Irregular Types — Columnar Transposition
Square units, in actual use, are less convenient than those
rectangular encipherments in which only one dimension of the block
is restricted, thus permitting that a single key govern messages of
many different lengths. We have a more practical cipher in the
columnar transposition of Chapter IV, and this can be rendered
somewhat safer if care be taken to avoid completing the rectangle.
The preparation of such a block is illustrated in Fig. 28, where the
key-word PARADISE is being used to encipher the following text:
REGRET CHANGE IN SYSTEMS BUT THOUGHT ADVISABLE ACCOUNT INCREASED
VOLUME SENT BY AIR.
Figure 28
P A R A D I S E
6 1 7 2 3 5 8 4
R E G R E T C H
A N G E I N S Y
S T E M S B U T
T H O U G H T A
D V I S A B L E
A C C O U N T I
N C R E A S E D
V O L U M E S E
N T B Y A I R X
X X X . . . . .
First, let us understand the purpose of the four nulls. It
is customary, when cryptograms are to be transmitted by wire
or radio, to make them evenly divisible into five-letter groups.
This usually means the addition of from one to four nulls, and
since the nature of the cipher makes it inadvisable that additional
letters be added to the enciphered cryptogram, any desired nulls
must be added in the block before the columns are
taken off. Another precaution usually recommended is the
avoidance altogether of key-lengths which are divisible by 5,
so that an encipherer is practically never compelled to add a
complete five-letter group in order to leave his rectangle incomplete.
It might be added that our use of letters _XXXX_ is for emphasis only; a
better series would be one of the nature _AAEO_.
The decipherer’s only problem is illustrated in Fig. 29. Knowing the
key, the decipherer knows that there must be eight columns. The
number of letters, 75, divided by 8, results in 9, with remainder 3;
thus, the short columns are to contain nine letters, and there will
be three which contain ten letters. He lays out an 8 x 10 block,
cancels the last five cells, writes his key-numbers across the tops
of the columns, and then begins to copy letters, filling the column
numbered 1, then the column numbered 2, and so on, finally reading
his message by straight horizontals.
Figure 29
P A R A D I S E
6 1 7 2 3 5 8 4
_ _ _ _ _ _ _ _
_ _ _ _ _ _ _ _
...............
...............
_ _ _ _ _ _ _ _
_ _ _ x x x x x
The cryptogram from this block is shown as Fig. 30, and illustrates
the manner in which the decryptor will number the letters of
practically all cryptograms in order that he may quickly locate any
desired letter, or learn, by subtraction, the distance apart of any
two letters. The decryptor, of course, does not know how many columns
the cryptogram contains, and even after he finds out the key-length,
he still does not know exactly the point at which any one column ends
and another begins.
This form of transposition is among the most fascinating of
decryptment problems, and we shall look at it from several angles.
The simplest case is that in which the decryptor correctly assumes
the presence in his cryptogram of some word or phrase whose length is
greater than that of the key; if this probable word is long enough,
he is able to learn, not only the key-length, but the order in which
to write his columns. Our present cryptogram, for instance, has
key-length 8, and contains two nine-letter words, ADVISABLE and
INCREASED. These two words, repeated in Fig. 31, will show what
happens when a word is long enough to overlap the block. With the
word ADVISABLE, the final _E_ falls below the initial _A_, and when
this column is taken off, the letters _A E_ will stand in sequence in
the cryptogram. Similarly, the word INCREASED will provide, in the
cryptogram, a digram _ID_. Should the decryptor suspect the presence
of either of these words, he would look at once for sequences of this
kind in his cryptogram, and the presence of _AE_ (or _ID_) would tell
him that the key-length is probably 8, which is the distance apart of
the two letters in his probable word.
Figure 30
The Cryptogram Prepared for Examination
5 10 15 20 25 30
E N T H V C C O T X R E M U S O E U Y E I S G A U A M A H Y
35 40 45 50 55 60
T A E I D E X T N B H B N S E I R A S T D A N V N X G G E O
65 70 75
I C R L B X C S U T L T E S R
Figure 31
6 1 7 2 3 5 8 4
. . . . . . . .
. . . . . . . A
D V I S A B L E
. . . . . . . I
N C R E A S E D
V O L U M E . .
The ideal case is that in which the probable word is long enough to
furnish more than one of these overlapping letters, as shown in Fig.
32 in connection with the “word” INCREASED VOLUME. Suppose that we
have suspected the presence of this expression in our cryptogram, and
have ascertained that the necessary letters are present for forming
it. We consider its letters one by one, in the order _I_, _N_, _C_,
_R_ . . . . and go through the cryptogram, underscoring (or otherwise
noting) all cases in which the given letter is followed immediately
by another of the letters found in the same probable “word.” But, in
considering any one letter, say the letter _N_, we ignore such
sequences as _NT_, _NB_, _NX_, whose second letters, _T_, _B_, _X_,
do not occur in the expression INCREASED VOLUME. Fig. 33 shows
exactly what digrams of this kind can be found in connection with
letters _I_, _N_, _C_, _R_, _E_, and also the distance (or distances)
apart of the two given letters as found in the probable word. Notice
that in connection with every letter there is one digram in which
this distance is 8, the correct key-length of our present cryptogram.
And when these digrams are selected from the tabulation, and set up
vertically with top letters in the order _I N C R E_, the lower five
letters prove up in the order _D V O L U_. In actual work, the
tabulation must sometimes be made, though ordinarily it will suffice
to start directly with the “proving up.”
Figure 32
I N C R E A S
E D V O L U M
E
I N C R E A S E
D V O L U M E
I N C R E A S E D V
O L U M E
Now let us go ahead and solve the cryptogram, as shown in Fig. 34. We
will assume, to begin with, that our cryptogram has been prepared at
the top of a sheet, and that our various trials are being made on the
blank space beneath it. We will assume also that, having discovered
key-length 8, we have divided this cryptogram roughly into eight
segments, three of which contain ten letters and the rest nine.
First, we are in possession of a series of embryo columns, shown at
(a), and these can be set up without looking at the cryptogram at all.
Having done this, we turn to the cryptogram, find each one of the
sequences again, and lengthen the columns of our beginning block by
adding to each pair of letters a few of the letters which immediately
precede and follow it. Thus, our block begins to build up as at (b);
and, for each time that a partial column is set up in (b), the segment
which contained it is promptly circled out of the cryptogram itself,
which now begins to assume the appearance indicated at (c). Thus some
words have automatically formed on the new lines which tell us
plainly that the final column must contain a sequence _L T E_,
followed by _S_ or _W_, and the appearance of the cryptogram tells us
plainly where to look for it; the final segment is the only one
having enough letters to furnish another nine- or ten-letter column.
Figure 33
Letter Examined & Distance Apart
Sequences Found In Word
I IS 6
ID 8
IR 3
IC 2
N NS 5
NV 8
C CO 8
CR 1
CS 4
R RE 1 4 11
RA 2
RL 8
E EM 9 (6)*
EU 8 (5)*
EO 6 (3)*
ES 2
(*) Distances from the second E
Proving up: I N C R E . . .
(8) D V O L U
At stage (c), we are practically in possession of the numerical key,
and to show this, the cryptogram segments have been numbered. The
first one, containing _V C C O T X_, has been set up in the partial
block as column 3; thus the third column of (b) should have
key-number _1_. The second segment, containing _S O E U Y E_, has
been set up as column 5, showing that the fifth column of (b) should
have key-number _2_. And so on with the rest, until the eight
key-numbers are standing in the order 4-6-1-7-2-3-5-8. This is shown
at (d), and directly below this, at (e) is the encipherer’s original
key. It can be seen that we are now in very much the same position as
the legitimate decipherer; by making a few trials, each time shifting
one key-number from the left side to the right, we need do little more
than decipher. Usually, however, it is quicker simply to go on and
rough out the block we have already started, and then make the
necessary adjustments, approximately as shown in Fig. 35. Having
noted, in the cryptogram, that there are some unused letters,
_E N T H_, on the left side of segment 1, we assume temporarily that
all other unused letters belong to the segment which follows them,
and add them all, indiscriminately, at the top of the block. Where
this is shown, at the left side of Fig. 35, the true key-numbers, as
found in the cryptogram, have been added above the original reference
numbers, and similarly with the adjusted block on the right.
Figure 34
Investigating the Key-length 8
(a) (b)
1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8 4 6 1 7 2 3 5 8
A D V I S A B .
I N C R E A S E E A C C O U N .
D V O L U M E ? I N C R E A S E
D V O L U M E .
E N T B Y A I .
(c) X X X X E H R .
(1) (2) (3)
E N T H/V C C O T X/ ¦R E M U/S O E U Y E/¦ I S G/A U A M A H/Y ¦
(4) (5) (6)
T/A E I D E X/T N¦B H/B N S E I R/A¦S T /D A N V N X/G¦G E O
(7) (8)
/I C R L B X║C S U T L T E S R
x
(d) Apparent key-numbers: 4 6 1 7 2 3 5 8
(e) True key-numbers: 6 1 7 2 3 5 8 4
With the block roughed out, and knowing that a cryptogram of 75
letters using key-length 8 cannot have columns of any other length
than 9 and 10, the first obvious maladjustment is seen in column 1
(key 4), which has only 8 letters. Since this is the 4th segment of
the cryptogram, its remaining letter (or its remaining two letters)
will have to be found at the end of the third segment or at the
beginning of the fifth (keys 3 and 5), that is, at the bottom, or at
the top, respectively, of the columns originally set up as columns 6
and 7. The selection of _H_ from the bottom of column 6 leaves this
column too short, while the top row of the block shows a gap in
sequence, and evidently needs the _E_ at the end of the second
segment. The lone _R_ which remains at the bottom of column 7 is then
erased and written at the top of column 2, and thus we arrive at the
adjustment shown on the right side of the figure, where the only
remaining operation will be that of transferring the misplaced
nine-letter column to its own side of the block. This final
adjustment shows us the segments of the cryptogram in their key
order: 6-1-7-2-3-5-8-4.
Figure 35
Forming and Adjusting a Tentative Block
_4 6 1 7 2 3 5 8 4 6 1 7 2 3 5 8_
1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
E G R T C R E G R E T C
A N G E I N S H A N G E I N S
Y S T E M S B U Y S T E M S B U
T T H O U G H T T T H O U G H T
A D V I S A B L A D V I S A B L
E A C C O U N T E A C C O U N T
I N C R E A S E I N C R E A S E
D V O L U M E S D V O L U M E S
E N T B Y A I R E N T B Y A I R
X X X X E H R X X X X
Column 1 must have another letter, top. (Found at bottom of Column 6).
(4) (3)
Column 6 must then have another letter, top. (Found a bottom of Column 5).
(3) (2)
Column 7, bottom, then shows an extra letter, which must be transferred to
(5)
Column 2, top. When these transfers have been made, as shown on
(6)
the right, all that remains is to transfer the short column (1)
(4)
to the right-hand side of its block, raising it by one position.
Having seen the ideal case, the student will understand how the less
perfect example would be handled, or the case in which the probable
word is not long enough to overlap at all. For the latter, he would
attempt to find some word like CRYPTOGRAM, in which there are letters
such as _C_, _Y_, _P_, _G_, _M_, not likely to appear more than once
or twice in a short text. We need not discuss this latter case, since
we are to see something very much like it before the present chapter
ends.
Now, as a preliminary to those cases in which we are unable to find a
probable word, suppose we turn to the back of the book, and make an
inspection of the tool chest. First in importance, and valuable in
ciphers of all kinds, is the digram chart which O. Phelps Meaker has
been kind enough to prepare especially for this text. To learn how
often he encountered any given digram in his 10,000-letter count,
note its first letter in the horizontal alphabet, at the top of a
column, then note its second letter in the vertical alphabet, at the
beginning of a row, and observe the figure which occupies the cell at
the intersection of this column and row. If the digram is _TH_, its
frequency was 315; if the digram is _JN_, the cell is blank. This
does not mean that the digram _TH_ will appear exactly 315 times in
any other 10,000-letter text, or that _JN_ will never be found
(occurring, say, as initials). It merely shows that the digram _TH_
is of remarkably high frequency, while a digram _JN_ is so rare that
it practically never appears. The most commonly occurring digrams of
this chart have been listed on another page in the order of decreasing
frequencies. A list of the principal reversals is also given, with
other data which will be found useful in the majority of ciphers.
Meaker’s digram chart shows also the frequencies found for single
letters in the same text. These are shown at the extreme right, and
were obtained by adding the figures found on the 26 rows of the chart
proper. When such counts are made, every letter in the text is
considered to be the first letter of a digram, and no attention is
paid to the separations between words. Thus the single-letter
frequencies can be found by totalling either the columns or the rows,
which, except for minor discrepancies, will check against each other.
So much for _frequencies_. Now let us take a closer look at
_sequence_. Certain letters, ordinarily those of lowest frequency,
are peculiar in their _contacts_ with other letters. The shining
example, in most languages, is the letter _Q_, followed, almost 100%
of the time, by _U_ plus another vowel; and if it seems, in the
present text, that the significance of _QU_ is being overlooked, this
is simply because the individuality of this digram, like that of the
German _CH_ (_CK_), is so well advertised that even the novice
encipherer finds a way to avoid using it. It is impossible, however,
to avoid all letters having individual preferences. We still have _J_
and _V_, practically sure to be followed by vowels, and _Z_, almost
as sure. We have _X_, nearly always preceded by a vowel, but more
often followed by a consonant. If these are missing from the
cryptogram, we may have letters like _K_, _B_, and _P_, which confine
an enormous percentage of their contacts to vowels; or to vowels and
liquids; or to letters from the high-frequency group _E T A O N I R S
H_. Even among the high-frequency letters themselves we find that _H_
is followed about 75% of the time by either _E_ or _A_, and that it
is preceded largely by _T_, with _S_, _C_, and _W_ as the next
favorites; or we find that _N_ is inordinately fond of vowels on its
left, though with some preference for consonants on its right. All
information of this kind is present in the digram chart, and usually
is known to the decryptor without recourse to a chart.
For the beginner, however, who might like to have it in a more
visible form, another chart, of a kind which we believe has never
before been published, appears on page 220. This is F. R. Carter’s
contact chart, on which every letter of the alphabet has been listed
in the center of the page, with its favorite contact-letters beside
it. The arrangement here is from the center outward; the letters
shown on the left of any given letter are those which most often
precede it, with percentages as found in Ohaver’s digram chart;
letters shown on the right are those which most often follow, with
percentages from the same digram chart. This information was not
completed to the end for every letter, since the only information
wanted is the actual preferences of each letter, or the fact that it
has none. However, the outermost columns will show the complete
percentages of vowel and consonant contacts for all letters as these
were found in one 10,000-letter text. With such a chart before us, it
becomes very easy, in the absence of _Q_, and other particularly
vulnerable letters, to make good use of whatever letters we happen
to have; and it is hoped that this new “contact chart” will prove
sufficiently valuable to justify Carter’s labor in having compiled
it for us. As to the other data in the appendix, the student will do
well to look it over. The list of trigrams is that of the Parker Hitt
Manual, where _THE_ was shown as having been found 89 times in 10,000
letters, the others graduating downward to _MEN_, found 20 times. Now
let us return to our columnar transposition.
Figure 36
Key-length Columns, 75 Letters
Key 5: (Impossible under system)
Key 6: 3 columns of 13
3 columns of 12
Key 7: 5 columns of 11
2 columns of 10
Key 8: 3 columns of 10
5 columns of 9
Key 9: 3 columns of 9
6 columns of 8
Key 10: (Improbable under system)
Key 11: 9 columns of 7
2 columns of 6
(Etc.)
When a digram _QU_ is actually present in a text, or when it is
fairly certain that some other digram may be present, such as the
_YP_ of CRYPTOGRAM (that is, one composed of two infrequent letters),
it is possible to discover (or limit) the key-length by observing the
distance apart of these two letters in the cryptogram. To approximate
such a case, using the foregoing cryptogram (Fig. 30), we will make
use of the digram _VI_, and, in order to be brief, we will assume
that the letter _V_, position 5, is the only one in the cryptogram,
and that the only _I_’s present are those at positions 46 and 61. In
one case the interval which separates _V_ from _I_ is 41, and, in the
other, 56. As a preliminary step, we may discard all key-lengths
which are factors of 75: 3, 5, 15, 25. In addition, we may discard,
for the time being, the key-lengths 10, 20, etc., which are multiples
of 5. Of those left, any very short length, as 2 or 4, is very
improbable. We may consider, then, possible key-lengths of 6, 7, 8,
9, 11, etc., as far as we care to take them.
To make ready for the investigation, we first prepare a sheet of the
kind shown as Fig. 36, where each possible key-length has been used
as a divisor in order to learn the column-lengths for each one in a
75-letter cryptogram.
Figure 37
Q U I T E A F *
E W F A N S W
I L L B E D E
L I G H T E D **
T O S E E
*) From F in a short column
to E in a long column is an
interval of 5; but from E
to F is an interval of 4.
**) From T to E is an
interval of 5; from E to T
is an interval of 4.
Now let us picture any text written into any block, as in Fig. 37,
where long columns have five letters and short columns have four.
Considering any digram in the text, as _QU_ at the beginning, its two
letters are separated by exactly one column of length, provided the
letters are counted straight down the columns and columns are taken
in one straight direction, or provided the counting is done strictly
upward with columns always taken in one direction. In the case of
_QU_, this column of separation is a long one (five letters), while,
in the case of _AF_, on the right-hand side of the block, it is a
short one (four letters), but in both cases it is a full column. This
is true, also, of the digram _FE_, which is on two different lines,
presuming that, having counted all the way to the end of the last
column, we start again with the first. If both letters are in short
columns, the interval which separates them is that of a short column,
and if both are in long columns, this interval is that of a long
column. But if one letter is in a long column and the other in a short
column, the separating interval may be long or short, according to
whether the columns are taken in straight order or in reverse order.
If the columns of Fig. 37 should be cut apart, and placed in some
other order, then other columns might be placed between the one
containing _Q_ and its neighbor containing _U_, but these would be
_full columns_, never fractional columns, so that the interval from
_Q_ to _U_ would always be an exact number of full columns.
This is what happens in columnar transposition. If the digram _VI_,
which we intend to consider, was actually present in the original
encipherment block, then, in the cryptogram, its letters _V_ and _I_
are separated by an exact number of columns, long or short or mixed.
Also, if the column containing _V_ was taken off first, the distance
from _V_ to _I_ may include the full number of long columns permitted
by the key-length, but must fall one short of including all of the
short columns; but if the _I_ comes first, the opposite is true. Now,
assuming that the only _V_’s and _I_’s in our cryptogram are those
appearing at positions 5, 46, and 61, we find that if the first of
the _I_’s is considered, the distance from _V_ to _I_ is 41, while,
if the other is the one considered, then this distance is 56. We will
investigate, first, the interval 41.
If _V_ and the first _I_ stood in sequence in the encipherment block,
either as _VI_ or as _IV_, then the interval 41 represents a certain
number of complete columns, and if the digram was _VI_ (since the
_V_-column was evidently taken off first), this interval 41 must not
include the full number of short columns, but may include the full
number of long ones.
Consulting Fig. 36, we find that key-length 6 calls for columns
having 12 and 13 letters, and it is impossible to divide an interval
41 into columns of such lengths. The key-length 7 calls for columns
having 10 or 11 letters, of which only two columns may have the
shorter length; an interval 41 can be divided into the right lengths,
but only if three of the columns are short. Thus, if the first _I_ is
the correct one, the key-lengths 6 and 7 are totally impossible, as
is also key-length 8. The key-length 9, however, calls for columns
having 8 and 9 letters, of which six have the shorter length. An
interval 41 can be divided to produce four short columns and one long
column. Again, the key-length 11 calls for columns having 6 and 7
letters, of which two columns may be short; and an interval 41 will
provide five long columns and one short column. These two key-lengths,
then, 9 and 11, are possible, presuming that the first _I_ is the one
which actually followed _V_. When the other _I_, interval 56, is
investigated in the same way, it is found that the only key-lengths
possible are 8 and 11.
So that if the digram _VI_ is present at all, the key-length must be
8, 9, 11, or something longer. Since the key-length 11 is possible in
both cases, this is the one which tempts; when it fails, the remaining
two can be tried. The student may decide for himself whether a trigram
_IVI_ is possible, considering the distance apart of the two _I_’s. It
will be readily understood how this method, in combination with the
one first explained, could be used, say, in a cryptogram where the
suspected word is CIPHER, with the low-frequency letter _P_ occurring
only once.
* * *
Totally aside from analysis, there are many ways in which the
key-length can become known, or suspected. If the correspondence is a
military one, it may have been learned by espionage, perhaps through
careless talk on the part of an enlisted man; or, because of careless
habits on the part of the authority providing the keys, in having
confined himself always to certain lengths. Knowing the key-length is
two-thirds of the battle. It enables us, as in our former case, to
mark off the cryptogram into its approximate column-lengths, making
it easier to know the approximate whereabouts of any several letters
supposed to form a sequence. It even enables us to prepare a block,
which, cut apart to form paper strips, will effect a mechanical
solution almost as easily as in the case of the completed unit.
Such a block, for our foregoing cryptogram (Fig. 30), can be studied
in Fig. 38, and is explained as follows: An 8-unit key, used on a
75-letter text, calls definitely for three 10-letter columns and five
9-letter columns, and these columns have become eight segments in the
cryptogram. If all three of the long columns were taken off first,
then the arrangement shown at (a) has every letter in its proper
column. And if all three of these were taken off last, then the
arrangement shown at (b) has every letter in its proper column. With
blocks shown for the two extreme cases, it can be seen that the block
at (c) is a combination-block, in which one of the two extremes has
been superimposed upon the other, so that every column in block (c)
shows every letter which it could possibly have contained. By
concealing the letters of the “cap,” we have a duplicate of block
(a); and by changing the alignment, so as to bring all of the topmost
letters into the same row, we have block (b), with a “cap” attached
at the bottom.
Figure 38
Preparation of Strips for a Known Key-length
(a) Long Columns at Left (b) Long Columns at Right
E R I T B S G C E X Y A X I X X
N E S A H T E S N R E H T R G C
T M G E B D O U T E I Y N A G S
H U A I N A I T H M S T B S E U
V S U D S N C L V U G A H T O T
C O A E E V R T C S A E B D I L
C E M X I N L E C O U I N A C T
O U A T R X B S O E A D S N R E
T Y H N A G X R T U M E E V L S
X E Y N B R
(c) Combination Block (d) Matching Strips
1 2 3 4 5 6 7 8 . . | 8 | .
a x i 3 . . .
y h t r x . 5 . 4
x e y n a g x y x x a
E R I T B S G C e t C h ←
N E S A H T E S I n S y
T M G E B D O U S B U T
H U A I N A I T G H T A
V S U D S N C L ** A B L E **
C O A E E V R T U N T I
C E M |X I| N L E A S E D
O U |A T R X| B S M E S E
T |Y H N A G X| R A I R X
|X E Y H R T ←
Y A N
Comparing block (c) with the two above it: If the first column of (a)
was actually a short one, then its last letter, _X_, belongs at the
top of the second column. The making of this transfer would cause the
second column to have eleven letters, so that it would become
necessary also to transfer the last letter of the second column to
the top of the third; this third column would then have too many
letters, and its last letter would have to be transferred to the top
of the fourth, which at present has only nine and may have another.
But if the second column was also short, then there are two of its
letters which belong at the top of column 3. And if this column, too,
was a short one, it has three transferable letters at the bottom.
To prepare such a block, first write the cryptogram as at (a), and
mark off its transferable (uncertain) letters by the following rule:
One for the first column, two for the second, and so on, until the
number is _equal to the number of long columns_, which is the maximum
number possible. But if the final row is more than half filled, _the
maximum will not be reached_, and a check may be made by marking off
letters from right to left: zero for the last column, one for the
next-to-last, two for the third-to-last, and so backward to the number
which equals the number of long columns. Having marked off the
transferable letters, form the “bonnet” by copying these, in each
case, at the top of the following column, preferably making some
clear distinction to show the duplication. For this latter purpose,
many solvers use red ink. In this kind of work, as we saw in a
previous case, the spacing must be accurate both laterally and
vertically, since many of the letters belonging to the same sequence
are not found on the same row. A few of the strips cut from block (c)
have been matched at (d), where the beginning was made from the
common suffix -_ABLE_. The duplicated letters _A H Y_ have shown up
plainly, partly by the style in which the letters are written, and
partly, too, by the fact of consecutive column-numbers, 3 and 4. This
same thing is true of the letters _X T N_, column-numbers 4 and 5.
These numbers, it must not be forgotten, are also the serial numbers
of the cryptogram segments, and thus are the key-numbers. With the
eight strips correctly matched, and any misplaced columns transferred
to their own side of the block, the strip-numbers as they stand
across the top will reproduce the numerical key.
The matching of strips is generally a purely mechanical process, in
which impossibilities are not considered. However, having before us a
block (a) or (b), it is possible to apply the principle used with our
former digram _VI_, and find out in advance whether certain letters
found on two strips can possibly have stood in sequence. Nor is the
cutting apart of the strips really necessary; it is merely a
convenient method for dispensing with mental effort.
Now suppose we consider this same cryptogram on the theory that its
key-length cannot be determined, or restricted to certain
possibilities. Our first step is to select, somewhere in the
cryptogram, a segment which is to be set up vertically on a sheet of
paper to act as a _trial column_. If we select it from the body of
the cryptogram, we shall have to make it a rather long segment, since
we are uncertain as to whether it represents one column or parts of
two. We should do this, however, if the body of the cryptogram shows
_Q_, or any other letter or series of letters likely to be vulnerable.
Otherwise, we know definitely that one of the columns begins with the
first letter of the cryptogram, and that another column ends with the
final letter of the cryptogram, and one or the other of these two
segments is usually chosen, preferably the one containing the largest
number of vulnerable letters. If we have a probable word, and find
that its letter _P_, or _M_, or _G_, is the only one in the
cryptogram, we select the segment which contains this _P_, or _M_,
or _G_.
Figure 39
Tests, in Attempting to
Judge Column-Length
E G 20 ( 15)
N G 75 ( 83)
T E 94 ( 74)
H O 46 ( 42)
V I 19 ( 14)
C C 12 ( 6)
C R 7 ( 12)
O L 17 ( 36)
T B 14 ( 8)
X X - ( -)
R C 14 ( 7)
E S 145 (115)
M U 13 ( 7)
U T 45 ( 35)
S L 6 ( 9)
(First column of figures
is taken from Meaker's
chart; the second is from
Ohaver's).
Wherever the trial segment is taken, there is always the question
as to how many letters ought to be included. In Fig. 39, the decryptor
has decided to take the beginning segment of the cryptogram, and has
started with 15 letters. He has written beside it another 15-letter
segment, chosen because of _NG_, _HO_, _VI_, and is attempting to
tell, by the appearance of his digrams, and their frequencies as
taken from two different digram charts, just about how far his digrams
are uniformly good. If the nulls in use are actually _XX_, he knows
immediately that this is the end of his two columns; otherwise, his
digrams are acceptable throughout. If he sets down beside each digram
its frequency as taken from Meaker’s chart, he might decide that his
digrams are good as far as _UT_, depending somewhat on the letters
represented in our _XX_. Using Ohaver’s frequencies, he would feel
sure that his digrams are good as far as _OL_. In many cases the
frequencies shown for the lower digrams will grow so erratic as to be
plainly unlikely; and in other cases, more difficult than the present
one, a check on the probable column-length can be had by preparing a
similar set-up for the end-segment of the cryptogram, in which the
lower digrams are excellent, while those extending upward may grow
erratic. This decryptor is safe, however, in accepting as much or as
little of the length as he likes; there will be a more definite line
of demarcation when he attempts to write beside these a third column
of 15 letters. The only cases which ever give trouble are those in
which a short text has been enciphered with a long key. Key-lengths,
generally speaking, hardly ever run outside of limits 5 to 15, that
is, lengths which come from single words. Thus a tentative key-length
10, 11, 12, lying half-way between these extremes, is always safe to
try. The key-length 10, applied to 75 letters, gives columns of 7 or
8, and, in the discussion which follows, the tentative column-length
was fixed at 8 letters.
Figure 40
(The numbers assigned to these set-ups merely indicate the order in which the
second segments were taken).
1 8 9 10 11 2 3
E U 7 E Y 17 E B 11 E X 17 E G 20 E S E O*
N S 51 N T 110 N N 9 N G 75 N G 75 N O N E
T O 111 T A 56 T S 32 T G 1 T E 94 T E T U
*H E 251 *H E 251 *H E 251 *H E 251 *H O 46 H U H Y*
V U - V I 19 V I 19 V O 6 V I 19 V Y* V E
C Y - C D - C R 7 C I 15 C C 12 C E C I
C E 55 C E 55 C A 44 C C 12 C R 7 C I C S*
O I 13 O X - O S 37 O R 113 O L 17 O S O G
488 508 410 490 290
4 5
(1) (8) (9) (10) (11)
488 508 410 490 290 E E E I
H E 251 H E 251 H E 251 H E 251 H O 46 N U N S
237 257 159 239 244 T Y T G*
H E H A
Ranked in the order: 8, 11, 10, 1, 9 V I V U*
......... C S* C A
C G* C M*
O A O A
Same Test, Using Mr. Ohaver's Digram Frequencies:
6 7
1 8 9 10 11
E S E H*
E U 6 E Y 24 E B 24 E X 14 E G 15 N G N Y
N S 47 N T 97 N N 8 N G 83 N G 83 T A T T
T O 92 T A 64 T S 27 T G - T E 74 H U H A
*H E 305 *H E 305 *H E 305 *H E 305 *H O 42 V A V E
V U - V I 14 V I 14 V O 9 V I 14 C M* C I
C Y 1 C D - C R 12 C I 19 C C 6 C A C D*
C E 46 C E 46 C A 36 C C 6 C R 12 O H* O E*
O I 15 O X 1 O S 35 O R 99 O L 36
512 551 461 535 282
(Set-up No. 2
(1) (8) (9) (10) (11) would have
512 551 461 535 282 been tested.)
H E 305 H E 305 H E 305 H E 305 H O 42
207 246 156 230 240
Ranked in the order: 8, 11, 10, 1, 9 (as before).
Usually these trials are made by setting up the trial column (in
pencil) several times in succession, so that several of the possible
combinations can be seen side by side, in order to determine which is
best. Sometimes this can be decided by simple observation. Otherwise,
the combinations can be subjected to a digram test. This is made by
setting down beside each digram, as formed by each pair of columns,
its frequency as taken from a digram chart. These figures are then
added in each of the set-ups, and the supposition is that the
combination furnishing the highest frequency-total will be the
correct one, provided this high total has been produced by all of its
digrams collectively, and not by some one or two individual digrams.
With short columns, such tests are never conclusive, but with as many
as ten or twelve digrams they are nearly always dependable, and even
with only five or six digrams they will often select a correct
combination.
Figure 41
1 2 3 4
Y E 12 D E 39 R E 148 E E 39
E N 101 E N 120 A N 172 O N 145
I T 88 X T 1 S T 121 I T 88
S H 30 T H 315 T H 315 C H 46
G V - N V 4 D V 4 R V 5
A C 39 B C - A C 39 L C 8
U C 17 H C 2 N C 31 B C -
A O 2 B O 11 V O 6 X O 1
308 492 836 332
.....
Same Test, Using Mr. Ohaver's Digram-frequencies:
1 2 3 4
Y E 8 D E 64 R E 139 E E 57
E N 120 E N 101 A N 168 O N 162
I T 90 X T 4 S T 119 I T 90
S H 40 T H 377 T H 377 C H 53
G V 1 N V 1 D V 1 R V 6
A C 35 B C - A C 35 L C 1
U C 11 H C - N C 34 B C -
A O 2 B O 13 V O 9 X O 1
288 560 882 370
It was decided here to choose as the trial column the first eight
letters of the cryptogram: _E N T H V C C O_. This column is filled
with consonants, indicating that those which follow or precede it
might contain a number of vowels; and of the six consonants present,
practically every one could be called a “vulnerable” letter, or, as
we say in the Association, a “clue-letter.” If we wish, for instance,
to choose a column which will fit well on the right-hand side of this
trial column, we can search the rest of the cryptogram for two
consecutive vowels to follow, respectively, _H_ and _V_, and these
two vowels we should expect to find followed, either immediately or
at interval 2 by some letter (usually a high-frequency one) which
will follow at least one of the _C_’s. This kind of pattern,
unfortunately, was found eleven times. In practice, we should
probably abandon it rather than copy down and test eleven
combinations; here, however, the eleven set-ups can all be seen in
Fig. 40, accompanied by serial numbers to show the order in which
their second columns were taken from the cryptogram. Some of these
have not been tested. Of the five retained, particular attention is
called to the fact that the one having the very lowest total is
actually the correct one, as may be seen by turning back to the
encipherment block. But when a single row of corresponding digrams
(_HE_ in the first four set-ups and _HO_ in No. 11), has been
subtracted throughout, it is seen that No. 11 moves upward toward its
proper rank, having now the second highest total. In practice, it
might even be selected in preference to No. 8, which grows erratic
after its fifth digram (frequencies of 0, 55, 0). But the
column-length 5, in practice, is not unlikely, so that a test made
on the right-hand side of our trial column has not been at all
conclusive.
Figure 42
Trigram Observation
1 2 3(*) 4 5
REU REY ¦ REB REX REG
ANS ANT ¦ ANN ANG ANG
STO STA ¦ STS STG STE
THE THE THE THE THO
DVO DVI DVI DVO DVI
ACY ACD* ACR ¦ ACI ACC
NCE NCE NCA ¦ NCC* NCR
VOI VOX* VOS ¦ VOR VOL
(*) Acceptance of combination 3 would
entail shortening columns.
Postponing the decision, then, let us take a fresh sheet of paper and
make some tests for columns which can be fitted on the left-hand side
of our trial column. Here, we find that the best “clue-letters” are
_N_ and _H_, standing at interval 2. To precede _N_, we should like
to find one of the vowels of which it is so fond, and to precede _H_,
we hope to find either _T_ or one of the letters _S_, _C_, _W_. That
is, we hope to find a pattern in the rest of the cryptogram in which
some vowel, other than _Y_, is followed at interval 2 by one of the
letters _T_, _S_, _C_, _W_. This time we find only four segments, and
when the test is made for these, as shown in Fig. 41, the resulting
totals point decisively to the correct combination, which is No. 3.
Notice, in both of these tests, that results are identical whether
the frequency-figures are those counted by Meaker or those counted by
Ohaver: In the test of Fig. 40, the five combinations (using either
chart) are ranked in the order 8, 11, 10, 1, 9, while the test of
Fig. 41 has ranked its four combinations in the order 3, 2, 4, 1.
Selecting, then, combination No. 3 of Fig. 41, let us return to the
doubtful tests of Fig. 40 and attempt to effect a combination between
our No. 3 and some one of the five previously considered worth
retaining. Thus we can make an observation of trigrams, as shown in
Fig. 42.
Figure 43
/E N T H V C C O/T X R E M U X O E U Y E I S G A U A M A H Y
T A E I D E X T N B H B N S E I/R A S T D A N V/N X/G G E O
I C R L/B X C S U T L T E S R
Here, we must be guided by our judgment, since trigram tests, even
with figures available, would never be feasible on columns of this
length. The acceptance of No. 3, evidently, would mean the cutting of
our column-length to 5 letters, which, as we have said, is not at all
unlikely in an actual case. The two highest tests from Fig. 40,
however, are those included in Nos. 2 and 5. With reference to No. 2,
where the right-hand digrams have the higher total, it is not
impossible that the trigrams _ACD_ and _VOX_ were actually in use, or
that the set-up should be cut, above the trigram _ACD_; but No. 5 is
the one which _carries word-suggestions all the way to the end_.
Figure 44
(a) (b) (c) (d)
A R E G A R E G R A R E G R E X A R E G R E T
H A N G H A N G E H A N G E I C H A N G E I N
Y S T E Y S T E M Y S T E M S S Y S T E M S B
T T H O T T H O U T T H O U G U T T H O U G H
A D V I A D V I S A D V I S A T A D V I S A B
E A C C E A C C O E A C C O U L E A C C O U N
I N C R I N C R E I N C R E A T I N C R E A S
D V O L D V O L U D V O L U M E D V O L U M E
...S
...R
(e)
/E N T H V C C O/T X /R E M U S O E U/Y/E I S G A U A M/A H Y
T A E I D/ E X/T N B H B N S E/ I/R A S T D A N V/N X/G G E O
I C R L/B X C S U T L T E S R
(f) (g) (h)
X A R E G R E T X * R E G R E T * * R E G R E T C H
C H A N G E I N C H A N G E I N C H A N G E I N S Y
S Y S T E M S B S Y S T E M S B S Y S T E M S B U T
U T T H O U G H U T T H O U G H U T T H O U G H T A
T A D V I S A B T A D V I S A B T A D V I S A B L E
L E A C C O U N L E A C C O U N L E A C C O U N T I
T I N C R E A S T I N C R E A S T I N C R E A S E D
E D V O L U M E E D V O L U M E E D V O L U M E S E
S E N T B Y * I S E N T B Y A I S E N T B Y A I R X
R X X X R X X X * R X X X X
6 1 7 2 3 5 8 4
.....(Key).......
With the adding of other columns, which can be done on either side of
the set-up, further digram tests can be made (taken only on the two
extreme right-hand or left-hand columns), but in most cases no further
tests are needed. Considering, for instance, that No. 5 is the
combination tentatively accepted, we need a segment from the
cryptogram containing the _U_ which ought (apparently) to follow
_THO_, then the _S_ or _C_ which ought (apparently) to follow _DVI_;
that is, we want to find a sequence _US_ or _UC_ in the rest of the
cryptogram; and this (apparently) should be followed by two vowels in
succession, to fit after the sequences _ACC_ and _NCR_. In other
words, we know exactly what kinds of letters ought to make up the
column which can be added on the right side of combination 5, and even
the specific letters. Or, if it is the left side on which we have
chosen to fit the new column, we need a segment containing the _A_ of
the apparent _ADVI_, followed at interval 2 by the vowel, probably
_I_, which ought to precede a trigram _NCR_.
In Fig. 43, the three segments of set-up No. 5 have been circled out
of the cryptogram (to prevent further use of their letters), and the
segment chosen to fit on the left side of set-up No. 5 has been
underscored, ready to be circled out in case it is found to fit. It
is now possible to see the suggested nine-letter words, ADVISABLE and
INCREASED, the guessing of which would permit us to apply the easy
method first described.
With or without these guesses, the rest of the solution, as outlined
in Fig. 44, is now plain sailing. At (a), the underscored segment of
Fig. 43 is in place. At (b), the column containing the desired _US_
and following vowels has been set up on the right, where we seem to
need the _E_ or _A_ of ADVISE or ADVISABLE, followed immediately by
the _U_ or _R_ of ACCOUNT or ACCORD. At (c), we have found the
segment, and at (d) (usually earlier), we are introduced to the
actual lengths of our columns.
Figure 45
The OHAVER CV-VC Test:
RATEB OWSTT EETOP UUIMC YUAOG
AIOIA OBSTB BAKAR YYEDT UWYNT
NNFKG FJSOT WYQAR IROIH.
TT TO cv TM TE cv
WO cv WP WC WD
YP vc YU YY YT vc
(QU) (QU) (QU) (QU)
AU AI AA AW vc
RI cv RM RO cv RY cv
IM vc IC vc IG vc IN vc
4 2 2 5
The following is the original cryptogram
used by Mr. Ohaver for demonstration:
TVYIE TRROR EHNIA EUDSR IEONI
ORENA EEORP TEALO LTSUH LHQNO
UCADD CSAAE TDVFU GNNYC YI.
(Reprinted from Detective Fiction Weekly
of October 8, 1927, with permission of
The Frank A. Munsey Company).
This latter can be seen by looking at the cryptogram (e), where all
segments, as soon as selected, have been circled out. In finding a
column which would complete the very evident word SYSTEM and, at the
same time, furnish a letter suitable to precede _HA_, we find that
this is the end-segment of the cryptogram, and would leave only two
letters — far fewer than the number needed for furnishing another
column.
At (f), we have extended the rest of the columns by two (and one)
letters, except that there is a gap in sequence on the next-to-last
line. At (g), we have transferred the letter which will fill this gap,
leaving a misplaced _X_ at the top; and, at (h), we have placed this
_X_ where it belongs and are now ready to transfer the two misplaced
columns and recover the key. This key, as before, is found by
numbering the segments of the cryptogram, and assigning these
key-numbers to the correct columns in the adjusted block. It is
usually possible to go further, and learn the long words on which
such keys might have been based.
Concerning digram-tests, Ohaver suggests another which is more
quickly made than the frequency test, and which the writer, so far,
has found fully as reliable. Using “C” for “consonant” and “V” for
“vowel,” he speaks of this as his VC-CV or “mixed” test. A digram
like _HA_ is a _cv_ digram, one like _AT_ is a _vc_ digram, and
others are _vv_ and _cc_ digrams. His theory is this: Since almost
two-thirds of the digrams used in the language will be of _mixed_
formation, that is, either _vc_ or _cv_ digrams, it stands to reason
that the set-up containing the largest number of “mixed” digrams
would probably be the correct choice. The student may look it over in
Fig. 45.
* * *
Figure 46
The Myszkowsky Cipher
(a) Keyword: CURTAINS
C U R T A I N S C U R T A I N S C U R T A
4 19 11 16 1 7 9 14 5 20 12 17 2 8 10 15 6 21 13 18 3
R E G R E T C H A N G E I N S Y S T E M S
Cryptogram: E I S R A S T N C S G G E H Y R E M E N T.
(b) Keyword: PARADISE
P A R A D I S E P A R A D I S E P A R A D
14 1 17 2 7 12 20 10 15 3 18 4 8 13 21 11 16 5 19 6 9
R E G R E T C H A N G E I N S Y S T E M S
Cryptogram: E R N E T M E I S H Y T N R A S G G E C S.
As to possible variations, a cipher with a new name is not necessarily
a different cipher. Fig. 46 shows a cipher originated many years ago
by the cryptologist E. Myszkowsky, and advertised by its inventor as
non-decryptable. The key-word here is repeated often enough to furnish
one key-letter for each text-letter, nulls being added, when
necessary, to prevent the complete unit which would result if key-word
and text were allowed to end at the same point. This long series of
key-letters is then treated as a single word, and is converted to a
numerical key in the usual way, all _A_’s receiving the first numbers,
all _B_’s the next numbers, and so on. The message of the figure is
very short: REGRET CHANGE IN SYSTEMS. Try enciphering this in the
ordinary columnar transposition, using first the key-word CURTAIN,
which contains no repeated letters, and afterward the key-word
PARADISE, which has a repeated letter _A_. In the second case, what
happens to the two columns belonging to the _A_-numbers? Suspecting
a Myszkowsky encipherment, how could you go about unscrambling the
two? Suppose there were three?
Fig. 47 plays another variation on the columnar theme. This cipher,
originated by a member of the _American Cryptogram Association_,
follows the rules of columnar transposition in all respects except
that pairs alternate throughout with single letters (The text is:
CHIEF WANTS YOU TO INTERVIEW SMITH). Can you pick out at a glance the
really vulnerable feature of this cipher, and formulate a special
method for its solution?
Figure 47
The "AMSCO" Cipher (A.M.Scott)
R A C K E T
5 1 2 4 3 6
CH I EF W AN T
S YO U TO I NT
ER V IE W SM I
T HX
Order for taking off:
I YO V HX / EF U IE /...
Cryptogram: I Y O V H X E F U I.....
29. By NEMO. (A military message).
A O T O I N E H T C T O T L I I A W G E L P R V L R I I R I U A D E O
W L R R R L C M E O N P E P T A V T S O H O E E N L S N P S S B Y T S
L R O P D R G E T S S T S Y A W N E.
30. By PICCOLA. (Hostilities?)
T A M L R I T E D W E E D H H N P W O S W R S H C N O I E D O H I L T
C S T N I W A A R C D H H D A I E T P T R L R O W A S E E T A K F P W
G M A T X E K A H D P I L E O F H W G I N H A K S F S S A A A H E H N
D H H E H.
31. By AMSCO. (The "AMSCO" Cipher).
N W L E L N T L C S L W D L Y L N S O O I D F I N R U C H A L N D C B
S I D E A I T E T I K S T B E E O U T J A T I L I A C O R E A Y E E G A O.
32. By PICCOLA. (Can you recover this nice long keyword from the numbers?)
Y K I E T N T H H E X I A E N U B A K E E W S C S I H T N L N E N E A
K I E O B O L I E E A M C I F T I N A H S K A N I D L G S O E E I T T
S W H L L E U A D H F S H A B E O E N O A N O S C P H S N O D H T X R
N H R E A.
33. By PICCOLA. (An easy Myszkowski. Probable words: SOLVE, CIPHER, COLUMN).
V I N S R C F E A E O O H S E F H L E T F H U N S T N C L T S L C I A
E E S H R H S I R E T T M T S E T E P D T S O I N M R T T H T L O L R U B E.
34. By PICCOLA. (Nothing like a bit of "philosophy" - oyeah?)
E L O S W E A H X P N N T R N H L W I E G E I G E A E Q A G L E A R R
Q L O N K E S Q L O R N X A R S P X S E E A E I P A G L R E P R Y M T
H N K S E I X X A Y.
35. By PICCOLA. (Not so easy; still, it's just another columnar).
H R O T E T E T E H I W E O T T D A O D K G DT C E R A I W O S Y N H
Y R H T W.