CHAPTER IV

Geometrical Types — The Nihilist Transposition

In the preceding chapter, we glanced at the most elementary form of _columnar transposition_: a text is written into a block by rows and taken off by columns in such a way that even though all or part of the columns may be reversed in direction, these columns are always left standing one after another in regular order. Columnar transposition becomes less crude when the order for taking off the columns is an irregular one, governed by a changeable numerical key, the length of this key governing also the width of the rectangle. This process can be examined in Fig. 11. In this figure, the numerical key, 4 1 6 5 3 2 7, was first derived from a _keyword_, HALIFAX, according to the following very common plan: The two _A_’s, taken from left to right, receive the first two numbers; the third number, in the absence of _B_, _C_, _D_, and _E_, is assigned to _F_; and so on, following the alphabetical rank of the letters present, and taking repeated letters from left to right. The presence of seven numbers implies seven columns, and it is said that the _key-length_ is 7. When a text has been written into a block of that width, with a key-number standing above each column, these columns can be taken off in the order shown by the numbers, and not in regular sequence.

Figure 11

Usual Plan for Transposing Columns

H A L I F A X 4 1 6 5 3 2 7

L E T U S H E A R F R O M Y O U A T O N C E C O N C E R N I N G J E W E L S X X X X

Cryptogram: E R U C I L H M N E

E X S O O C J X L A O E N E U, etc.

The key, used exactly as described, is a “taking off” key, and this is the common way of using one. It can, however, be used for “writing in” the successive units, placing the first letter of a given unit beneath number 1, the second letter beneath number 2, and so on until the seventh letter has been written below number 7, afterward beginning with the first letter of another unit below number 1 again. Under this plan the first unit of our figure, _L E T U S H E_, would have been _written in_ in the order _U L H S T E E_. Since all units would follow exactly the same pattern, the resulting _columns_ would be identical with those of the present block; the only essential difference would be that the new columns are already transposed, and can be taken off in straight order. The two resulting cryptograms, however, would not be the same. The unit which was _written in_ in the order _U L H S T E E_, would have been in the order _E H S L U T E_ had the method been that of _taking out_ (or “off”).

The Nihilist transposition is ordinarily accomplished by “writing in,” and its numerical key is applied to _both columns and rows_. Thus its major unit is a square, and the seven-letter keyword HALIFAX, applied to both dimensions of a rectangle, demands a unit of 49 letters, while the shorter word SCOTIA, key-length 6, requires a unit of 36 letters.

Theoretically, this cipher is a _double transposition_, requiring two successive operations as shown in Fig. 12. But in practice, these two transpositions can take place simultaneously as pointed out in Fig. 13. The operator, having laid out his key-numbers at top and side of his square, begins his writing in the cell at which the column headed by number 1 crosses the row headed by number 1. He _writes in_ his first unit, proceeds to the row numbered 2 for the writing in of his second unit, then to the row numbered 3, and so on, taking rows in the order shown by the numbers at the left, and placing the letters of his unit by following the numbers across the top. Thus, with only a little concentration, he has the entire major unit at one continuous writing. The decipherer, too, having restored his cryptogram unit to its block and written his two series of numbers, may read, or copy, continuously. The decipherer, in fact, uses the exact method which would produce a Nihilist cryptogram if a key were used in the “taking out” manner. What we have described is the encipherment of a single major unit; and all cryptograms must contain an exact number of these major units.

Figure 12

Nihilist Plan

(a) Transposal of Columns (b) Transposal of Rows

S C O T I A S-5 E U J W T O 5 2 4 6 3 1 C-2 R A F O R E O-4 A N E B C O S E U H T L (Let us h) T-6 X L X X S E R A F O R E I-3 A Y U T O M A Y U T O M A-1 S E U H T L (Let us h) A N E B C O E U J W T O (c) Cryptogram: E U J W T O R A F O R E A N E X L X X S E B C O X L X X S E A Y U T O M S E U H T L.

The second operation, that of taking off the cryptogram, is not always done by straight horizontals as we have shown this under (c) of Fig. 12. This, of course, is the expected way; but the Nihilist square is quite frequently taken off by some other one of the forty-odd routes possible to rectangular transpositions. The decipherer, knowing this route, merely writes his units back into their blocks; but the decryptor is often faced with a preliminary problem of discovering how they were taken off. Sometimes he must also discover how many units a cryptogram contains.

To understand how such problems are solved, it is necessary to pause and consider the make-up of ordinary written plaintext. English vowel-percentage, as mentioned, is about 40%, and practically never varies out of its limits 35%-45%. Each 40 vowels are fairly _evenly_ distributed throughout their 100 letters. Take any English text whatever, not composed of initials or otherwise distorted, and, beginning where you please, mark it off into ten-letter segments and count the vowels in each of the segments. You will find that the majority of these have exactly the normal number of vowels, which is 4. Others will have 3 or 5, which, though outside of the limits 35%-45%, are the closest variations possible. It will be a rare segment indeed which contains fewer than 3 vowels or a greater number than 5.

But suppose, having marked off such a text into ten-letter units, or segments, we take each of these segments individually and mix up the order of its letters, though still allowing it to stand where it is. And suppose, having done this, we erase the original division-marks and, beginning at some point in the midst of a former segment, we again mark off a series of ten-letter units, and count the vowels of these new segments. This time, we are just as likely as not to find seven or eight vowels in one segment and none at all in the next, depending on just what we did to the old units, and still we have not actually mixed the units; we simply have our division marks in the wrong places. Imagine, then, how the vowel distribution can vary when a transposition is one so planned as to break up units and scramble their letters.

This fact of uniformity in vowel distribution is of enormous assistance in dealing with the simpler transpositions. For instance, it may be that what we want to know is the length of the units, and that what we have is a cryptogram of 144 letters, which could be a single square, or a series of 36-letter squares, or even a series of 16-letter or 9-letter squares. We may start at the beginning of this cryptogram and mark it off into equal segments of any length we like, afterward counting the vowels per segment. If every segment shows approximately a 40% vowel count, the chances are that we have a series of intact units, each one merely transposed within itself; but if one segment shows 50%, another 30%, another 28%, and so on, we may be quite sure that our division marks are in the wrong places.

Figure 13

5 2 4 6 3 1 5 2 4 6 3 1 5 2 4 6 3 1

5 5 5 2 2 . A . . . E 2 R A F O R E 4 4 4 6 6 6 3 3 3 . Y . . . M 1 . E . . . L 1 S E U H T L 1 S E U H T L

Returning, now, to the Nihilist cipher, suppose we consider the make-up of its major unit, that is, of any one block. This major unit is a series of minor units, and each of these minor units, at the time of encipherment, was written by itself on its own line. In the beginning, it was a small fragment of plaintext, presumably conforming closely to a 40% vowel count. It is true that we placed it on the line in transposed order, but we did not remove any of its letters or add any new letters. Even in the transposal of the lines themselves, we merely removed a number of intact units from one place to another. There has never been a time, throughout the entire encipherment, when we took any letter out of its original minor unit and put it with some other unit. Thus, as we first see our completed Nihilist square, we still have, on each horizontal line, a small fragment of an English sentence in which all of the original vowels are still present. If such a block is now taken off by straight horizontals, it is no more than a series of intact units. To break up these units, we must at least take it out by verticals; and they will, of course, be much more thoroughly mixed when taken out by diagonals or spirals.

The decryptor, hoping for the best, writes his cryptogram into a square (or series of squares) by straight horizontals and counts the vowels per horizontal line. If his block is wide, he may estimate the actual number of vowels represented by 40%; if it is narrow, he may only roughly approximate the number; but in either case what he hopes to see is _evenness of distribution_. More than half of his units must be exactly normal, and any which are not exactly normal must show the smallest variation possible. If he finds that this is the case, he assumes that his block arrangement is the encipherer’s original square, with only the minor possibility that half of his lines may be written in the wrong direction. If his distribution is not uniform, he counts the vowels per _column_ so as to find out what kind of distribution he would get from a vertical arrangement (ascending or descending). If this, too, fails to show him a uniform vowel distribution, he writes out a new block by the route of alternating verticals (or gets this count from his first block; this is possible, though a little confusing). Afterward, he may go on to the diagonals and spirals until finally he reaches the arrangement in which more than half of his horizontal lines show a 40% vowel count, and the rest a minimum variation.

Now let us consider a concrete example of decryptment. The (purely imaginary) history of the cryptogram shown as Fig. 14 is meager. It was taken from the body of an unnamed man, killed in attempting to dynamite a bridge in an American town called Baysport.

Figure 14

I Y W B B O R T A F T I X D G S S E G H N A T O O I T O X T L U T R E

L X F A Y S D R C H T O M E D E I O V I K F T V T L A E U.

To begin with, the cipher appears to be transposition. Its cryptogram shows 37½% of vowels, very close to the number expected of English or German. It is too short to provide any reliable distinction between these two languages, but the source of the cryptogram points to English. Again, the encipherer, although he has grouped his message in the usual fives, has neglected to complete his final group with a null, and from this we judge that 64 letters is the actual length of the message. The fact that 64 is a _square_ is promptly noticed. But it is also the sum of several smaller squares, and the unit might be 16. To investigate this possibility, we may mark the cryptogram off into four equal segments of 16 letters each, and count the vowels per segment. The normal number of vowels in a 16-letter segment should be about 6, and segments of this length are long enough to afford reliable information, so that we may promptly discard the possible unit 16 when we find that the first segment shows 5 vowels (31%), the second, 7 vowels (44%), and the remaining two, respectively, 4 and 8. Such a distribution does not prove that the unit 16 is a total impossibility, because many things are not average in single examples, but it is an extremely bad one and would never be accepted. On the other hand, a satisfactory distribution does not prove absolutely that a given unit-length, or block arrangement, is correct. Here, had there been no question of the ever-present _square_, we might have been led astray by the unit 32, which divides the vowels of the present cryptogram into two equal halves. In this connection, we can only say that the decryptment of any cipher, even the simplest, will at times include a number of wanderings which we shall have to overlook in demonstrating principles.

Assuming, then, that the large unit, 64, is correct, we must get it back into its block — presumably square — in the encipherer’s original arrangement. Fig. 15 shows the same cryptogram written into two different blocks. For an 8-letter unit, the normal number of vowels is about 3 (actually 3.2). In block (a), a count taken on the horizontal lines shows half of the units normal, two of the others with the smallest possible variation, and two greatly outside the 35%-45% limits. When the unit is so short, and when the line containing only one vowel may be the one which was completed with nulls, and most particularly when we have no other units to act as a check, we cannot confidently discard a block of this kind. In practice, we might waste some time giving it a trial, or we might look for something better. Notice that its distribution is “ragged.” We expected to find _even_ distribution, with _more_ than half of the units exactly normal. This block (a) is the simple horizontal arrangement. To find out what the simple vertical arrangement would give us, we have only to examine the columns of this. Here the count is obviously bad.

In block (b), we have one of the diagonal rearrangements from which two sets of vowel counts can also be taken. Here, the horizontal lines have given us exactly what we hoped for: Evenness of distribution, more than half of the units normal, and only one unit outside of limits. This, almost surely, is the encipherer’s original block, in which every line contains one intact unit.

From our meager history of the case, we do not, of course, know that this is specifically the Nihilist cipher. It becomes a case of considering the various ciphers with which we happen to be acquainted, and a _columnar_ transposition of the general kind shown in Fig. 11 is an exceedingly common case. Moreover, a series of juggled columns is suggested here in the fact that intact units are standing on their own lines and still have not resulted in plaintext.

Figure 15

(a) Horizontal Rearrangement, (b) Diagonal Rearrangement, With TWO Vowel-Counts With TWO Vowel-Counts

I Y W B B O R T 3 I W O F G N O L 3 A F T I X D G S 2 Y B A D H T E R 3 S E G H N A T O 3 B T X G I R D E 2 O I T O X T L U 4 R I E O T S M V 3 T R E L X F A Y 3 T S O U Y O O T 5 S D R C H T O M 1 S T L A T I F L 2 E D E I O V I K 5 A T F H E K T E 3 F T V T L A E U 3 X X C D I V A U 3

4 3 2 3 1 3 4 4 3 1 4 3 4 2 4 3

In Fig. 16, we have the successive steps which would be taken in order to investigate this probability. At (a), the diagonal rearrangement of our cryptogram, selected as the most likely of those which were examined, has been repeated with its eight columns set wide apart, and consecutively numbered for identification. These presumed columns are now cut apart, and thus we have eight paper strips which can be moved about and rearranged in various manners in the hope of causing words to form on some of the lines.

Since we lack that most powerful of decrypting tools, a _probable word_, we are forced to begin with probable letter-sequence. If the magic letter _Q_ were present, we should look for a companion _U_, and after that for a vowel to follow _QU_. But this, too, is lacking.

Familiarity with English digrams (or, in the case of the beginner, an inspection of the digram chart or the list of digrams) shows that _TH_ is by far the most frequent combination used in the language, and that _HE_ and _HA_, also including an _H_, are very prominent among the leaders. Further than this, the list of trigrams informs us that both _THE_ and _THA_ are of outstanding frequency. Of the four letters included, three are so frequent, and appear in so many different combinations, as to be confusing; but _H_, though belonging to the high-frequency group, does not appear in many _different_ combinations, and is less frequent than the other three.

Looking, then, for _H_, we find it twice in our present cryptogram, once on the second row and once on the seventh; and, since the seventh row shows two _T_’s and the second only one _T_, suppose we try the second row, placing together the two columns (strips) which are headed by the numbers 6-5 in order to set up a digram _TH_ on the second row, as shown at (b).

Figure 16

(a) (b) 1 2 3 4 5 6 7 8 6 5

I W O F G N O L N G Y B A D H T E R T H B T X G I R D E R I R I E O T S M V S T T S O U Y O O T O Y S T L A T I F L I T A T F H E K T E K E X X C D I V A U V I

(c) 6 5 7 ........ 6 5 7 4 ... 1 6 5 7 4

N G O N G O F I N G O F T H E T H E D Y T H E D R I D R I D G B R I D G S T M S T M O R S T M O O Y O O Y O U T O Y O U I T F I T F A S I T F A K E T K E T H A K E T H V I A V I A D X V I A D

(d) 6 5 3 ........ 6 5 3 4 ...

N G O N G O F (Abandoned in T H A T H A D R I X R I X G favor of c.) S T E S T E O O Y O O Y O U I T L I T L A K E F K E F H V I C V I C D

The formation of this digram _TH_ on the second row has automatically set up a digram _NG_ on the top row, a digram _RI_ on the third row, and so on; and we find, upon examining these newly-formed digrams, that the whole series is made up of good English combinations. Thus, it looks as if our combination 6-5 is correct, and we will proceed with a possible _HE_ or _HA_, attempting to complete a trigram _THE_ or _THA_ on the second row.

Both _E_ and _A_ are present on the second row, and we may observe at the steps marked (c) and (d) in the figure just what would be the result of adding strip 7 or strip 3. At first glance, it appears that combinations 6-5-7 and 6-5-3 are about equally probable. But it so happens that both set-ups have formed a sequence _YO_ on the fifth line, suggesting _YOU_; and when the only _U_ on that line is tried in both places, it becomes evident that combination 6-5-7-4 is going to give better results than combination 6-5-3-4, where we find poor sequences like _KEFH_. At this point, or earlier, a decryptor will probably proceed on the left side of his set-up, completing the syllable _ING_ and the series of column-numbers 1-6-5-7-4, as shown. When this setting together of columns automatically brings out on the third row a sequence _BRIDG_, we have our first suggestion of a _probable word_, since the man who had this cryptogram on his person had just attempted to blow up a _BRIDGE_. After this, all is plain sailing; the necessary _E_ happens to be on the same line, and even if it were not, we have only three strips left, and these may be placed by trial. Thus our eight paper strips arrive at the stage indicated on the left-hand side of Fig. 17.

Figure 17

Strips in order Adjustment of rows

2 1 6 5 7 4 8 3

1 W I N G O F L O 2.... B Y T H E D R A 2 B Y T H E D R A 1.... W I N G O F L O 3 T B R I D G E X 6.... T S I T F A L L 4 I R S T M O V E 5.... S T O Y O U T O 5 S T O Y O U T O 7.... T A K E T H E F 6 T S I T F A L L 4.... I R S T M O V E 7 T A K E T H E F 8.... X X V I A D U C 8 X X V I A D U C 3.... T B R I D G E X

"Taking-out" Key: 2 1 6 5 7 4 8 3 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 "Writing-in" Key: 2 1 8 6 4 3 5 7

If we have previously met the Nihilist transposition, we can see now what the cipher is, and, if it is a true Nihilist, we can finish the reconstruction _by decipherment with the key_. To do this, we simply number the rows from 1 to 8 and then disarrange these rows so that their numbers will reproduce the series of column numbers. This is shown on the right-hand side of Fig. 17, where the plaintext is easily read: “By the drawing of lots, it falls to you to take the first move. Viaduct bridge.” The gentleman required three nulls, and thriftily made use of them as punctuation. If we have not previously met the Nihilist encipherment, or if this cryptogram is of a kindred type but governed by two separate keys, one for columns and another for rows, the only difference is that we may have to experiment a little with rows before finding their correct order.

In completing our solution, we have obtained a key, 2 1 6 5 7 4 8 3, shown in the series of column-numbers, and should other cryptograms be intercepted having the same key as the first, we need merely decipher them with our key. It is, however, a “taking out” key, while the Nihilist, as we have seen, is ordinarily _written in_. Having either of the keys, we may find the other easily enough as suggested in the figure. Simply “number the numbers” and put them back in serial order. The new set of numbers, now disarranged, will show you the other key. It would not be impossible for the student who is a good guesser to find the keyword on which our present writing-in key was based. This kind of work, with paper strips, is much more rapid than it probably seems, and is often done at random. The keen eye needs no digram list for the spotting of _HT_, merely reversed, with _GN_ above it.

Speaking now of the ordinary columnars (Fig. 11), one minor point should perhaps be brought to the attention of the very new student. Quite often, a digram, such as the _QU_ of Fig. 18, is not written on a single line, and it may be necessary to match this valuable digram in the manner shown at (b) of that figure, coming out in the end as at (c). In such event, we can later on transfer columns 5-6-7 to the other side of the block, raising them all by one position. (Column numbers, in this case, are for reference only.) The same would not apply to a Nihilist block in which the whereabouts of the “next” row is unknown; the digram_ QU_ would have to be abandoned in favor of something else.

Figure 18

(a) (b) (c)

1 2 3 4 5 6 7 1 1 2 3 4 7 5 6 7 T H I S I S Q T T H I S U I T E T R U Q U I S Q U I T E E B U T W E D U E T R U E B U T D W E D

We mentioned briefly, too, the possibility of finding alternating horizontals, so that only half of the rows can be “anagrammed” together. Such minor problems, and they are numerous, can all be ironed out easily enough once the student is familiar with his type, and columnar transposition, encountered frequently and in all sorts of disguises, is surely the most fascinating of all types. In