H R
H T A T M R O F V ? (a) S X F U R R W X I Q S S.
(q) U F V H C N T I T T F O E J X O G N S G X U S O E H I V L X
E A T ? (a) U R Q Q T W E X U W I T O S.
(q) H C W R U Q U I T T F Y O Q I U D R S G X Z W H G F E T P C
J E M K Q F I N D O E E Z B L ? (a) U R Q Q T W.
(q) U R S U F U G J R E D V T O V E C Z X Z U Z G X S D S H C Q
K E X Z W I O V I R M H D W B D Z R R M ? (a) U R Q T D H T T H I
J S S.
(q) X C Q H R E M Z L T T O P A V H U L E B O Y O G N U F V X T
Z L E K S W F A V N ? (a) X R L L Q E W L T C O S P W V C Z L E B
D W Y I Z F I S R D T W C E Z T T A S G O L E.
143. By PICCOLA. (Can you guess what cipher? "Foregoing" refers to No. 141).
R N N G T R I O O H E I T T A F N D E N O G E L G E Y F I R D A I S E.
CHAPTER XX
Vigenère with Key-Progression
Before leaving the study of multiple-alphabet ciphers, we will
consider briefly the process which, in its simplest form, would be
that shown in Fig. 154. The initial key, in each of three examples,
is _A_, and a long key has been formed by causing the initial one to
progress in the normal alphabet according to an agreed index. In the
first example, the progression index is 1, in the second, it is 2,
and in the third, it is 25 (or minus 1). The resulting long key will
govern a period, which is 26 or 13, according to whether the
progression index is odd or even. This encipherment, logically, would
be applied with a cipher disk. The initial key, as _A_ of the
examples, would indicate the starting position of the revolving disk,
the first letter being enciphered with the disk in this initial
position, after which the disk is made to revolve, so many angles at
a time, without further reference to key-letters. For this kind of
cryptogram, the solution is purely mechanical. A series of alphabets
may be extended, with each cryptogram letter as a beginning, and the
message can be found following a diagonal path in the resulting
set-up.
Figure 154
Forms of Key-Progression
Keys: A B C D E F G... A C E G I K M... A Z Y X W V U...
S E N D... S E N D... S E N D...
S F P G... S G R J... S D L A...
This type of key-progression can be decrypted by "running down the
alphabet," and watching the diagonals for plaintext.
A much commoner scheme, when using a cipher disk, is that of following
a series of irregular shifts in accordance with a numerical key. If,
for instance, the initial position has been established and the first
letter enciphered in that position, and if the numerical key is
3-5-2-1-6, the disk will now be revolved 3 positions for encipherment
of the second letter, 5 positions for encipherment of the third
letter, 2 positions for encipherment of the fourth letter, and so on,
so that the disk must move 17 positions during encipherment of five
letters. This can produce a very long period indeed, especially when
the collective shifts result in an odd number.
Figure 155
Progressing Key: C U L P E P E R D V M Q F Q F S E W N R ...
Plaintext: T H E R E I S O T H E R C A U S E F O R ...
Partial Encipherment: V B P G . . . . W C Q H . . . . . . . I ...
Substantially the same encipherment as the foregoing can be had with
a slide and a key-word, as indicated in Fig. 155. The progression
index, in this figure, is 1. The preliminary key-word, CULPEPER,
enciphers the first eight letters, then moves forward in the alphabet
and becomes _D V M Q F Q F S_ for the encipherment of the next eight,
_E W N R G R G T_ for the encipherment of the third eight, and so on.
In its practical application, one column could be taken at a time.
Notice, however, in Fig. 156, that when a key-letter progresses in
the alphabet, the possible substitutes for any one letter will also
progress, and to exactly the same extent. If the encipherment is
Vigenère or Beaufort proper, this progression is in the same
alphabetical direction as that of key-letters, while the variant
encipherment causes the substitutes to progress in the contrary
direction. Probably, then, the most convenient method of application,
and the one least likely to result in errors, would be that of Fig.
157. The cryptogram is first enciphered as an ordinary periodic, and
the progression is added later, using group-by-group encipherment.
Thus, as we receive the cryptogram, our repeated _ther_ has been
enciphered once as _V B P G_, again as _W C Q H_, and possibly, later
on, as _A G U L_, and the only period we shall be able to find, using
the regular methods, will be 26 x 8, or, if the progression index is
an even number, 13 x 8. But notice, in the same figure, comparisons
(a) and (b).
Figure 156
Vigenère and Beaufort Progression Variant Progression
Progressing Key: A B C D E..... Progressing Key: A B C D E.....
Plaintext letter: H H H H H..... Plaintext Letter: H H H H H.....
Cipher letter (V) H I J K L..... Cipher letter: H G F E D.....
(B) T U V W X.....
Vigenère, it will be remembered, has been compared to the mathematical
process of addition. If the key-digram _CU_ be added to the plaintext
digram _TH_, their sum is the cipher-digram _VB_. The alphabetical
distance from _C_ to _U_ is 18, the alphabetical distance _TH_ is 14,
and the alphabetical distance _VB_ is 6 or could be 26 plus 6, 52 plus
6, and so on. It is a fact that when we “add” the two digrams _CU_
and _TH_, we actually do add their separating intervals, 18 and 14,
since we obtain a sum 32 in that of the cipher digram _VB_. It is
also an easily verified fact that the same reasoning applies to the
subtractions of the two Beauforts. The student who cares to
investigate may make use of the tableau shown as Fig. 158; to find
quickly the distance from one letter to another, find the first of
these at the left, the second at the top, and the alphabetical
interval between the two is shown in the cell of intersection. If it
is desired to know the reverse interval, find the first letter at the
top and the second at the side. Now notice, carefully, that when any
digram progresses in the alphabet, as _CU_ would become _DV_, _EW_,
_FX_, and so on, in a series of periods, _it does not change its
alphabetical interval_; in all of these digrams, the distance apart
of the two component letters is still 18. Thus, while our period
vanishes, the alphabetical intervals which represent it are still
present in the cryptogram; we have only to find these intervals,
subject them to a Kasiski examination, and convert the cryptogram to
an ordinary Vigenère.
Figure 157
Initial Key-word: C U L P E P E R C U L P E P E R C U L P E P E R
Plaintext: T H E R E I S O T H E R C A U S E F O R T H I N...
PRIMARY Cryptogram: V B P G I X W F V B P G G P Y J G Z Z G X W M E...
Progression Key: A B C
FINAL Cryptogram: V B P G I X W F W C Q H H Q Z K I B B I Z Y N G...
(a)
C U (key) plus T H (plaintext) equals V B (cipher).
(b)
Interval 18 plus interval 14 equals (32 less 26) = 6
Figure 158
Tableau for Finding ALPHABETICAL INTERVALS
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
A 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 A
B 25 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 B
C 24 25 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 C
D 23 24 25 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 D
E 22 23 24 25 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 E
F 21 22 23 24 25 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 F
G 20 21 22 23 24 25 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 G
H 19 20 21 22 23 24 25 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 H
I 18 19 20 21 22 23 24 25 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 I
J 17 18 19 20 21 22 23 24 25 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 J
K 16 17 18 19 20 21 22 23 24 25 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 K
L 15 16 17 18 19 20 21 22 23 24 25 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 L
M 14 15 16 17 18 19 20 21 22 23 24 25 0 1 2 3 4 5 6 7 8 9 10 11 12 13 M
N 13 14 15 16 17 18 19 20 21 22 23 24 25 0 1 2 3 4 5 6 7 8 9 10 11 12 N
O 12 13 14 15 16 17 18 19 20 21 22 23 24 25 0 1 2 3 4 5 6 7 8 9 10 11 O
P 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 0 1 2 3 4 5 6 7 8 9 10 P
Q 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 0 1 2 3 4 5 6 7 8 9 Q
R 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 0 1 2 3 4 5 6 7 8 R
S 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 0 1 2 3 4 5 6 7 S
T 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 0 1 2 3 4 5 6 T
U 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 0 1 2 3 4 5 U
V 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 0 1 2 3 4 V
W 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 0 1 2 3 W
X 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 0 1 2 X
Y 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 0 1 Y
Z 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 0 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 Z
Fig. 159 shows the preparation of the cryptogram: The alphabetical
interval from _K_ to _O_ is 4, that from _O_ to _S_ is 4, that from
_S_ to _X_ is 5, and so on. If these numbers are placed directly
below the first letter, as shown, the computation of their _lineal_
intervals apart is less confusing than when they are placed between
the two. As to repetitions, each repeated single number may represent
a repeated digram, each repeated sequence of two numbers may represent
a repeated trigram, and so on. Only the longer of these possibilities
have been underscored.
Figure 159
(5) (10) (15) (20)
K O S X M Y M M Q Y T K N G Z W L T Z L
4 4 5 15 12 14 0 4 8 21 17 3 19 19 23 15 8 6 12 22
(25) (30) (35) (40)
H C G F A P J Y K W A T Z P Q X U J Z P
21 4 25 21 15 20 15 12 12 4 19 6 16 1 7 23 15 16 16 6
(45) (50) (55) (60)
V C Z Q A R F P V Y U Y C R C X M X G I
7 23 17 10 17 14 10 6 3 22 4 4 15 11 21 15 11 9 2 21
(65) (70) (75) (80)
D U X Q Y M T E V V S C X J L J D A E Y
17 3 19 8 14 7 11 17 0 23 10 21 12 2 24 20 23 4 20 4
(85) (90) (95) (100)
C S F P J W F V J V Q V E G A N G K B B
16 13 10 20 13 9 16 14 12 21 5 9 2 20 13 19 4 17 0 17
(105) (110) (115) (120)
S C P Z B H G I D Z J A N Z I Y E Z P T
10 13 10 2 6 25 2 21 22 10 17 13 12 9 16 6 21 16 4 -
Figure 160
Repeated Intervals Lineal Interval Possible Factors
4-4 KOS-UYC 51 - 1 = 50 2 5 10
15-12 XMY-JYK 27 - 4 = 23
21-17-3-19 YTKNG-IDUXQ 60 - 10 = 50 2 5 10
23-15 ZWL-XUJ 36 - 15 = 21 3 7
21-15 FAP-CXM 55 - 24 = 31
7-23 QXU-VCZ 41 - 35 = 6 2 3 6
16-6 ZPV-IYE 115 - 39 = 76 2 4 19
17-10 ZQA-BSC 100 - 43 = 57 3 19
10-17 QAR-ZJA 110 - 44 = 66 2 3 6 11
15-11 CRC-XMX 56 - 53 = 3 3
9-2 XGI-VEG 92 - 58 = 34 2 17
2-21 GID-GID 107 - 59 = 48 2 3 4 6 8 12
17-0 EVV-KBB 98 - 68 = 30 2 3 5 6 10
13-10 SFP-CPZ 102 - 82 = 20 2 4 5 10
20-13 PJW-GAN 94 - 84 = 10 2 5 10
9-16 WFV-ZIY 114 - 86 = 28 2 4 7
Fig. 160 shows the application of a modified Kasiski examination.
Notice the prominence of small factors 2 and 3, caused, often, by
repeated alphabetical intervals in the key itself. In the given case,
the period 10 would probably be the choice, though period 5 is
correct; in practice, we should probably consider possible digrams as
well as longer sequences. Accepting period 10, we have still to learn
the progression index, and for this we must consider letters, all of
which are shown in the second column of the same figure. Taking the
longest repetition, most likely to be reliable, the two first letters
are _Y_ and _I_; their alphabetical distance apart is 10, and their
lineal distance apart in the cryptogram is 50. If the accepted period,
10, is correct, it has taken five periods to produce the alphabetical
shift of 10, therefore the shift per period (the progression index)
is 10 divided by 5; or 2. This, of course, has taken for granted that
the encipherment is either Vigenère or Beaufort. Considered as a
possible variant Beaufort, where the progression is backward, the
alphabetical interval from _Y_ to _I_ is 16, which is not divisible
by 5, the number of periods. But this progression might have covered
the entire alphabet and then included 16, or it might have covered
the alphabet twice, and so on, before including 16. We must make it
divisible by 5, adding 26, then another 26, and so on, until we
obtain a total progression of 120. This, divided by 5, gives the
progression index as “minus 24” — the same as a normal progression of
2. In Fig. 161, the cryptogram has been re-written into the accepted
period 10, and the figures in parentheses at the right of each group
will indicate the amount of alphabetical shift when the progression
index is 2. A constant progression of 2 per group would correspond to
the application of a Vigenère key _A C E G_. . . . . , so that the
Saint-Cyr slide will serve for quickly converting the cryptogram to
its periodic form, and this is shown in Fig. 162. The period, as
mentioned, is actually 5, though this makes no difference in the
final results.
Figure 161 Figure 162
K4 O4 S5 X15M12Y14M0 M4 Q8 Y21 (A) K O S X M Y M M Q Y
T17K3 N19G19Z23W15L8 T6 Z12L22 (2) (C) R I L E X U J R X J
H21C4 G25F21A15P20J15Y12K12W4 (4) (E) D Y C B W L F U G S
A19T6 Z16P1 Q7 X23U15J16Z16P6 (6) (G) U N T J K R O D T J
V7 C23Z17Q10A17R14F10P6 V3 Y22 (8) (I) N U R I S J X H N Q
U4 Y4 C15R11C21X15M11X9 G2 I21 (10) (K) K O S H S N C N W Y
D17U3 X19Q8 Y14M7 T11E17V0 V23 (12) (M) R I L E M A H S J J
S10C21X12J2 L24J20D23A4 E20Y4 (14) (O) E O J V X V P M Q K
C16S13F10P20J13W9 F16V14J12V21 (16) (Q) M C P Z T G P F T F
Q5 V9 E2 G20A13N19G4 K17B0 B17 (18) (S) Y D M O I V O S J J
S10C13P10Z2 B6 H25G2 I21D22Z10 (20) (U) Y I V F H N M O J F
J17A13N12Z9 I16Y6 E21Z16P4 T- (22) (W) N E R D M C I D T X
Figure 163
MATHEMATICAL FORMULA - C. H. PRICE
X = AD x P
LD
P = Period AD = Alphabetical Distance
X = Progression Index LD = Lineal Distance
As Applied to the Supposed Repeated Trigram KOS-UYC, Positions 1, 51:
X = 10 P = P
50 5
BUT: P and X must be integers • • If P = 5, then X = 1
(and P must be a divisor of 50) • If P = 10, then X = 2
(Periods of 25, 50, are unlikely)
For those who like mathematics, Fig. 163 shows a method used by one
of our collaborators for determining both the period and the
progression index directly from the cryptogram. Price also preferred
to find alphabetical intervals by writing the normal alphabet into a
block, five letters to the line, with _Z_ standing alone on the last
line; thus, except for watching _Z_ occasionally, the distance from
one letter to another could be counted by fives. It is understood, of
course, that we do not accept the evidence obtained from only one of
the supposed repeated sequences; too many of these will be accidental,
and many of those which are actually periodic have not represented
repeated digrams, but merely repeated intervals. Naturally, too, the
progression index need not be a small number; the disk encipherment,
mentioned in the beginning, showed a progression of 17 for each period
5. This disk encipherment, incidentally, has been dealt with in a most
interesting manner in Givierge’s _Cours de cryptographie_.
* * *
We have seen, then, all of the essentials of polyalphabetical
encipherment. With the cipher alphabets known to the decryptor,
practically all of the multiple-alphabet ciphers will be solved by
suitable modifications of processes described for Vigenère. When
alphabets are not known, his problem, always, is that of collecting
as many as possible of the substitutes belonging to each alphabet,
so that he may determine both the order of the letters and the
relationship of alphabets to one another.
144. By THE SQUIRE.
S O V F O G S G U F V I J R I F M O U I C F T T I K Z Y Z Z Z U I F Q
Q L O W U V A F J F I W W L N C R G J F E M V V N N C D H W T A J N W
A R D B. yallyyayyayalyyaaallayaaaylalllaylyayyallyyayaylalllyyall.
145. By NEMO.
Y Y I Z C U O F Y V H Q Y H T B E B S X P T S Y C R M R X L X E A G U
Y L P U Q B U U Q N Y U S O Q M O O S P U G I J I I F F F A L I R G G
F G E H H N T E G Y Z S M C O F U D E M X O G I K K V B N K W K P Q X
M G D L A I F N H M X T U M E Z X Y Z G N A P D W C D M N C T T H N J F D.
146. By PICCOLA. (We wouldn't throw monkey-wrenches for anything!)
X E I T B B B B V M X R S J P Y L K E N Y K S Z K F R W L G S A Y E A
V I X I X D U V D U R J G E I B A N Z F H D C C Y C O Y R V A B K W B
R H F K K F X S E J Y T F N L R N I V K V K Q H I Q H I J L P G O U J
V F C F T S H L I D V D D M P.
147. By DAN SURR. (Might try this without bothering about its progression!)
E C G M H T Y T A J B T H N G A W K L I B E M N R H T D G N G P D A O
Q A X R P Z P F H D D X I E A U B S Y C I X C W V R H P B O I X Y P Y
D V W N R N X O O K K I H F O X D S V L V W W C L I H Z H V W R L H W
M M I E E A H G Q Y R S R L K L W Z T J A Y W F N S S U C V Z L P X P
S E E E Y R T H D H T Z N U P U R M G K Z N T Y E Q D E Z E N N H W M
I N R L P S S W P Y M C R U B J Z Y C R N L M A S M E U C L R M D Y R
N E S T O B V J E U D V L O T S Q B J H B N R L B V D X J P X N I G F
I C Q J Y Q Z X Q G K B L F Q U B Q K N E L S S L Y G T L F L T D Z Z
Y K E E R H K L W L I M R N J S O O J P Q C A U D M E I B B Q X A H C
V A J C M G X B I C D K V C L G Q I B S C F V F W Q N A X I D R Z S X
R B I W R C Q R.
148. By PICCOLA. (When is a tramp not a tramp?)
E R N I C D M R A S T A A P H T P I L T Q V A A S N E A E E R O O L R.
CHAPTER XXI
Polygram Substitution — The Playfair Cipher
Polygram substitution contemplates the encipherment of several letters
collectively: Digrams are to be replaced with other digrams, or with
three-digit numbers; trigrams are to be replaced with other trigrams,
or with four-digit numbers; and so on, the substitute for an
individual letter being entirely dependent upon the combination in
which it happens to occur. Many devices have been contrived for
accomplishing this. For pair-encipherment, tableaux of the general
kind shown in Fig. 164 are fairly common.
Figure 164
A M E R I C N B D F G H J K L O P Q S T U V W X Y Z
E AA BA CA DA EA FA GA HA IA JA KA LA MA NA OA PA QA RA SA TA UA VA WA XA YA ZA
Q AB BB CB DB EB FB GB HB IB JB KB LB MB NB OB PB QB RB SB TB UB VB WB XB YB ZB
U AC BC CC DC EC FC GC HC IC JC KC LC MC NC OC PC QC RC SC TC UC VC WC XC YC ZC
A AD BD CD DD ED FD GD HD ID JD KD LD MD ND OD PD QD RD SD TD UD VD WD XD YD ZD
L AE BE CE DE EE FE GE HE IE JE KE LE ME NE OE PE QE RE SE TE UE VE WE XE YE ZE
I AF BF CF DF EF FF GF HF IF JF KF LF MF NF OF PF QF RF SF TF UF VF WF XF YF ZF
T AG BG CG DG EG FG GG HG IG JG KG LG MG NG OG PG QG RG SG TG UG VG WG XG YG ZG
Y AH BH CH DH EH FH GH HH IH JH KH LH MH NH OH PH QH RH SH TH UH VH WH XH YH ZH
B AI BI CI DI EI FI GI HI II JI KI LI MI NI OI PI QI RI SI TI UI VI WI XI YI ZI
C AJ BJ CJ DJ EJ FJ GJ HJ IJ JJ KJ LJ MJ NJ OJ PJ QJ RJ SJ TJ UJ VJ WJ XJ YJ ZJ
D AK BK CK DK EK FK GK HK IK JK KK LK MK NK OK PK QK RK SK TK UK VK WK XK YK ZK
F AL BL CL DL EL FL GL HL IL JL KL LL ML NL OL PL QL RL SL TL UL VL WL XL YL ZL
G AM BM CM DM EM FM GM HM IM JM KM LM MM NM OM PM QM RM SM TM UM VM WM XM YM ZM
H AN BN CN DN EN FN GN HN IN JN KN LN MN NN ON PN QN RN SN TN UN VN WN XN YN ZN
J AO BO CO DO EO FO GO HO IO JO KO LO MO NO OO PO QO RO SO TO UO VO WO XO YO ZO
K AP BP CP DP EP FP GP HP IP JP KP LP MP NP OP PP QP RP SP TP UP VP WP XP YP ZP
M AQ BQ CQ DQ EQ FQ GQ HQ IQ JQ KQ LQ MQ NQ OQ PQ QQ RQ SQ TQ UQ VQ WQ XQ YQ ZQ
N AR BR CR DR ER FR GR HR IR JR KR LR MR NR OR PR QR RR SR TR UR VR WR XR YR ZR
O AS BS CS DS ES FS GS HS IS JS KS LS MS NS OS PS QS RS SS TS US VS WS XS YS ZS
P AT BT CT DT ET FT GT HT IT JT KT LT MT NT OT PT QT RT ST TT UT VT WT XT YT ZT
R AU BU CU DU EU FU GU HU IU JU KU LU MU NU OU PU QU RU SU TU UU VU WU XU YU ZU
S AV BV CV DV EV FV GV HV IV JV KV LV MV NV OV PV QV RV SV TV UV VV WV XV YV ZV
V AW BW CW DW EW FW GW HW IW JW KW LW MW NW OW PW QW RW SW TW UW VW WW XW YW ZW
W AX BX CX DX EX FX GX HX IX JX KX LX MX NX OX PX QX RX SX TX UX VX WX XX YX ZX
X AY BY CY DY EY FY GY HY IY JY KY LY MY NY OY PY QY RY SY TY UY VY WY XY YY ZY
Z AZ BZ CZ DZ EZ FZ GZ HZ IZ JZ KZ LZ MZ NZ OZ PZ QZ RZ SZ TZ UZ VZ WZ XZ YZ ZZ
The tableau proper includes a full list of the 676 possible two-letter
combinations, while two external alphabets will furnish another
possible 676 two-letter combinations. With the plaintext marked off
into pairs, the encipherment of a pair is usually accomplished by
finding its two letters in the two external alphabets, where they act
as co-ordinates, and replacing this pair with the one which is found
at the cell of intersection. Thus, using the tableau of the figure,
and the order row-column, the substitute for _th_ would be _LG_; or,
using the order column-row, _TN_. With a tableau like that of the
figure (notice the straight unshifted alphabets), it is also possible
to encipher by what is ordinarily considered the decipherment process,
finding the plaintext pair inside the tableau and replacing it with
the two co-ordinates. But many of these tableaux are filled in a
thoroughly haphazard manner, and when this is the case, only the
ordinary encipherment plan is really feasible; in fact, the decipherer
has trouble in finding his cryptogram pairs, and it is usually
necessary that a second tableau be prepared especially for
decipherment purposes. On the other hand, it is very easy to construct
a mixed tableau in such a way that all of its encipherment is
reciprocal, and in this case there is no need for a second tableau,
since encipherment and decipherment are the same process. In most
forms of tableau, one or both of the external alphabets may be made
to slide and for the most part external alphabets are readily
changeable. But the tableaux themselves will have to be of more or
less fixed nature. Those which are safest are least readily
reconstructed from memory, and even those most easily remembered are
not set up very rapidly.
Another common method for pair encipherment can be understood from
the following description: Picture a chart of 100 cells which is
divided sharply into four quarters; that is, having much the
appearance of a 100-cell Fleissner grille. Each of the four quarters
is a 5 x 5 square and contains a 25-letter alphabet. At least two of
these alphabets are mixed, usually with different keys. To encipher
a pair, find its two letters in two different squares, and substitute
two others which occupy certain relative positions in the other two
squares.
Figure 165
The "SLIDEFAIR" Cipher -(H. F. GAINES)
Key: H E R C U L E S
Plaintext: SE ND DI AM ON DS TO AM
CIPHER:..... XZ ZR RU KC TI HO KX US
ST ER DA MM ON DA YW EE
..... MZ NI JU KO TI PO SC MW
KW IT HO UT FA IL
..... PR PM XY RW GZ AT
Cryptogram taken off:
X Z Z R R U K C T I H O K X U , etc.
The writer’s own contribution, accomplished with a slide, may be
examined in Fig. 165; the slide used in the example was the Saint-Cyr,
and details are self-explanatory, except for the method of enciphering
pairs, which was as follows: To encipher those of the _H_-column,
bring the _H_-alphabet into position on the slide; then, for each
pair, find its first letter in the upper alphabet and its second
letter in the lower one; imagine these to be standing at the two ends
of a diagonal, and substitute the two letters from the two ends of
the corresponding cross-diagonal, taking the upper one first. Where
the two plaintext letters happen to coincide (as would be the case
with a pair _EL_ using the _H_-alphabet of the Saint-Cyr slide) use
the two letters which have also coincided immediately to their right
(as _FM_ for _EL_ in the given case).
Occasionally, such a tableau as that of Fig. 164 is made to serve
(not very successfully) for trigram encipherment. A third external
alphabet is added beside one of the others, so that the two which are
parallel will make provision for the encipherment of a third letter
simultaneously with the encipherment of each pair. But for trigram
encipherment, another type of tableau is commoner: The tableau proper
is not a list of pairs, but an alphabet square such as could be
prepared for one of the mixed-alphabet slides of Chapter XVIII, and
is accompanied by four external alphabets, two on the left and two
across the top. The exact details of construction are not always the
same, and the methods prescribed for using such a tableau are
sometimes quite devious, but results are fairly uniform: We obtain
cryptograms in which enciphered pairs have alternated with enciphered
(sometimes _not_ enciphered!) single letters.
In all of the foregoing, no system has been mentioned which the
student will not be able to analyze for himself. At worst, he has a
case very much like simple substitution, except that he would require
a great deal more material, and would use a digram chart instead of a
frequency list. Otherwise, he usually finds that he has merely a
variation of pure periodic encipherment. There are, of course, more
effective methods. Trigrams, tetragrams, and other polygrams, found
in alphabet squares, can always be considered as geometrical figures,
and replaced with other series of letters in related geometrical
figures; and Lester S. Hill, in the American Mathematical Monthly,
has described an “algebraic” method which appears practical, provided
a machine is used. These methods, however, even granting that some
are practical and not too unwieldy for use, are entirely beyond our
present scope, and we will spend our few remaining pages on a cipher
of a less cumbersome nature and presenting far more points of interest.
Figure 166
The PLAYFAIR Cipher
KEY Encipherment Some Equivalent Keysquares
C U L P E (1) bl = IB L P E C U N O Q S T Y Z V W X
R A B D F ez = FE B D F R A V W X Y Z P E C U L
G H I K M (2) cl = UP I K M G H C U L P E D F R A B
N O Q S T ce = UC Q S T N O R A B D F K M G H I
V W X Y Z (3) th = OM X Y Z V W G H I K M S T N O Q
ht = MO
Example:
SE ND DI AM ON DS TO AM ST ER DA MX MO ND AY WE EK WI TH OU TF AI LX.
TP SR BK FH QO KY NQ FH TN CF FB IZ HT SR DW ZU PM XH OM WA ZM BH BL.
* * *
_The Playfair cipher_, which may be examined in Fig. 166, requires
no apparatus other than pencil and paper. Its key is the usual
5 x 5 square, based on a key-word, and filled in by any agreed plan
(preferably not by straight horizontals). For encipherment, the
plaintext is marked off into pairs, and these pairs are enciphered
according to three very simple rules:
1. If the two letters of the pair are found in the same column in
the key-square, replace each letter with the one directly beneath it;
and if one letter stands at the bottom of the column, use the one
standing at the top of the same column. With the key of the figure,
_ha_ becomes _OH_; _wa_ becomes _UH_.
2. If the two letters of the pair are found in the same row in the
key-square, replace each letter with the one immediately to its right;
and if one letter stands at the extreme right end of the row, use the
one standing at the extreme left end of the same row (_os_ becomes
_QT_; _st_ becomes _TN_).
3. If the two letters of the pair have a diagonal relationship in the
key-square (and these are usually in the majority), consider them to
be standing at the diagonally opposite corners of an imaginary small
rectangle, and substitute for each letter that letter of the other
diagonal which stands on the same row with itself (_bu_ becomes _AL_,
not _LA_). The decipherment rules, as usual, are the same rules in
reverse.
* * *
Notice that this encipherment is _cyclical_. So long as the order
1-2-3-4-5 is maintained in both columns and rows, it makes no
difference whatever how many columns are transferred from one side
to the other, or how many rows are transferred from top to bottom.
This may be investigated in the three equivalent squares of the
figure. Notice, too, that our three rules do not make any provision
for the case in which the two letters of a pair are the same. If,
in marking off the plaintext into pairs, we encounter a pair which
is a double, it becomes necessary to dispose of this, usually by
inserting a null which will throw the second letter into the next
pair. Occasionally we find a sequence such as LESS SEVEN, in which
it is necessary to do this twice in succession: _LE Sx Sx SE VE
Nx_. An unpaired final letter also requires a null (unless left
unenciphered), and when five-letter groups are to be used, it often
becomes necessary to complete a final group by adding nulls.
The foregoing description and rules are those of the original
Playfair cipher. Many encipherers, however, will vary the rules,
especially the one concerning doubles; perhaps one letter will be
omitted or replaced with a null; sometimes one double is replaced
with another; occasionally an encipherer will separate every doubled
letter in the message whether or not this is necessary. We meet,
too, with variant forms. A 24-letter alphabet will be used in a
4 x 6 rectangle, or a 27-letter alphabet (with character &) will
be used in a 3 x 9 rectangle. One variation, attributed to W. W.
Rouse-Ball, uses the standard key-square with the standard rule 3,
but varies the two rules for lineal encipherment. Rule 1: If the two
letters of the pair stand in a column, use the two letters immediately
to their right. Rule 2: If they stand in a row, use the two letters
immediately beneath them. In all of these cases, presuming the method
to be known, the degree of difficulty would be the same as if the
standard system had been used; otherwise, it is only necessary to
keep in mind the fact that variations occasionally occur. We will
give our attention, then, to the standard encipherment. But before
entering into the subject of decryptment, let us look carefully at
the system itself.
Primarily, we have a _fixed_ substitution. No plaintext pair ever has
more than one substitute pair; and no substitute pair ever changes
its original. We might say that the Playfair is, in effect, a “simple
substitution” based on an “alphabet” of 600 pairs; and, just as in
simple substitution proper, the Playfair cryptograms will very often
contain long repeated sequences which represent whole words. Again,
the reversal of a plaintext pair means the reversal of its substitute
pair, and vice versa, so that the discovery of any one equation (as
_th_ = _OM_) always means the discovery of another (as _ht_ = _MO_);
and if, in addition, the encipherment was a rectangular one (rule 3),
we obtain also the two reciprocal equations (as _om_ = _TH_, and _mo_
= _HT_). The two lineal encipherments, however (rules 1 and 2), are
not reciprocal. But notice particularly that, in spite of the polygram
theory, each letter has its individual substitutes. No letter in the
key-square may have more than five of these; the four which are
standing on its own line, and the one which stands directly beneath
it. It may be learned, too, by writing out the 24 (or 48) possible
pairs for any one given letter, that the letter standing immediately
to its right in the key-square is twice as likely as any one of the
other four to act as its substitute; and, further than this, that any
letter which is paired with it will be limited to eight possible
substitutes, all of which must be found either in the column or on
the row of the letter itself. To clarify this important point, let us
assume that the letter in question is _E_, and that the key-square is
that of Fig. 166. The letter _E_ may have only the substitutes _C_,
_U_, _L_, _P_, and _F_, with _C_ twice as likely to be used as any
one of the other four. Any letter which is paired with _E_ must take
one of the following substitutes: _U L P E F M T Z_.
Naturally, then, those letters which, in the key-square, are standing
on the same row or in the same column with the normally frequent
letters will have high frequencies in the cryptograms; in fact, the
two or three which predominate in a given cryptogram will practically
always be letters which, in the key-square, were standing in the same
row or column with _E_ or _T_ (in English). Moreover, if any letter
has been identified once as the substitute for _E_, there is a most
excellent chance that it can be identified again as the substitute
for _E_. Say, for instance, that _CF_ has been identified, or assumed,
as the substitute for _er_. This means that _C_ is individually the
substitute for _e_, and when another pair _CT_ is found to be of some
frequency, it can be tried as the substitute for _en_, _es_, _et_,
and so on. Single-letter frequencies, then, will play an important
part in the decryptment of the Playfair. But the process will rest
fundamentally upon the frequencies of digrams, and will follow, in
general, three steps repeated over and over in the same rotation:
1. Certain pairs are identified, or assumed, as the substitutes for
certain digrams.
2. These pairs and their supposed originals are set together in such
a way as to start the reconstruction of the key-square.
3. Substitutions are made on the cryptogram and further pairs are
identified.
When probable words exist, the work of solution becomes more or less
mechanical, as we shall see. At worst, we may begin at the beginning
of the cryptogram and work straight through until we find the word.
But very often, a really probable word is repeated, and even repeated
more than once. In the latter case, we are sure to find the long
repeated sequence in the cryptogram; while a word repeated only once
may have been divided into two different sets of pairs, as:
_ex_-_ec_-_ut_-_io_-_n_ and _e_-_xe_-_cu_-_ti_-_on_. But notice, here,
what the two encipherments would be, using our key-square of the
figure: _LZ CU EO HQ x_ and _x ZL UL QM QO_. These two sequences have
five letters in common, _L Z U O Q_, and, in addition, when considered
together, show the letters _E C U O_ of the word “execution.” This
does not invariably happen, but is far from uncommon. Nor is the word
“execution” the only one which produces reversal (_ex_ in one sequence,
_xe_ in the other). Then, too, there are many words like “commission”
which, regardless of the point at which the division begins, will
always end in the same set of pairs: _mi_-_s?_-_si_-_on_.
Granting an absence of probable words, the difficulties of solution
are almost entirely dependent upon the amount of material available.
A pair-count will be made in the usual chart-form (but only on the
divided pairs, and not “straddling” from one pair to another), and
pairs will be identified by frequency, by the frequency with which
they are found reversed, by the possibility of their letter-combinations
in a key-square, and so on. We will not attempt, here, to go into a
detailed demonstration, since every case is individual in its details,
and success, in all of them, is dependent largely upon the decryptor’s
own persistence. But in order to see sketchily what some of the
routine might be, we will make use of the very short example shown
in Fig. 167.
In the usual case, there has been a preliminary frequency count on
single letters in order to find out what the cipher is. The appearance
of this frequency count has more or less negatived the possibility of
simple substitution, and the next step has been a Kasiski tabulation
in the hope of finding a period. This tabulation, in any pair-system,
will bring out a predominant factor 2, and, since many of the supposed
digram systems actually do produce periods, the two supposed alphabets
would have been examined for that possibility. But pair-systems, as a
rule, will leave a wide-open trail: Repeated sequences, in the
majority of cases, will include an _even_ number of letters (that is,
an exact number of pairs), and will begin largely at the _odd_ serial
positions (that is, at the beginnings of pairs). The Playfair shows
this a little less distinctly than some of the others, because of the
fact that substitutes for single letters are so limited in number.
It is sometimes said of the Playfair that it can be distinguished
from other ciphers by (1) the fact that cryptograms contain an even
number of letters, (2) the fact that only 25 letters are represented
in its general frequency count, (3) the fact that when the cryptogram
is marked into pairs, no pair will be a doubled letter, and (4) the
presence of long repeated sequences at irregular intervals. As
conclusive evidence, these are debatable points, but all are good
supporting evidence, provided a proper confession can be extracted
from the pair-chart: (5) When the cryptogram has been marked off into
pairs, and the pairs counted, the result should bear much resemblance
to a count made on the same number of normal digrams. Even on an
extremely long cryptogram, over half of the cells will be blank,
since a normal text never uses more than about 300 of the possible
676 combinations; there will be a certain group of predominant pairs
followed by a group of moderate frequencies; and, with any appreciable
length, there will be a generous sprinkling of reversals. In preparing
the cryptogram, a great deal of convenience may be had by placing
frequency figures beside their digrams, by marking long repeated
sequences, noticeable reversals, and so on; and many persons like to
list the most prominent pairs and the most prominent reversals.
Figure 167
HR5 KY3 LD ZX NQ2 EO ND EC TC TI2 AD CT AK RH LB2 GT SN AN
UN2 ON DR HX PE BN ZC DT KV EQ2 HD AO HR5 DU RP TQ OB DE2
QD2 HR5 KY3 YA2 HZ HB BU KZ EQ2 XG TI2 BI KY3 RI CQ HR5 CE CO
SX RM BC TH CG QD2 RK NQ2 IT DC WT FV2 UB2 YA2 GU HE CZ NU2
LB2 IQ2 YK FV2 UB2 IQ2 WD QB UN2 KM DE2 TD KA HR5 NU2 OU
Frequency Count - Rearranged:
D H B C N Q R T K U E A I O Y Z G X L V F M P S W J
14 12 11 11 11 11 11 11 10 10 9 7 7 6 6 5 4 4 3 3 2 2 2 2 2 -
List of REVERSALS "Chart of Probable Position"
HR 5 - RH 1 EC - CE E T/ D H B C N Q R K U
KY 3 - YK 1 TC - CT D H
UN 2 - NU 2 AK - KA B C N Q R
UB 2 - BU 1 ZC - CZ K U
TI 2 - IT 1 DT - TD
The Playfair has also a rather characteristic frequency count. Notice,
in the figure, where the general count has been rearranged in the
order of decreasing frequencies, that the gradation from high to low
is somewhat less even than in a periodic; frequency 8, for instance,
is skipped altogether, and we have a sort of modified high-frequency
group. Sometimes we find from one to three letters of great prominence
before the downward gradation begins.
Concerning the “chart of probable position,” most solvers prefer
simply to keep this in mind, while others will actually set it down
and make it the basis of their solution. With 176 letters of text,
the average frequency of letters is about 7 (176 divided by 25). Any
letter whose frequency is above that average is very likely to have
been standing on the same row or in the same column of the key-square
as _E_ or _T_, and the two or three which lead the list are
practically sure to have been substitutes for one of these two
letters.
With cryptograms of the present length, or even with those of 400 to
600 letters, it is very uncertain as to whether or not the leading
pair will represent _th_, or the leading reversal _er_-_re_. Here, in
fact, we have no reversal of a definitely frequent character, and our
one prominent pair, _HR_, might just as well represent _st_, _at_,
_it_, _on_, _re_, _se_, or any other normally frequent digram capable
of being used at the beginning of a sentence. Presuming, however, that
it might represent _th_, we know that this digram is followed almost
altogether by vowels, and is followed with remarkable frequency by
_e_ and _a_; we know also that letters have individual substitutes.
Thus, we might begin solution by listing (or noting) those pairs
which, in the cryptogram, have followed the supposed _th_: _KY_,
_DU_, _KY_, _CE_, _NU_, assuming that their first letters, _K_, _D_,
_C_, _N_, have probably represented vowels, and that, of these, _D_,
_C_, and _N_, which rank high in the list of single-letter
frequencies, are very likely to have represented _e_. We may attempt
to identify these five pairs by working down the list of normal
digrams, taking only those of _v_-_c_ formation. If, in addition, it
is assumed that the key-square has been filled by straight
horizontals, certain assumptions can be made through possible
alphabetical sequence; for instance, the _U_ of _DU_ and _NU_ may
have stood on the same line with _R_ _S_ _T_ (_U_). There is a
further field for suggestions to be found in _patterns_, such as
_TI BI_, in which the two _I_’s could represent the same letter.
And where the square is filled by straight horizontals, it is
often possible to identify such a sequence as _HZ_ _HB_ as a “split
double,” since the null used in these cases is often _X_, and _Z_
may well represent _X_ by alphabetical position, It is even possible
to guess here a doubled _L_, since _H_ and _L_ are not far apart in
the alphabet. (It may be, of course, that the two _H_’s represent two
different letters.) The foregoing, then, has indicated the general
path. If the student desires to follow out a detailed demonstration
made on a cryptogram of only moderate length, a most excellent
exposition can be found in the appendix to the Macbeth translation of
Langie’s “Cryptography” (Dutton). It was written by Lt. Commander
W. W. Smith of the U. S. Navy, and generally speaking, attacks the
identifications of pairs as follows:
Having placed frequency figures beside their digrams, find those
points at which two pairs of high frequency are consecutive (not
necessarily a repeated sequence), and attempt to identify these
tetragrams as frequent tetragrams of the language: _ther_, _ered_,
_ened_, _tion_, _atio_, _ment_, _beca_, and so on. We have one here,
provided a frequency of 3 can be considered important: _HR_ _KY_.
Since this happens also to be repeated, it probably represents a word,
as _that_, _this_, _they_.
Another good demonstration, provided the student has access to it in
his public library, is found in Colonel Parker Hitt’s “Manual for the
Solution of Military Ciphers.” This manual is an elementary work
intended for the preliminary instruction of soldiers, and the attack
is made on the assumption of a key-square filled by straight
horizontals. With a square of the kind we are using, most of the
vowels and high-frequency letters will be standing on the upper two
rows, and letters on the first two or three rows will have a much
higher frequency than those of the last two or three. In fact, it can
often be detected that the letters _V W X Y Z_ were standing on the
bottom row as an intact alphabetical sequence, for the simple reason
that they have no frequency in the cryptogram.
Colonel Hitt’s demonstration begins with the usual pair-count, made
on a chart. He selects from this chart the (approximately) ten letters
having the _widest variety of contact_, including, if necessary, the
vowel or so which would have to be present in a key-word; and these
letters are assumed to have stood on the upper rows of the key-square.
The remaining (approximately) fifteen letters are then set up in
their alphabetical sequence and are assumed to have stood on the
lower rows in about that order. They are not, of course, known to be
correctly placed; the set-up merely gives a concrete idea as to where
letters ought to have stood. Then, following the military case of
abundant material, it is assumed that the leading pair will represent
_th_ (sure to be followed often by _e_), or, if _th_ is not the
leader, then _he_ (sure to be preceded often by _t_). With a few
obvious identifications made in the usual way, letters begin to
arrange themselves on the upper rows, and a gradual adjustment takes
place which corrects the few wrong assumptions of the lower rows, so
that the key-square is restored far in advance of solution. When a
short key-word has been used, it is not impossible, by following
Colonel Hitt’s suggestions, to pick out all of the key-letters, guess
the word, and decipher with the key-square. Other demonstrations,
based, respectively, on French and Italian language characteristics,
can be found in General Givierge’s _Cours de cryptographie_ and in
General Sacco’s _Manuale di crittografia_. (In the French work, the
cipher is referred to simply as “orthogonal and diagonal substitution.”)
* * *
It will be seen from the foregoing that the initial difficulty lies
in the correct identification of the first few pairs, and this, in a
short cryptogram, is no small difficulty. By whatever means it is
found possible to make these first tentative identifications, the
operation which is to admit or disprove their correctness is step
No. 2, in which we set them up as equations and then attempt to
replace them into their connected relationships in the key-square. If
this cannot be done, they cannot be correct; and, on the other hand,
it would be an extremely rare case indeed in which we could combine
as many as five or six such equations into one framework and then
find them incorrectly matched. To understand “equations,” suppose we
look at Fig. 168.
Figure 168
POSSIBLE RELATIONSHIPS
(a) A 3-letter Equation (b) A 4-letter Equation
Vertical Horizontal Vertical Horizontal Rectangular
I
T K I . K
H T H R . I K . S Y . .
R S Y . S
Y
(c) Impossible Equations: (d) Possible Equations:
co nd em na ti on -c on de mn at io n-
EO ND EC TC TI AD EO ND EC TC TI AD CT
Assuming that the beginning pairs of our cryptogram, _HR KY_,
represent the word _this_, we have two equations, _HR_ = _th_, and
_KY_ = _is_. The first of these has only three different letters,
since _H_ is common to both members, while the second has four
different letters. With the first case (a), one of the lineal
encipherments must have been used, and the common letter, _H_, must
have stood between the other two, with its plaintext partner coming
first and its cryptogram partner coming last. We do not know whether
these three letters stood in a column or in a row, but we do know
that they were consecutive. This relationship may be expressed simply
as _T H R_, even though, in the actual square, the letters may have
been partly at the end of the row (or column) and partly at the
beginning: _H R * * T_ or _R * * T H_. Encipherment, remember, is
cyclical, and we may come out with any one of numerous “equivalent
squares.” With the second equation (b), the positions of letters are
not so definite. In either of the lineal encipherments, _IK_ must be
in direct sequence and _SY_ must be in direct sequence; either
sequence may have come first, and we do not know the exact location
of the fifth letter. Concerning their possible rectangular
encipherment, all we know is that there must have been a parallel
relationship; their distance apart, laterally or vertically, might
have been anything permitted by the key-square. As to the rest of the
figure, suppose that we have reason to suspect the presence in this
cryptogram of the word “condemnation.” The equations of (c) are
totally impossible, since, in Playfair, no letter may be its own
substitute. Those of (d) are not only possible, but probable, since
we find many letters from the word itself.
Figure 169
Equations of (d),(Fig.168) Possible Combinations
1 2 3 4 5 6 7 8
O D O
N O N D E D E C N O N D O
D C D E N O N D E C
E C D E C
C
9 10 11 12
M
T M . T O N D E C
. M T . N C . . . .
N C . N T M
C
13 14 15 16
A O N D E C O N D E C
T A T I . . . .
I A A T I M
T M
I
To learn whether or not the word “condemnation” does (or could) occur
here, we proceed as in Fig. 169. The first of the five equations may
have had either one of the relationships marked 1 and 2, and the
second may have had either of the relationships marked 3 and 4. These
two equations have a letter _D_ in common, and it must not be
impossible to form a combination which will represent both. This, as
it happens, can be done in four different ways, marked 5, 6, 7, and 8,
and we do not know which of the four is most likely. The third
equation, which has four different letters, may have had any one of
relationships 9, 10, and 11. These, fortunately, show two letters,
_N_ and _C_, which are also present in combinations 5, 6, 7, and 8,
and with two common letters, there will not be so many possible
adjustments as when we had only one. Nos. 9 and 10, for instance,
cannot possibly combine with any one of combinations 5, 6, 7, and 8;
both of these have demanded that the letters _NC_ be in direct
sequence, where the first four combinations will not permit this. We
may begin, then, by discarding Nos. 9 and 10. But No. 11, which we
have retained, demands of _C_ and _N_ only that they be on the same
row. This is not permitted by any one of combinations 5, 6, or 7, and
these also may be discarded. But No. 8 shows them on the same row;
thus Nos. 8 and 11 may be further combined, and we have the
combination marked 12. The fourth equation, another lineal one, may
have had either of the relationships marked 13 and 14, and both of
these will combine easily with the combination marked 12, so that
again we have more than one possibility, as indicated under numbers
15 and 16. As to which of these is correct, the fifth equation,
_io_ = _AD_, is impossible to one, and has automatically been set up
in the other.
* * *
We are safe, now, in making substitutions on the cryptogram. This
means not only the five pairs originally identified, together with
their reversals and possible reciprocals, but all others which can be
derived from combination 16, such as _om_ = _CA_, or _dm_ = _CI_, or
_en_ = _DC_, together with their reversals and possible reciprocals.
Then, too, there will be many partial equations, such as those
indicated in Fig. 170, where one letter of a pair can be identified.
Usually time is saved by taking cryptogram pairs just as they come
and filling in as many letters as possible; in this way, patterns are
sometimes brought out, and thus we come back to step 1: the
identification of more pairs. With the key-square beginning to shape
up, the “chart of probable position” may be used to good advantage.
For instance, what about the letters _H_ and _B_ which were very high
in the frequency list?
Figure 170
O * * E N * E DE EC
. . . .
A T I M
Once a beginning is made, the cipher is broken, though just how
rapidly we may proceed with the solution depends chiefly upon the
manner in which the square has been filled. The presence of
alphabetical sequences (either horizontal or vertical) will often
enable us to complete the key-square independently of the cryptogram;
but the badly mixed square must usually be built up to the very end,
and we must sometimes be satisfied with one of the “equivalents” in
place of the square originally used. If the student cares to make a
fresh beginning of his own, this same cryptogram contains the word
RECONSTRUCT.
* * *
Figure 171
S E N D D D S T O A R D A M M
I A M O N M X S T E O N D A Y
The Playfair has been, in its day, a very effective cipher, and is
still good for many purposes. It can be rendered much safer if
subjected to the process called _seriation_. This process may be
examined in Fig. 171. Here, the text is “Send diamonds to Amsterdam
Monday,” and the agreed _seriation index_ is 5. The text is written
in pairs of five-letter lines, so that each ten-letter segment forms
five _vertical pairs_, _SI_, _EA_, _NM_, etc., and these are the
pairs which undergo the digram encipherment (notice the treatment of
the doubled _S_ in the second group). If the key-square is that of
Fig. 166, the first ten-letter segment is enciphered _QK_, _UF_,
_TG_, _SA_, _RS_, and the cryptogram may be taken off in that order,
or by taking the upper and lower lines separately:
_Q U T S R_ _K F G A S_. Seriation, it will be noticed, adds a
transposition to a substitution, so that what we have here is
combination cipher. This case, in short examples, is extremely
difficult; it is mentioned only by way of general information, and
is not included in the practice cryptograms which follow.
149. By NEMO. (Playfair. Probable words: ENEMY AGENTS).
OS CF WD OG DR AN PO AS OA DH SD EH XK FU CN DR PF UK SD.
150. By NEMO. (Playfair. Probable words: AUTHORIZED, EMERGENCY).
PK HL PG RI YH YN HQ IF YF GY ZL EB YF UK NK NG FL FG OL BD GX GK FC PK HG NV AC ZL
KH PK FG FK RZ FG RQ XO IB PB BD LE MV KG GY OL AD FK OR FC LK YN HL LK KZ IF EF AX
NG ON BV IK BI PK IO HQ IF AG LX YA FK AD YG KO AK EG TO OH BI RB OL ON KM FO PK KY
PR EF DZ IF AY QH CZ OK IQ WP FG LF DM CA VO GW PK NG KX KO LH AG NB RT NG KO KH HK
OX ML GP ML PB QD RB OH EB LH NK NG FC PK KY QS LH NE IQ WP FG.
151. By PICCOLA. (Figure 164; new external alphabets. THE, THIS, CHARACTER).
NF BJ HT MD NF WJ GD UC HN FW QI CE HP NF IA SE HS HS LG QA IY QD HV CC LB NF IA IA
CL GA RJ BD NM MA SY KU RD FT HC US HN GH VJ UA SY UL HN XJ EG QQ CJ LB NF KJ CN CS
UK GJ MD NF IA FK NQ GJ XX.
152. By PICCOLA. ("Slidefair." Probable word: DESCENT).
AA FS AF XY GJ BD UI AA PW GN IV QZ RC NK CC WA FT QQ PR GP TT WF PS JS QC HM DI XC
AH JP FB DC EW OX UG GP UI US CV GP MH QR OG JI ZR.
153. By DAN SURR. (Playfair. History: Detective Gettamann, investigating the
murder of Francis V. Bacon, well-known traveler, explorer, and
connoisseur, has found this message in an envelope addressed
to Wm. K. Pierce, former traveling companion of the deceased.
Death had occurred during sleep; caused by strangulation with
red silk thread. Only clues found: a few grains of sand on the
kitchen floor and what appears to be an oriental turban).
YG NG CR FV FZ RI OU KZ CW OW BQ GQ IH HL YW EG NG QM WX RT KP VE CA IG QI VD QI GN
GZ IZ QY QR HY NG XN AB AK OX NY WC WC TN OX DH NE IH IH YR IS QY WC HI UI UI IR QE
WS RW LG WR AB GW VW CA RQ XM ER QM RE CW ZI RQ XW QW GH YC AY YO VO NE RL PG CG WI
NX VW CA NX QM LH IG RQ WT GO UI GZ EG XN IW OU XT WO LH IG RQ XM WS QY TX IR IQ XM
OG DU AB RM AK UM RG ZR XA PM RW LD KG HI XK LC RT KP VE FO NX XK WR WS QY UR ZX YL
AT UI RH TR AV WS DH WQ PM AK IW OU WT DE IR WX RQ XZ SI GU QN IR XN IR YN IG GY TR
ZX YU RU YL IQ YA RU KG QM PD QM IY HA WS FE RW GH RB HA QI QM GI QC QR UL WV AB NX
GO HA FR IY QY BM QM YH NG IQ RU YL IQ BL PO QM RU GU IR TX SI GQ LQ DX XO EV BM CR
FV GV AB GE RZ GQ YH HA RW YM NE YM BL VW PS.
CHAPTER XXII
Highlights of Fractional Substitution
Fractional substitution requires a cipher alphabet of the “multifid”
type; that is, one in which the symbols are composed of two or more
units, as in the Bacon and Trithemius alphabets (Chapter II: Figs. 3
and 4), the various “checkerboards” (Chapter XI), and so on. Polygram
“alphabets” are also of this type, and seriation is a forrn of
fractional substitution.
Among the older fractionals, we find a system called the “Pollux,” in
which the basis was the Morse telegraphic alphabet. There were three
units, the dot, the dash, and a separator (made necessary by the
irregular lengths of the substitutes). There was a first substitution
in which the letters of the text were replaced with their Morse
symbols, including the space. The resulting cryptogram, composed
entirely of the units dot, dash, space (. — x), was then subjected to
a second substitution, using a small cipher alphabet (either digits
or letters) in which each one of the three units might have any one
of several different substitutes, chosen at will. For instance, a dot
might be replaced with any one of digits 1, 8, 5, 6, a space with any
one of digits 3, 9, 0, and a dash with any one of digits 2, 4, 7.
Figure 172
Delastelle's "BIFID" Substitution -(Keyword Feature Added by M. E. OHAVER)
Preparation of Alphabet: Checkerboard Key: Substitutes:
G E N • R A L 1 2 3 4 5 S = 43
B C D F H I K E = 15
M O P Q S T U 1 G B M V E N = 24
V W X Y Z 2 C O W N D D = 25
3 P X F Q Y
4 R H S Z A
5 I T L K U
Preliminary Substitution:
S E N D S U P P L I E S T O M O R L E Y S R I G H T A W A Y.
4 1 2 2 4 5 3 3 5 5 1 4 5 2 1 2 4 5 1 3 4 4 5 1 4 5 4 2 4 3
3 5 4 5 3 5 1 1 3 1 5 3 2 2 3 2 1 3 5 5 3 1 1 1 2 2 5 3 5 5
Re-Substitution:
41 22 45 33 54 53 51 35 51 45 21 31 53 22 12 45 13 43 21 35 53
R O A F K L I Y I A C P L O B A M S C Y L
45 14 54 21 11 22 53 43 55 Transmitted:
A V K C G O L S U.
R O A F K L I Y I A C P L O B, etc.
We find also a number of systems called “Collon” in which the basis
is some one of the “checkerboards.” The text is subjected to a simple
substitution in the agreed alphabet, and the resulting cryptogram is
then subjected to a transposition, usually seriation, this being the
final operation.
A similar system called the “Mirabeau” uses an alphabet of the same
type as that of the _Polybius square_, in which only the digits
1-2-3-4-5 are significant. The remaining digits are all null, and
numbers like 67 or 88 may be inserted at will. Numbers are written
vertically (tens below units); then, in the taking off of the
cryptograms, the whole series of units is taken first, and the second
half of the cryptogram includes all of the tens-digits. In all of
these forms, the undesirable features are self-evident. The later
devices have added another operation: the regrouping of the scattered
units, and their reconversion into letters.
Classic examples are those described by Delastelle as “bifid” and
“trifid” (terms, incidentally, which some of our own writers find
objectionable, as they do also the term “multifid”). Delastelle’s
“bifid” cipher was of the kind shown in Fig. 172. A two-unit alphabet
must be used, and all possible two-unit combinations must be
convertible into letters. Any desired seriation-length may be agreed
upon, though it should not be divisible by 2. In the figure, the
key-word GENERAL, 7 letters, governs the seriation-length as well as
the mixing of the key-square, a feature suggested by Ohaver. The
substitution is identical with that of the Polybius square, except
that the two units of the substitute are written vertically below the
original. Digits are then grouped horizontally in pairs, treating one
seven-letter group at a time (if the seriation index is 7), and these
pairs are replaced with letters from the same key-square. It will be
noticed that we have here a form of polygram substitution, in which
one seven-letter group has been replaced with another. Also, that
possible errors have been confined by the seriation feature to their
own seven-letter group.
Figure 173
A Fractional Substitution Based on Morse Symbols - M.E.OHAVER
The Alphabet, Arranged by Group-Lengths:
E . S ... H .... B -...
T - U ..- V ...- X -..-
R .-. F ..-. C -.-.
W .-- ü ..-- Y -.--
I .. D -.. L .-.. Z --..
A .- K -.- ä .-.- Q --.-
N -. G --. P .--. ö ---.
M -- O --- J .--- ch ----
S E N D S U P
... . -. -.. ... ..- .--. Reverse digits, and re-group:
3 1 2 3 3 3 4 4 3 3 3 2 1 3
.... -.- ... ... .- . --.
H K S S A E G
Delastelle’s “trifid” cipher was of the same kind, except that a
three-unit alphabet was required, resulting in three rows of units.
It would have been the same as that of Fig. 4, Chapter II, but with
the French accented _E_ replacing the character _&_. All combinations
of three units must be re-convertible into letters.
Fig. 173 shows a form of “mutilation” cipher once published by
Ohaver. Beyond stating that its only key is the group-length (7 in
the example), we leave the student to figure it out for himself.
As an example of recent use (1918), we are told on excellent
authority that the Germans, for quite a long time during the World
War, used a field cipher of the following description: There was a
preliminary substitution using a key-square of the Nihilist type,
except that the external co-ordinates were letters, and not digits,
and were chosen in such a way that the five or six letters used were
letters having very distinctive Morse symbols; this was for the
avoidance of telegraphic errors. In some cases a 5 x 5 square was
used, containing only a 25-letter mixed alphabet, and in others a
6 x 6 square containing a 26-letter mixed alphabet and all of the
digits. The preliminary cryptogram obtained from this first
encipherment was then written into a transposition block and taken
off by columns, using key-word columnar transposition. The
cryptograms were not afterward shortened by resubstitution, but were
always twice as long as their messages, and never contained any other
letters than the five (or six) originally used as co-ordinates. This
German Field Cipher proved very effective until finally broken by the
great French analytical genius, Georges Painvin.
We shall make no attempt, here, to go into the decryptment of these
ciphers. The Delastelle “bifid” is, perhaps, a practical cipher, and
the student may try his own hand at analyzing the example. The other
examples should give no trouble.
154. By PICCOLA. (Delastelle's "Bifid." - Repeated words: AMERICA(N), ATTEMPT,
REPORT, THAT, THE, OF, TO. Other short words: FROM, WITH,
BEEN, HAVE. Likely words: REPORT, AGENT, CONFIRM, CABLE, etc).
Q I N H P R M L M G R N B M A H G T O L O O E L O A O D R I N H W R O
A A B M M I M M W I B M D A B T H D I L T H T H I N T L A Q M C A M F
I V N K Y N O F H B I I T R F Q L A D K V Q I N H P R M R B H S L L U
A B M E T S O A A B M M I M M I B P I V R Q F T K H I R D F G N I E M
A B E N I L M M P A S I F I O P L Y C C R C I T W I V W M F G I O O S
O E R O I K Q I E F O V N V M Q T D R S I O E R I B U Q C D O A L L A
P L A A O O C A Q O M E I D C N T I U L O L Z D G.
The mixed alphabet here was placed in the square by straight horizontals. History:
Message intercepted following a report that on the tenth of August an attenpt had
been made to enter the American embassy in a country where Royalists are opposed to
a group of radicals.
155. By PICCOLA. (Fractional. - Not so hard).
3 3 3 2 3 1 1 1 2 3 2 2 1 3 1 1 1 1 3 1 3 3 1 1 3 2 2 1 2 2 1 1 2 3 1
2 3 3 2 1 2 3 3 1 1 3 2 1 1 2 1 2 2 2 3 1 2 2 2 3 1 1 2 2 1 2 3 2 3 2
2 1 2 3 1 3 3 2 3 1 1 2 2 1 3 2 1 2 2 3 2 1 3 1 2 2 2 2 3 2 3 2 2 2 2
3 1 1 1 3 1 2 3 2 1 1 2 2 2 3 2 3 1 3 2 2 2 2 1 2 3 1 2 2 1 2 1 2 2 1
1 2 2 3 2 3 2 2 3 2 2 2 3 2 2 3 3 1 2 2 3 1 2 1 3 1 1 1 1 2 1 3 3 3 3
1 2 3 3 3 2 1 3 3 1 1 1 1 2 2 3 1 1 3 1 1 1 1 1 1 1 1 3 2 2 1 2 3 2 2
2 1 2 1 2 1 2 2 2 3 3 2 2 1 3.
156. By PICCOLA. (Fractional. - Nor is this very hard).
E D C Y B A Z C B Z A V W X C X B A E Y D C B V A E D W B X A E Y Z D
A E Z V W D C A E D X C B Y D Y Z V C B W B A Z V E W X B X A E Y D C
B V A E W D C X A E Y Z D C E Z V D W C B E D X C B Y A Z D C B V W A
A E D C B A E E W D C B X Y D C Y Z B V A B A Z V E W X A E W D C X Y
E D Y Z C B V E D C W B A X E D Z V C B A C B V W A X Y X B Y A E D Z
E Y Z D V W C W E D X C B A D Y Z V C B A D C W B A X E E D C B A V E
D C X Y B Z A E D C B A E W D Y Z C B V A B A Z V E W X E D V W C X Y
X D C Y B A Z C Z B A V E W B A E W D X Y E D X Y C Z V V E D W C B A.
CHAPTER XXIII
Investigating the Unknown Cipher
When the type of encipherment is unknown, the decryptor’s first
problem may concern the probable language used in the plaintext, and
this he is usually able to determine from the source and history of
the cryptogram.
His second problem is the major classification, and this, too, is
usually simple, since transposition, as a rule, can be recognized by
its appearance. It must, however, respond to a group-test, and for
cases in which this is needed, the approximate percentages for English
can be taken as follows:
Vowels, with or without Y, about 40% (Variation limits: 35% to 45%)
Consonants L N R S T about 30% (Variation limits: 25% to 35%)
Consonants J K Q X Z about 2% (May be influenced by nulls).
The 5% variation is suggested in the Parker Hitt Manual. In this
connection, it should be pointed out that an apparent transposition
with exactly 40% of vowels and 100% evenness in their distribution
is suspicious. Many of the checkerboard systems result in this way,
and also some of the codes based on pronounceable five-letter groups.
Then, too, it is easily possible to construct a simple substitution
cipher alphabet in such a way that the resulting cryptograms will
resemble transposition, and even respond satisfactorily to a
group-test. It should be carefully ascertained that a supposed
transposition cryptogram does not contain the many repeated sequences
which belong to simple substitution. As to those transpositions which
do show an appreciable number of repeated digrams, they will probably
have undergone one of the route transpositions, especially one in
which columns were taken off in alternating directions.
* * *
Concerning the characteristics of simple substitution, these have
been seen throughout the text; we have normal frequencies attached
to the wrong letters, and we have those numerous repetitions of
various lengths, occurring at all kinds of intervals, which are never
found in a transposition. Here, too, we may apply a group-test, based
only on the relative frequencies of letters. The five most frequent
are supposed to represent the letters _E T A O N_ or their equivalents,
and should total about 45% of the text. The nine most frequent should
total about 70%; the eleven most frequent well over 75%; the five of
lowest frequency (which would include all of those totally absent)
should correspond to the normal behavior of the group _J K Q X Z_.
* * *
If the simple substitution frequency count is present without the
repeated sequences, then we probably have a combination of simple
substitution with transposition. It becomes necessary to rewrite
the cryptogram into various new arrangements until one is found which
will bring back the repeated sequences. Ordinarily, the simplest
kinds of transposition will have been used; sometimes the
transposition will have taken place in a complete-unit block, and
there will be a clue in the total number of letters present in the
cryptogram.
* * *
When all letters are present in the frequency count (or all but one
or two in the possible cases of 25-letter and 24-letter alphabets),
a period-investigation is usually indicated. The case of periodics
has been seen at considerable length, though a final hint might be
added for the detection of a possible Porta encipherment. One of our
many collaborators, F. R. Carter, suggests that any Porta cryptogram,
periodic or otherwise, ought to show from 52% to 53% of letters _N_
to _Z_ — the opposite of normal.
The characteristics of digram-encipherment have been mentioned. Other
polygram ciphers show corresponding characteristics, according to the
polygram length, though the trail grows fainter as polygrams grow
longer. A trigram-system, for instance, might be present when the
cryptogram is evenly divisible into three-letter groups; it might
suggest period 3, and might even show repeated sequences whose length
is a multiple of 3 and which begin at serial positions such as 1, 4,
7, 10, which are the beginnings of trigrams. A great many of the
trigram systems will show only repeated digrams beginning at these
serial positions, or separated by intervals which are divisible by 3.
A 5 x 5 square is often suggested in the fact of a missing letter;
but the fact of 26 letters does not deny one, since the careful
encipherer may make use of his missing _J_ instead of using _I_
exclusively. Great evenness in frequencies may suggest one of the
key-lengthening devices, such as autokey and progressing key; and
the practical absence of repeated sequences will usually mean that
a transposition has been added to a substitution. It is never a bad
idea, in a puzzling example, to make the various digram-counts (in
chart form): An actual digram count, in which every letter is
considered the beginning of a digram; a pair-count on separated
pairs, as in Playfair; the two counts which could be made with the
cryptogram marked off into three-letter groups; and the kind of
pair-count which could be made in Playfair if the first
cryptogram-letter were omitted. Many devices, as mentioned, may be
uncovered simply by “running down the alphabet.” And if the
cryptogram has come from an amateur “inventor,” it may be a case
of digging into one’s memory for previous “inventions.” With this
last case, however, the “inventor” very often fails to submit
material in proportion to the amount of complication he has introduced.
* * *
Of the examples to follow, there is none in which the system may not
be learned through analysis, unless perhaps the final unnumbered
cryptogram, and the material, in every case, should be suffcient for
solution.
No. 163 follows Mr. Berkley’s encipherment plan, illustrated just
above it.
No. 164 is said to have been taken from a German spy serving in the
American army in France. This applies, however, to the first fifty
groups only; the remainder was added to increase the length and to
emphasize the plan followed by the spy.
No. 166 was accompanied by a plot:
“Supposed to have been found on the body of a man floating in San
Diego Bay. Autopsy shows death by drowning. Victim was a local banker
who had disappeared a few days earlier. Wife says no financial
worries. No money missing. Banker had prospered during depression.
Was yachting enthusiast. Our hero solved the cipher with the
unconscious assistance of a radio crooner. Tragedy occurred in
_August, 1932_.” The date was doubly underscored, but those who have
read the message have found no reason for this and no explanation
for the “crooner.”
157. By PICCOLA.
C S R Z V Y P Q Z J K H K V Q U U C V M R T W Z N G H Q S A K O X P M
H D R W A J D F Q D F S R Z Z C G X P A J J T Z U L H T G S A H X J J
L T R N N Z P B Z G R E B N F Y G E J N M T N J J Q H P J X M O B J A
L X I A I C P F J O O F R N H.
158. By PICCOLA.
O C E E A T T I T K S N D T D S T H O O Y E A O E E P E B O T Y T A O
A D S E O E T F T T T H R V W C T H O Y L T O O H L R B T T U H R R V
R A W O B R U A O Y E H H L A B N E R L R K V C R I O N S E I D R U E
R I P.
159. By TITOGI.
A H Y N U H C E S T I T N D O R F E H R W E A T F N R F P A O T M A T
L H R E I O T N R L O D R H E E A T E S C T D I N W T S T O E A T S I
T E C D U T M S O T R L D O N G N I I S O F A E T L I T A S.
160. By PICCOLA. (Veiled reference to crypt No. 166?)
(a) H Z M Q L D N N D Z S P R F S K L L L L. (b) L I L V M S T Z U G
D H Z U Q X L L L L. (c) T V I U M F R U O Y U Q Y P S F W X L L.
(d) L I L V M F P O E Y Z K F D V U E L L L. (e) G K P V D Z T A Y T
B F Y Y C F I U L L. (f) Q B F P W Y C L U D V P Z Z O S W Y N C.
(g) Q B R T F F G V T U E N S Z H B Q E R L.
161. By PICCOLA.
E G W G W G E G T U C L C U O X G K Z T E G O B G B Y L W M I Q N K Q
Y E N F S C L H M N Y B X S E T N I W O C E G C B F C T C S Z T V G B
E A E G T U R K F K B E G K X B C T G Z Y L X C H Y E G C U O X Y T Q
F A D Q T T C U N B O G C O H X C E W E C U V E G C O C X Y X G B E A
Y T K X F Q C O T B X N E G T U C O N T O P E L E K U V U N T O C N G
N G B K W C E E C S Z K W N H E I K C C R E G C T E G T U R K F K T B
R G M W X C F G Q N I C E B P E E W E N B K I Y F K F D O F E G N U C
B G M T Z T F X C E W E C F V D T T U Z T E N E G W G L F M C T O V L.
162. By PICCOLA.
S P P A S T A S E F U N M T E H S O O A E S L E I C T R C H V U G S E
L Y R E M E N E E R O S N E H I R A E T O R N S H M O D R O P E A O R
P O S R Y P D O I N O C K G T.
The "NICODEMUS" Cipher (Harold Berkley) SPECIMEN ENCIPHERMENT
Key: M E T H O D I C M E T H O D I C M E T...
6 3 8 4 7 2 5 1 6 3 8 4 7 2 5 1 6 3 8...
Cryptogram:
T H I S I S E N A N S P O S E D E T C...
C I P H E R E D U S I N G T H E PFGVT VUHDG LMRIV
I V V I G E N E S A M E K E Y F ZOPUH MMVNB FOUDQ WSURF
R E A N D A F T O R B O T H O P BIOTP FGHRU VWHKR RWEVV
E R W A R D T R E R A T I O N S WULVA MPGWV MGEAQ CUYHW LBFUT.
163. By PICCOLA.
T Y D Q V W P A Z O M B W B I R K F I O O G W C O G E F L T Q M S R F
X T C J C M A W P P Q M E X V O Q C O C Z F S F W V F E V E R S A B E
C V J J W S I P P H M M K O X V Y I D B D B C I S Y N L J C Y F K C W
E N Z E I T J V L Z M I L I I R W K R O O S Z A W E K J V J G F M Q K
G F N C K H P B R D L V I A P E S L V M D J Z Z V F Z F F R D B A D P
Q W E N L A L O E K M F M F W X O K D W D G C K K K C Q R V.
164. By VULPUS.
P E N A R C P F T I Q E V A T E N B L A T K Q F O A R E N E U I P E P
F U K X I L C N F Q E P C V B T A W A O B N C O E T I N D W B N A R D
Q F O F N B V C P E P G V G P A V A P B P F O A O B S C L B V B T F W
A N E W B T C S D N F M A N A O E V A R A R C T K Q E N B M B Q F V E
V B X K O A P E T B U I P F O F Q E L E O B R D R B Q F U A W A S C U
K L F P E W B O C O D N A M E L G V F V A N C N D M F N B V D T D L E
P F V I T I Q E Q F O C O A U C L F L A O B M E P E N A S D L B T K L
H N E P D. ..... U I L A L B O B M A V K M G U K R F P F U B U D M
F W E T A T I Q E V B R C M B W A N F Z I L E N A Q F W B T C R D T B
T K O E P E U A V A O F N B S C Z K V B W C U B O A L F O B M E X I T
D Q C Q D W A P F Q E N A L A.
165. By PICCOLA. (Again that No. 166?)
R O V L L A B T L D L B C Q M P X L B A F B T C T A T C O R L T O L C
R H P D T X L Y O A E L B X P H L X B T X X Q L D R G L T K X R L G D
B K L D P P L O H L Y O A E L K O M X B L H O E L V C R R C R J L T K
D T L R C I N X P L L L T K X L R C I N X P L V D B L V O R L P O R J
L D J O L F Y L I O P O R X P L M D E N X E L K C T T L V K O L O H H
X E X G L T O L I O Q M E O Q C B X L H O E L T V O L I X R T B L B C
R I X L K X L V D B L D F P X L T O L B X R G L T K X L B O P A T C O
R L F Y L E X T A E R L Q D C P L L B T C P P L C T L V O A P G L F X
L V O E T K L D R O T K X E L R C I N X P L T O L H C R G L O A T L T
K X L N X Y L L T K C B L Q A B T L F X L T K X L X W M P D R D T C O
R L O H L T K X L E X H X E X R I X L T O L D L I E O O R X E L D R G
L T K X L X Q M K D B C B L O R L D L G D T X L L M L B L T K X L T V
O L I X R T B L K D B L R O T L Y X T L F X X R L M D C G L.
166. By CACHE. (Contributor, C. H. Price, died without explaining his key).
03 65 12 45 58 28 06 41 72 14 22 03 02 17 36 88 25 20 55 77 74 51 23 45 41 42 30 24
36 61 96 09 07 78 05 44 08 06 55 92 16 93 02 15 36 37 40 87 41 01 33 77 06 36 27 54
48 29 16 78 92 66 03 10 38 17 45 23 72 96 73 01 49 25 72 38 92 72 24 55 48 08 40 92
28 01 72 96 02 04 74 61 06 99 30 45 72 69 74 93 77 23 55 36 24 93 47 84 76 35 32 89
87 76 77 64 51 96 58 43 76 02 81 38 87 69 89 55 99 23 79 55 51 06 99 71 74 69 89 84
27 25 22 39 42 53 19 93 41 66 09 75 87 37 91 87 90 91 43 19 40 30 38 16 96 22 69 38
78 02 74 92 47 25 77 91 15 40 24 45 07 07 96 48 44 15 12 06 99 44 93 19 25 23 55 30
45 87 96 18 01 78 44 29 45 86 47 69 48 30 66 44 03 41 66 37 38 22 06 42 41.
59.
Here is one which nobody has ever been able to decrypt:
V Q B U P P V S P G G F P N U E D O K D X H E W T I Y C L K X R Z A P
V U F S A W E M U X G P N I V Q J M N J J N I Z Y K B P N F R R H T B
W W N U Q J A J G J F H A D Q L Q M F L X R G G W U G W V Z G K F B C
M P X K E K Q C Q Q L B O D O Q J V E L.
APPENDIX
ENGLISH FREQUENCY AND SEQUENCE DATA
(Compiled from the MEAKER Digram Chart)
Order and Frequency of Order and Frequency of
Single Letters Leading DIGRAMS
E 1231 L 403 B 162 TH 315 TO 111 SA 75 MA 56
T 959 D 365 G 161 HE 251 NT 110 HI 72 TA 56
A 805 C 320 V 93 AN 172 ED 107 LE 72 CE 55
O 794 U 310 K 52 IN 169 IS 106 SO 71 IC 55
N 719 P 229 Q 20 ER 154 AR 101 AS 67 LL 55
I 718 F 228 X 20 RE 148 OU 96 NO 65 NA 54
S 659 M 225 J 10 ES 145 TE 94 NE 64 RO 54
R 603 W 203 Z 9 ON 145 OF 94 EC 64 OT 53
H 514 Y 188 EA 131 IT 88 IO 63 TT 53
TI 128 HA 84 RT 63 VE 53
AT 124 SE 84 CO 59 NS 51
Group Percentages: ST 121 ET 80 BE 58 UR 49
EN 120 AL 77 DI 57 ME 48
A E I O U 38.58% ND 118 RI 77 LI 57 WH 48
OR 113 NG 75 RA 57 LY 47
L N R S T 33.43%
List of Common REVERSALS:
J K Q X Z 1.11%
ER RE ON NO TE ET ST TS
E T A 0 N 45.08% ES SE IN NI OR RO IS SI
AN NA EN NE TO OT ED DE
E T A O N I S R H 70.02% TI IT AT TA AR RA OF FO
Order of the Leading TRIGRAMS
In 10,000 Letters of Semi-Military Text - PARKER HITT
THE ENT FOR NCE OFT
AND ION NDE EDT STH
THA TIO HAS TIS MEN
INITIAL LETTERS OF WORDS:
Order, as found by M. E. OHAVER ... T A O S H I W C B P F D M R, etc.
Order, as found by H. O. YARDLEY .. T O A W B C D S F M R H I Y, etc.
FINAL LETTERS OF WORDS:
Order, as found by M. E. OHAVER ... E S T D N R O Y, etc.
Order, as found by H. O. YARDLEY .. E T D N S R Y, etc.
NOTE: Lists of terminals (letters, digrams, trigrams); of common affixes,
short words, and common pattern-words, can be found in the booklet
"CRYPTOGRAM SOLVING", obtainable from the author, M.E.Ohaver, at
Columbus, Ohio.
X J M M T V O Z B N Q M F B T F S F N J U G P S U I J T B E ?
COMPARATIVE TABLE OF SINGLE-LETTER FREQUENCIES (Per 100)
ENGLISH GERMAN FRENCH ITALIAN SPANISH PORTUGUESE
A 7.81 A 5. A 9.42 A 11.74 A 12.69 A 13.5
B 1.28 B 2.5 B 1.02 B .92 B 1.41 B .5
C 2.93 C 1.5 C 2.64 C 4.50 C 3.93 C 3.5
D 4.11 D 5. D 3.38 D 3.73 D 5.58 D 5.
E 13.05 E 18.5 E 15.87 E 11.79 E 13.15 E 13.
F 2.88 F 1.5 F .95 F .95 F .46 F 1.
G 1.39 G 4. G 1.04 G 1.64 G 1.12 G 1.
H 5.85 H 4. H .77 H 1.54 H 1.24 H 1.
I 6.77 I 8. I 8.41 I 11.28 I 6.25 I 6.
J .23 J ... J .89 J ... J .56 J .5
K .42 K 1. K ... K ... K ... K ...
L 3.60 L 3. L 5.34 L 6.51 L 5.94 L 3.5
M 2.62 M 2.5 M 3.24 M 2.51 M 2.65 M 4.5
N 7.28 N 11.5 N 7.15 N 6.88 N 6.95 N 5.5
O 8.21 O 3.5 O 5.14 O 9.83 O 9.49 O 11.5
P 2.15 P .5 P 2.86 P 3.05 P 2.43 P 3.
Q .14 Q ... Q 1.06 Q .61 Q 1.16 Q 1.5
R 6.64 R 7. R 6.46 R 6.37 R 6.25 R 7.5
S 6.46 S 7. S 7.90 S 4.98 S 7.60 S 7.5
T 9.02 T 5. T 7.26 T 5.62 T 3.91 T 4.5
U 2.77 U 5. U 6.24 U 3.01 U 4.63 U 4.
V 1.00 V 1. V 2.15 V 2.10 V 1.07 V 1.5
W 1.49 W 1.5 W ... W ... W ... W ...
X .30 X ... X .30 X ... X .13 X .2
Y 1.51 Y ... Y .24 Y ... Y 1.06 Y ...
Z .09 Z 1.5 Z .32 Z .49 Z .35 Z .3
Vowel Percentages:
English German French Italian Spanish Portuguese
40% 40% 45% 48% 47% 48%
Percentages for L N R S T:
33% 34% 34% 30% 31% 29%
NOTES: ENGLISH frequencies, which may be compared with those of Mr. Meaker,
(A, 8.05; B, 1.62; C, 3.20; etc.), were taken from M.E.OHAVER.
FRENCH, ITALIAN, and SPANISH frequencies were taken from a count
made by the author. All four counts are based on 10,000 letters
of literary text, and the dropping of the decimal point gives
the actual count. The frequencies given for GERMAN and PORTUGUESE
are approximations, reduced from other texts, probably military.
Chart Showing Normal CONTACT PERCENTAGES - Compiled by F. R. CARTER
(Based on a Digram Chart by M.E.OHAVER)
% %
V. C. V. C.
19 81 P4 L4 C5 D5 M5 N6 S6 W7 T8 R8 E11 H14 A N21T17S12R10L8 D5 C4 M4 6 94
55 45 Y4 B4 N5 T5 U8 D9 O9 S10A16E16 B E34L17U11O9 A7 Y5 B4 R4 70 30
61 39 U4 O5 S8 N13A13I18E20 C H19O19E17A13I7 T6 R4 L4 K4 59 41
52 48 R4 I5 L6 A10N29E39 D E16I14T14A10O8 S6 U5 54 46
8 92 C4 B4 E5 M5 V5 D5 S5 L5 N6 T6 R11H24 E R15D10S9 N8 A7 T6 M5 E4 C4 O4 W4 21 79
69 31 S4 N5 F5 D5 A6 I7 E12O41 F T22O21E10I9 A7 R5 F5 U4 52 48
36 64 O4 D4 U5 R5 I9 E9 A10N48 G E14H14O12R10A8 T6 F5 W4 I4 S4 42 58
7 93 G4 E5 W5 S7 C9 T62 H E50A23I12O7 90 10
13 87 F4 M4 W5 E6 N6 L8 D8 S8 R9 H11T14 I N25T13S10O8 C7 R4 E4 M4 A4 L4 17 83
28 72 Y7 W7 T7 S7 N7 E7 C7 B7 A14M29 J U35O29A12E12M6 W6 88 12
53 47 Y5 U5 I5 N7 A11R13E13O15C18 K E34I21N10A9 T7 S6 68 32
52 48 N4 P4 T6 I7 B7 U7 O10E11L11A17 L E19I15Y12L12O9 A8 D7 U4 65 35
69 31 S4 D4 M5 R5 I12A13O16E24 M E26A17O12I11P5 M5 71 29
89 11 U7 E14O22A23I24 N D16T14G12E10A7 S7 O7 I6 C5 32 68
21 79 M4 O4 D4 L4 P4 H5 N6 E6 C7 F7 S8 I8 R9 T11 O N20F14R11U10T6 M5 L5 S4 W4 O4 18 82
47 53 R4 L4 T4 N4 I4 P6 M6 A7 O8 U10E16S17 P O17E16A15R15L8 U6 P6 T5 I5 S4 59 41
20 80 O10N10L10E10D10R20S30 Q U100 100 --
70 30 P5 I5 U5 T7 A13O16E30 R E23O12A11T11I10S7 Y4 61 39
48 52 D4 T4 O6 U6 R7 N8 S9 I11A16E18 S T19E11O10I9 S9 A8 H6 P5 U4 41 59
43 57 U4 O5 D6 T6 F7 R7 E8 I10N10S13A14 T H39I10O10E8 A7 T6 R4 38 62
35 65 P5 F5 T5 L5 B6 D8 S9 O30 U N18S13T13R12L10P7 B4 C4 8 92
88 12 R6 U10O16A16I16E30 V E65I14O9 A8 99 1
48 52 G4 D4 Y5 N9 S10T11O16E23 W A27H16I16E15O11N4 80 20
95 5 U5 N5 I16E74 X P29T19I14A14U10C5 K5 O5 38 62
24 76 B4 N8 A8 T13E14R15L25 Y A15O12S12T9 W7 H5 I5 E5 D4 M4 B4 38 62
88 12 O12N12A25I50 Z E43I43W14 86 14
All figures indicate PERCENTAGES. - Taking any one letter, as A: On the left, it was contacted
14% of the time by H, 11% by E, etc., and 81% of its total contacts on that side were consonants.
On the right, it was contacted 21% of the time by N, and 94% of the time by consonants.
Chart Showing FREQUENCIES of English DIGRAMS - Prepared by O. PHELPS MEAKER
(Actual Count Made on 10,000 Letters of Literary Text).
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
A 1 8 44 45│131│21 11 84 18 34 56 54 9 21 57 75 56 18 15 32 3 11 805
B 32 18│ 11│ 2 2 1 7 7 9 7 18 1 4 13 14 5 11 162
C 39 12 4│ 64│ 9 1 2 55 8 1 31 18 14 21 6 17 3 5 10 320
D 15 10│107│ 1 1 1 16 28 2 118 16 16 6 9 11 4 4 365
E 58 55 39│ 39│25 32 251 37 2 28 72 48 64 3 40 148 84 94 11 53 30 1 12 5 1231
F 10 1 12│ 23│14 3 2 27 5 8 94 6 13 5 1 1 3 228
G 18 2│ 20│ 1 1 10 1 75 3 6 6 1 12 5 161
H 46 3│ 15│ 6 16 5 1 9 3 7 3 30 315 2 48 5 514
I 16 6 15 57│ 40│21 10 72 8 57 26 37 13 8 77 42 128 5 19 37 4 18 2 718
J 2 1│ 1│ 1 1 3 1 10
K 10 8 │ 2│ 8 3 3 5 11 2 52
L 77 21 16 7│ 46│10 4 3 39 55 10 17 29 12 6 12 28 4 6 1 403
M 18 1 9│ 43│ 3 1 1 32 4 5 7 44 15 14 14 9 1 4 225
N 172 5│120│ 2 3 2 169 3 1 3 9 145 12 19 8 33 10 3 719
O 2 11 59 37│ 46│38 23 46 63 4 3 28 28 65 23 28 54 71 111 2 6 17 1 28 794
P 31 1 7│ 32│ 3 1 1 3 2 16 7 29 26 8 24 8 17 2 4 7 229
Q 1 1│ 14│ 2 2 20
R 101 6 7 10│154│ 4 21 8 21 2 5 113 42 18 6 30 49 1 5 603
S 67 5 1 32│145│ 8 7 3 106 2 12 6 51 37 3 39 41 32 42 3 17 659
T 124 38 39│ 80│42 13 22 88 1 19 6 110 53 14 63 121 53 45 6 1 21 959
U 12 25 16 8│ 7│11 8 2 4 8 13 12 96 7 20 6 30 22 1 1 1 310
V 24 4│ 16│ 1 14 2 4 13 5 2 4 1 3 93
W 7 1 9│ 41│ 4 2 7 1 3 5 2 15 36 1 10 27 16 2 14 203
X │ 17│ 1 1 1 20
Y 27 19 6│ 17│ 1 1 1 3 47 3 14 4 2 17 4 21 1 188
Z 1 │ │ 4 2 1 1 9
To learn the frequency of any digram, find its first letter at the top, find its second
letter at the side, and observe the figure in the cell at which the column headed by the
first letter crosses the row headed by the second. Frequency for EA, 131; for AE, zero.
SOME FOREIGN LANGUAGE DATA
NOTE: Frequencies of letters, and their order, are fixed quantities
in any language. Group frequencies, however. are fairly constant
in every language. (These may be computed from the Comparative
Table for any desired group in the languages given.) Of the material
which follows, portions came from Lange and Soudart, and from Valerio,
but exact sources were not in every case furnished to the author.
G E R M A N
Order of single letters: E N I R S A D T U G H O L B M C F W Z K V P (J Q X Y)
Order of digrams: EN ER CH DE GE EI IE IN NE ND BE EL TE UN ST DI NO UE SE AU RE HE
Order of trigrams: EIN ICH DEN DER TEN CHT SCH CHE DIE UNG GEN UND NEN DES BEN RCH
Order of tetragrams: ICHT KEIT HEIT CHON CHEN CHER URCH EICH DERN AUCH SCHA SCHE
SCHI SCHO SCHU (Furnished by JOSEPH ARTHOLD).
Peculiarities:
C is practically always followed by H (or K), and SC by H.
Word-length is normally greater than in English.
F R E N C H
Order of single letters: E A I S T N R U L O D M P C V Q G B F J H Z X Y (K W)
Order of digrams: ES EN OU DE NT TE ON SE AI IT LE ET ME ER EM OI UN QU
Order of trigrams: ENT QUE ION LES AIT TIO ANS ONT ANT OUR AIS OUS
Peculiarities:
Q followed by U and a second vowel.
Four and five vowels may be found in sequence ("J'ai oui dire.."), but
E seldom touches the other vowels. D and M contact E about 75% of the
time, and L contacts it over 50% of the time. It is unusual to find
more than four consonants in sequence; when five are found in
succession, one is almost surely the final S of a plural word.
Order of doubled letters: S L M R T N P E C F
Order of initials: P A S M C E D T V F R B L G J I Q N O H U Y X Z
Order of finals: E S T R N D A I X Z L C U P F Y
Average word-length: 4.3 letters.
Commonest short words, in order: DE IL LE ET QUE JE LA NE UN LES EN CE SE SON MON
PAS LUI ME AU UNE DES SA QUI EST DU
I T A L I A N
Order of single letters: E A I O N L R T S C D P U M V G H F B Q Z (J X K W Y)
Order of digrams: ER ES ON RE EL EN DE DI TI SI AL AN RA NT TA CO
Order of trigrams: CHE ERE ZIO DEL ECO QUE ARI ATO EDI IDE ESI IDI ERO PAR NTE STA
Peculiarities:
Q followed by U and a second vowel. H largely preceded by C, in CHE,CHI,
or sometimes by G in GHE GHI. Z most often part of ZIO or NZA.
The frequencies of the vowels E A I O often exchange places.
Doubling of consonants is very frequent.
Order of doubled letters: L T S C R G P N B M Z F V I D
Order of initials: S P A C D V T M F I G Q R E B L N O U Z H
Order of finals: O E A I (Others, if used: R L D N)
Average word-length: 4.5 letters.
Commonest short words, in order: LA DI CHE IL NON SI LE UNA LO IN PER UN MI IO PIU
DEL MA SE
S P A N I S H
Order of single letters: E A O S N I R L D U C T M P B H Q G V Y J F Z X (K W)
Order of digrams: ES EN EL DE LA OS AR UE RA RE ER AS ON ST AD AL OR TA CO
Order of trigrams: QUE EST ARA ADO AQU DEL CIO NTE OSA EDE PER IST NEI RES SDE
Peculiarities:
Q followed by U and a second vowel. The only doubles are
LL, RR, CC, EE, NN, OO, in the order given, but the latter three
are very rare. Group frequencies somewhat less stable than
in the other languages.
Order of initials: C P A S M E D T H V R U N I L B O F Q G J Z
Order of finals: O A S E N R D L I Z
Average word-length: 4.4 letters.
Commonest short words, in order: DE LA EL QUE EN NO CON UN SE SU LAS LOS ES ME AL
LO SI MI UNA DEL POR SUS MUY HAY MAS
P O R T U G U E S E
Order of single letters: A E O R S I N D M T U C L P Q V F G H B J Z X (K W Y)
Order of digrams: ES OS DE AS RO EN CO DO RE ER NT SE AD OR AO SA TE AR EM QU UE OD ST
Order of trigrams: QUE ENT NTE DES EST ODE ADO CON STA MEN ADE DOS ARA COM
Much like Spanish. Spanish cion becomes cao; ll becomes lh. Articles drop
the L: os, as, in place of Spanish los, las, etc.
BIBLIOGRAPHY
By W. D. Witt
An extended bibliography of cryptography would fill many pages and
is therefore beyond the scope of this work, but it is hoped that the
following short selected list will be found useful. Some of these
works are out of print or otherwise unobtainable, but may, in some
instances, be found in public libraries or in old bookstores. The
Riverbank Publications may be consulted at The Library of Congress,
Washington, D. C.
More or Less Elementary Works
Boyer, John Q. “The Cryptogram” in Real Puzzles by John Q. Boyer,
Rufus T. Strohm and George H. Pryor, pp. 147-154. Baltimore, 1925.
(Simple substitution ciphers only.)
Buranelli, Prosper, Margaret Petherbridge and F. Gregory Hartswick.
The Cryptogram Book. New York, 1928. (Simple substitution ciphers
only.)
Hitt, Parker (Colonel). The A B C of Secret Writing. New York, 1935.
Lysing, Henry, pseud. (John Leonard Nanovic). Secret Writing. New
York, 1936.
―. The Cryptogram Book. New York, 1937.
Mansfield, Louis C. S. The Solution of Codes and Ciphers. London,
1936.
―. One Hundred Problems in Cipher. London, 1936.
Ohaver, M. E. Cryptogram Solving. Columbus, Ohio, 1933, (Simple
substitution ciphers only.)
Thomas, Paul B. Secret Messages. New York and London, 1928. 2nd
printing, 1929.
Windolph, J. Fred (“Phil Down”). “Cryptograms: Their Construction and
Solution” in A Key to Puzzledom, pp. 53-64. New York, 1906. (Simple
substitution ciphers only.)
Yardley, Herbert Osborne (Major). Yardleygrams. Indianapolis, 1932.
(The London edition (1932) bears the title Ciphergrams.)
Advanced Works
Friedman, William F. (Lt.-Colonel). Elements of Cryptanalysis.
Washington, Gov. Printing Office, 1924. “For Offcial Use Only.”
(Contains a bibliography. Out of print and unobtainable.)
―. See also Riverbank Publications.
Givierge, Marcel (General). Cours de Cryptographie. Paris, 1st
edition, 1925, 2nd edition, 1932.
Hitt, Parker (Colonel). Manual for the Solution of Military Ciphers.
Fort Leavenworth, Kans., 1st edition, 1916, 2nd edition, 1918.
Riverbank Publications. Papers (except No. 19) by W. F. Friedman.
Department of Ciphers, Riverbank Laboratories, Geneva, Illinois.
No. 15, 1917. “A Method of Reconstructing the Primary Alphabet.”
No. 16, 1918. “Methods for the Solution of Running-key Ciphers.”
No. 17, 1918. “An Introduction to Methods for the Solution of
Ciphers.”
No. 18, 1918. “Synoptic Tables for the Solution of Ciphers, and
a Bibliography of Cipher Literature.”
No. 19, 1918. “Formulae for the Solution of Geometrical
Transposition Ciphers,” by Captain Lenox R. Lohr,
with an introduction by W. F. Friedman.
No. 20, 1918. “Several Machine Ciphers and Methods for Their
Solution.”
No. 21, 1918. “Methods for the Reconstruction of Primary
Alphabets.”
No. 22, 1922. “The Index of Coincidence and Its Applications in
Cryptographic Analysis.”
Sacco, Luigi (General). Manuale di crittografia. Rome, 2nd edition,
revised and enlarged, 1936. (The first edition was privately printed
under the title Nozioni di crittografia. Rome, 1930.)
Zanotti, Mario. Crittografia. Milan, 1928.
Miscellaneous (Primarily descriptive of systems, historical, special
essays, etc., usually with little on cryptanalysis.)
Anon. “Cryptography” in Encyclopaedia Britannica, Vol. 6, 14th
edition, New York and London, 1929. (Contains a bibliography.)
Ball, W. W. Rouse. “Cryptographs and Ciphers” in his Mathematical
Recreations and Essays. 7th edition, 1917 and later. (Latest edition
is the 11th, 1939.)
Blair, William. “Cipher in Diplomatic Affairs” in Rees’s Cyclopaedia.
1803-1819.
Candela, Rosario. The Military Cipher of Commandant Bazeries, An
Essay in Decrypting. New York, 1938.
Friedman, William F. (Lt.-Colonel). “Codes and Ciphers” in
Encyclopaedia Britannica, Vol. 5, 14th edition, New York and London,
1929. (Contains bibliography.)
―. “Edgar Allan Poe, Cryptographer” in American Literature, A Journal
of Literary History, Criticism and Bibliography, Vol. 8, No. 3, Nov.
1936, pp. 266-280. Duke University, Durham, N. C.
Hulme, Frederick Edward. Cryptography, or, The History, Principles
and Practice of Cipher-Writing. London, 1898.
Lange, André and E. A. Soudart. Traité de Cryptographie. Paris, 1st
edition, 1925, new edition, 1935. (Contains an extensive bibliography.)
Lange, André. Cryptography. Translated from the French by J. C. H.
Macbeth, London and New York, 1922.
Pratt, Fletcher. Secret and Urgent, The Story of Codes and Ciphers.
Indianapolis, 1939.
Yardley, Herbert Osborne (Major). The American Black Chamber.
Indianapolis and London, 1931. Reprinted, New York, 1933, London,
1934.
THE COMMONEST ENGLISH WORDS
Below are listed the hundred most frequently used words in English.
The figures give occurrences in 242,432 words of English text taken
from fifteen English authors and many newspapers. Compiled by Frank
R. Fraprie, after the rest of the book had been completed.
THE 15568 OR 1101 WHEN 603 ONLY 309
OF 9767 HER 1093 WHAT 570 ANY 302
AND 7638 HAD 1062 YOUR 533 THEN 298
TO 5739 AT 1053 MORE 523 ABOUT 294
A 5074 FROM 1039 WOULD 516 THOSE 288
IN 4312 THIS 1021 THEM 498 CAN 285
THAT 3017 MY 963 SOME 478 MADE 284
IS 2509 THEY 959 THAN 445 WELL 283
I 2292 ALL 881 MAY 441 OLD 282
IT 2255 THEIR 824 UPON 430 MUST 280
FOR 1869 AN 789 ITS 425 US 279
AS 1853 SHE 775 OUT 387 SAID 276
WITH 1849 HAS 753 INTO 387 TIME 273
WAS 1761 WERE 752 OUR 386 EVEN 272
HIS 1732 ME 745 THESE 385 NEW 265
HE 1727 BEEN 720 MAN 383 COULD 264
BE 1535 HIM 708 UP 369 VERY 259
NOT 1496 ONE 700 DO 360 MUCH 252
BY 1392 SO 696 LIKE 354 OWN 251
BUT 1379 IF 684 SHALL 351 MOST 251
HAVE 1344 WILL 680 GREAT 340 MIGHT 250
YOU 1336 THERE 668 NOW 331 FIRST 249
WHICH 1291 WHO 664 SUCH 328 AFTER 247
ARE 1222 NO 658 SHOULD 327 YET 247
ON 1155 WE 638 OTHER 320 TWO 244
ENGLISH TRIGRAMS
The ninety-eight most frequent English trigrams, combining a count
of 20,000 trigrams by Fletcher Pratt, in “Secret and Urgent,” supposed
not to include overlaps between words, and 5,000 by Frank R. Fraprie,
including overlaps. This table and the following one are not referred
to in the text, having been compiled since the completion of the book.
THE 1182 HER 170 HIS 130 ITH 111
ING 356 ATE 165 RES 125 TED 110
AND 284 VER 159 ILL 118 AIN 108
ION 252 TER 157 ARE 117 EST 106
ENT 246 THA 155 CON 114 MAN 101
FOR 191 ATI 148 NCE 113 RED 101
TIO 188 HAT 138 ALL 111 THI 100
ERE 173 ERS 135 EVE 111 IVE 96
REA 95 INE 73 ORE 65 ART 58
WIT 93 WHI 71 BUT 64 NTE 58
ONS 92 OVE 71 OUT 63 RAT 58
ESS 90 TIN 71 URE 63 TUR 58
AVE 84 AST 70 STR 62 ICA 57
PER 84 DER 70 TIC 62 ICH 57
ECT 83 OUS 70 AME 61 NDE 57
ONE 83 ROM 70 COM 61 PRE 57
UND 83 VEN 70 OUR 61 ENC 56
INT 80 ARD 69 WER 61 HAS 56
ANT 79 EAR 69 OME 60 WHE 55
HOU 77 DIN 68 EEN 59 WIL 55
MEN 76 STI 68 LAR 59 ERA 54
WAS 76 NOT 67 LES 59 LIN 54
OUN 75 ORT 67 SAN 59 TRA 54
PRO 75 THO 66 STE 59
STA 75 DAY 65 ANY 58
ENGLISH DIGRAMS
The one hundred and nine most frequent English digrams, compiled from
a count of 20,000 trigrams by Fletcher Pratt, in “Secret and Urgent,”
supposed not to include overlaps between words, and 5,000 by Frank R.
Fraprie, including overlaps.
TH 1582 RO 275 WI 188 SA 146 CT 111
IN 784 LI 273 HO 184 NI 142 TU 108
ER 667 RI 271 TR 183 RT 142 DA 107
RE 625 IO 270 BE 181 NA 141 AM 104
AN 542 LE 263 CE 177 OL 141 CI 104
HE 542 ND 263 WH 177 EV 131 SU 102
AR 511 MA 260 LL 176 IE 129 BL 101
EN 511 SE 259 FI 175 MI 128 OF 101
TI 510 AL 246 NO 175 NG 128 BU 100
TE 492 IC 244 TO 175 PL 128
AT 440 FO 239 PE 174 IV 127
ON 420 IL 232 AS 172 PO 125
HA 420 NE 232 WA 171 CH 122
OU 361 LA 229 UR 169 EI 122
IT 356 TA 225 LO 166 AD 120
ES 343 EL 216 PA 165 SS 120
ST 340 ME 216 US 165 IL 118
OR 339 EC 214 MO 164 OS 117
NT 337 IS 211 OM 163 UL 115
HI 330 DI 210 AI 162 EM 114
EA 321 SI 210 PR 161 NS 113
VE 321 CA 202 WE 158 OT 113
CO 296 UN 201 AC 152 GE 112
DE 275 UT 189 EE 148 IR 112
RA 275 NC 188 ET 146 AV 111
INDEX
Amateurs, devices proposed by, 132, 140, 152
“Amsco” transposition cipher, 51
Anagramming, 22, 29, 45-50
Anagramming, multiple, 32, 56-68
“Aristocrats,” 69, 72, 88
Auto-encipherment, 146-158
Bacon’s biliteral cipher, 6
Basic cipher alphabet, 164
Bassières processes for autokey, 147-151
Beaufort ciphers, 121
“Bifid” substitution, 210
Book cipher, 106
“Caesar” alphabets, 69, 71, 100, 108, 118, 130
Characteristics of systems, 53, 100, 103, 105, 130, 153, 168,
201-203, 213-214
“Checkerboard” alphabets, 102, 104, 164, 209
Cipher, meaning of, 1
Cipher alphabets, 69, 130, 159, 172, 185
Cipher disks, 108, 111
Cipher disks, special application of, 192
Classification of ciphers, 1
Classification of substitutions, 68
Code, 1, 106
Collon systems, 209
Columnar transposition, 11, 17, 23, 37-52
Columnar transposition, double, 54
“Columns” in autokeyed ciphers, 146
Combination block, 44
Combination cipher, 1, 4, 6, 68, 207, 209, 213, 216
Complements, complementary alphabets, 123
Complete-unit transposition, 9
Concealment cipher, meaning of, 1
Concealment ciphers, 4-8
Consonant-line method for simple substitution, 88
Contact count, 74, 77, 83, 173
Conventional writing, 4
“Crypt,” 72, 91
Cryptanalysis, meaning of, 3
Cryptogram, meaning of, 1
Cryptographic security, 2
Cycle (see period or unit)
Cyclical encipherment, 186, 200
Deciphering alphabet (or key), 71, 104
Decipherment, meaning of, 1
Decoding, meaning of, 2
Decrypt, meaning of, 1
Definitions, miscellaneous, 1, 68-70
Dictionary cipher, 106
Digram, meaning of, 1
Digram count, 83
Digram-solution method for simple substitution, 83
Digram tests, 45, 46, 47, 50
Dissimulated writing, 4
Double-key substitution, meaning of, 68
Double Myszkowsky transposition, 60
Double substitutions, 155, 164, 193
Double transpositions, 18, 54
Empty, or negative, words, 76
Encipherment, meaning of, 1
Encoding, meaning of, 2
“Entry,” 74
Equivalent slide (or alphabet), 169, 180
Factoring, in transpositions, 12, 42
Factoring of intervals to find a period, 128, 132
“Force Method,” 90
Fractional substitution, meaning of, 68
Fractional substitutions, 209-212
Frequency (see statistics)
Frequency counts, 74, 83, 129, 161, 173
“Frequential checkerboard,” 105
German field cipher (the ADFGVX), 210
Grandpré’s word-square, 104-105
Graph, graphic appearance, 106, 131, 173
Grille, Fleissner’s, 26
Grille, Richelieu’s, 4
Grille, Sacco’s indefinite, 12
Grille, turning, 26
Gronsfeld cipher, 117
Group-application in polyalphabetic ciphers, 185
Group percentages (see statistics)
Hermann cipher, 143
Index-letter (or cell), 111, 119, 124, 143, 169
Indicators, 101, 143
Intervals, alphabetical and lineal, 169, 180, 193, 194
Intervals in cryptograms, 43, 127
Inverse alphabets, 69
Irregular transposition, 37-67
Isolating a cipher alphabet, 122
Kasiski method, 127
Key, meaning of, 1
Key-alphabet, 108
“Key-frame” (or “skeleton”), 86, 175, 206
Key-interruption, 143
Key-length, 11, 17, 42, 127, 147
Key-lengthening devices in Vigenère, 143
Key-letter, 108
“Key-phrase” cipher, 103
Keys for simple substitution, 69
Keys, preparation of, 17, 26, 70, 104
Keys, recovery of, 23, 33, 39, 40, 45, 50, 58-63, 64, 85, 106, 124,
150, 157, 163, 164, 167, 175-183, 188-190, 205
Keywords, uses of, 17, 70, 146, 169
Knight’s tour, 10
Legrand’s open-letter cipher, 5
Levine’s concealment ciphers, 7
“Lining up” frequency counts, 162-163
Low-frequencies, assistance from, 99, 100, 129, 136, 138, 160
Low-frequency contacts, 78, 174
Magic squares, 10
Mathematical aspects of multiple-alphabet ciphers, 142, 151, 193, 196
Mechanical methods, 21, 32, 45, 56-58, 133, 138, 149
Military aspects of ciphers, 3, 55
Mirabeau’s cipher, 209
Mixed alphabets, 70, 169
Monoalphabetic substitution, meaning of, 68
Morse alphabet, 210
“Multifid” alphabets, 209
Multi-literal substitutes, 6, 7, 104
Multiple-alphabet ciphers, 108-197
Multiple-alphabet substitution, meaning of, 68
Multiple messages, 56, 85, 140, 146, 185
Multiple-substitutional encipherment, Multiple substitutes, 68, 99,
102, 103, 159
Myszkowsky’s transposition, 51
Negative, or empty, words, 76
“Nicodemus” cipher, 216
Nihilist number-cipher, 164-168
Nihilist transposition cipher, 17, 53
Novelty ciphers, 100, 214
Null cipher, 4
Nulls, legitimate uses of, 37, 55, 110, 201
Nulls, meaning of, 9
Numbers and symbols, encipherment of, 101
Numbers used as substitutes, 154-157, 159-168
Ohaver method for period finding, 160
Open-letter cipher, 4
Pair-count, 202
Pair-encipherment, 198, 199, 200
Parallel frequency counts, 162, 165, 177
“Pattern words,” 73
Pentagram, meaning of, 1
Periodic ciphers, 108-184
“Period” in autokey ciphers, 146
Period, periodicity, 108, 112-114, 127, 138, 166, 195
“Phillips” cipher, 185
Phonetic alphabets, 106
Playfair cipher, 200
“Pointers” for vowel-spotting, 78
Pollux systems, 209
Polyalphabetic ciphers, names of, 108
Polyalphabetic substitution, meaning of, 68
Polyalphabetic substitutions, 108-197
Polybius square, 164, 209
Polygram substitution, meaning of, 68
Polygram substitutions, 198-208
Porta cipher, 118
Primary cipher alphabet, 164, 169
Primary cryptogram (see double transpositions or double substitutions)
Probable word methods, 23, 29, 34, 37, 99, 113, 119, 124, 147,
202, 206
Progression, Progression index, 192
Puncture cipher, 4
“Quagmires” cipher, 182
“Rail fence” transpositions, 12
Reciprocal substitution, 70, 118, 121, 123
Rectangular transposition, 11
“Route cipher,” 14
Route transposition, 11
“Running down the alphabet,” 72, 100, 118, 124, 139, 185, 192, 214
“Running key,” 143
Saint-Cyr cipher, 110
Scytale, 14
Secondary cipher alphabets, 164, 169
Sequence (see statistics)
Seriation, Seriation index, 207, 210
Shift, Shifted alphabets, 69, 70, 72, 108, 119, 133, 172, 177
Short word method for divided cryptograms, 72
Simple substitution, meaning of, 68
Simple substitutions, 69-107
Slides, slide-rules
for decrypting, 141
for encipherment, 110, 119, 123, 169, 170
used in transposition, 13, 61
Specific key, 169
Statistics, some uses of, 14, 18, 21, 40-42, 45-50, 57, 73, 75,
78-85, 88, 113, 130, 141, 150, 153, 174, 187, 201, 202, 213
“Step-up” and “step-down” letters, 89
“Straddling” devices, 105
Strips
used in decryptment of transpositions, 15
used for encipherment of polyalphabeticals, 108
used for decryptment of periodics, 136
Substitutes, 68
Substitution cipher, meaning of, 1
Substitution ciphers, 68-212
Symbols, 68
“Symmetry of position,” 176
Tableaux:
Vigenère, 109
Porta, 118
Beaufort, 121
used for decryptment of periodics, 134
formed in autokey decryptment, 150
high-frequency co-efficients, 152
imaginary, or formed in key-recovery, 170-172, 178
Delastelle, 177
for finding alphabetical intervals, 194
for pair-encipherment, 198
“Taking off” (Transcription), 9, 17, 23, 60
Terminals (see statistics)
Tetragram, meaning of, 1
Transformation device, 102
Transposition, meaning of, 1
Transposition ciphers, 9-67
Trial key, 111, 150
“Trifid” substitution, 210
Trigram, meaning of, 1
Trigram encipherment, 199
Trigram method of solution:
applied to Beaufort, 125
applied to Gronsfeld, 117
applied to Porta, 119
applied to Vigenère, 113
with the use of a slide, 141
Trithème’s alphabet, 7
Unit, 9, 18
United States Army Cipher Disk, 108, 123
United States Army Double Transposition, 54
Variant Beaufort cipher, 121
Variety of contact, 75, 78, 88, 174, 204
Variety count, 75, 88
Vigenère cipher, 108
Vowel-distribution (see statistics)
Vowel-line method for simple substitution, 91
Vowel-solution method, 78, 174
Word-spacers, 100
“Writing-in” (Inscription), 9, 17, 23, 60
Transcriber's Note
The original figures were made with a typewriter, and then lines,
arrows, or circles were added. This file reproduces them as well
as is possible as plain text. Please see the HTML version if you
wish to see the figures with all of their details.
Minor corrections, such as removing extra spaces or punctuation, or
adding missing spaces or punctuation or diacritical marks, were made
without note. Archaic and inconsistent spelling was retained. In two
places, the number of a referenced figure was corrected. Grammar and
spelling were only corrected as in the following list of notable
changes.
In two places, “diagram” was corrected to “digram”.
In three places, “direct” was corrected to “directly”.
In Chapter III, we italicized the first occurrence of “magic square”
because it seemed like the author’s intention. In the same chapter,
“said be clockwise” was corrected to “said to be clockwise”.
In Figure 43, “ISGAY” was corrected to “ISGAU” to match previous
figures and to correct spelling in the plaintext.
In Chapter VII, “those who like method” was corrected to “those who
like this method”.
In Chapter XVIII, “misuse of the Type II slide” was corrected to
“misuse of the Type I slide”.
In Figure 139, “IQCIO” is corrected to “IQVIO”.
In the Appendix, in the list of English trigrams, “WHI 51” was
corrected to “WHI 71”.
Some corrections were made to the exercises in order to rectify the
spelling in their plaintexts or (in one case) to resolve an
inconsistency in its key.
In exercise 33, seven letters were missing. The uncorrected cryptogram
is VINSC FEAEO OHSEF HLEHU NSTNC LTSLC IAESH RHSIR ERMTS ETEPD TOINM
RTTHT TLRUB E.
Exercise 110: “LRSFQ” corrected to “IRSFQ”.
Exercise 115: “FBEHZHWU” corrected to “FBEHZHWG”.
Exercise 116: “HVIYF” corrected to “HVXYF”.
Exercise 128: “JQAOJ” corrected to “JQAOI” and “QNUSG” to “QNUSU”.
Exercise 130: “FBWXJ” corrected to “FRWXJ”.
Exercise 138: “XVKFV” corrected to “XVKEV”.
Exercise 141: “KFTFB” corrected to “KFTFX”.
Exercise 144: “TAJNI WRDB” corrected to “TAJNW ARDB”.
Exercise 145: “TTHNY” corrected to “TTHNJ”.
Exercise 146: “PGOUR” corrected to “PGOUJ” and “VDDMH” corrected
to “VDDMP” (we are not sure about “VFCFT”).
Exercise 147: “UVSYC” corrected to “UBSYC”, “IXYDY” to “IXYPY”,
“VXOOK” to “NXOOK”, and “NNHMM” to “NNHWM”.
Exercise 149: “MS” corrected to “PF”.
Exercise 153: “QY BN QM” corrected to “QY BM QM” and “BL PK QM”
to “BL PO QM”.
Exercise 40 has error(s) in the first 25 letters. We do not know
what was the intention of its creator.
The above list should not be assumed to be correct or complete.
We wish to thank the anonymous proof-readers who volunteered at the
British National Cipher Challenge (www.cipherchallenge.org).