Chapter 3 of 25 · 5079 words · ~25 min read

CHAPTER III

Transposition Types

Transposition has already been explained as a form of cipher in which the letters of a message are disarranged from their natural order in accordance with any pattern, or key, agreeable to the correspondents. The fact that _any_ plan may be followed will suggest the possible ramifications as to detail. Transpositions are, in fact, found in every conceivable degree of complexity. They are not even unanimous in their demand that there be two separate operations in the preparation of a cryptogram: (1) the writing down of the plaintext letters, and (2) the taking off of these letters.

Generally speaking, these ciphers follow two types, the regular (geometrical, symmetrical), and the irregular. The strictly geometrical type, sometimes called _complete-unit_ transposition, is based on one comparatively small _unit_, or _cycle_, repeated over and over, every unit having exactly the same number of letters and exactly the same disarrangement as the rest. This type always demands an exact number of units, and when a plaintext message is not evenly divisible into units, it must either be cut down to fit, or lengthened by the addition of extra letters called _nulls_. Some of these keys are actual geometrical figures, such as triangles, diamonds, hexagons, etc., or conventional designs like crosses. Any figure of this kind provides a number of cells, or points, for the _writing in_ of letters, and thus will serve as a mnemonic device, or key.

Figure 5

A D E H I L M P X X X X B C F G J K N O

Plaintext message: A B C D E F G H I J K L M N O P.

Cryptogram (a) A D B C E H F G I L J K M P N O.

Cryptogram (b) A D E H I L M P B C F G J K N O.

The two operations of writing-in and taking off may be governed by any agreed ruling, though the second of these must be made to result in five-letter groups if the cryptogram is to be transmitted by wire or radio. Fig. 5, in which an imaginary message has been represented as _A B C D E_ . . . . . , shows only one of the many ways in which a simple cross could be used as the key for the writing-in operation, together with only two of the many cryptograms which could be taken off from this one arrangement. This figure shows also, in its two cryptograms (a) and (b), two fundamentally different plans for the taking off of transpositions. The unit here is 4, the first unit containing the letters _A B C D_, the next unit _E F G H_, and so on. In cryptogram (a), the letters of every unit are still standing together in a group, while in cryptogram (b), the letters of any one unit have been mixed with letters of other units. In this latter case, the two correspondents will have to agree upon a certain number of crosses per line; otherwise, they run the risk of having to decrypt each other’s cryptograms.

The most popular of the geometrical figures appears to be the square, with or without a series of numbers 1 to 25, 1 to 36, and so on. Any device or game, which will provide a square, is likely to be seized upon as the source of a transposition key. We find two widely-known examples of this in the “magic square” and the “knight’s tour.”

A _magic square_, as most of us understand this term, is made up of a series of numbers, such as 1 to 25, 1 to 36, which are so arranged in their cells (positions) that the added numbers of any row, column, or diagonal, will always give the same total. A square of given size will provide more than one magic square arrangement; and these numbers, being a series, constitute an _order_, which, once it can be remembered or reconstructed, will serve either for writing in or for taking off a unit of 25, 36, etc., letters.

The _knight’s tour_ is based on the chessboard, a unit of 64 cells. In the game of chess, where each piece has certain prescribed moves, the piece called the knight must move diagonally across a 2 x 3 oblong. The “tour” consists in starting the knight at one corner and carrying him completely over the 64 cells of the chessboard, causing him to touch every square exactly once without having made any other move than the one allotted to him. Fig. 6 will show one of the many such tours which have been published. Such designs will serve either for writing in or for taking out. In either case, the text is made to contain exactly 64 letters or a multiple thereof. For writing in, the first letter is placed in the cell corresponding to No. 1, the next letter in the cell numbered 2, and so on. For puzzle purposes, the 64 letters are usually left standing in the form of a square. As cipher, they would be taken off, by rows, or by columns, or otherwise. Or the 64 letters may first be written in simple order into the form of a square, and then taken out one by one following the route of the knight.

Figure 6

1 4 53 18 55 6 43 20

52 17 2 5 38 19 56 7

3 64 15 54 31 42 21 44

16 51 28 39 34 37 8 57

63 14 35 32 41 30 45 22

50 27 40 29 36 33 58 9

13 62 25 48 11 60 23 46

26 49 12 61 24 47 10 59

Other ciphers of the regular type merely employ a unit of so many letters, to be arranged in some specified order, generally in accordance with a numerical key. If, say, the unit has a length of six letters, which we will represent as _A B C D E F_, and the specified order for these is 6 2 1 4 3 5, this unit may be transposed to read _F B A D C E_. Each unit will be transposed to have exactly this pattern, except that semi-occasionally we find a final unit slightly different from the others, owing to the fact that nulls were not added to complete its length (Accurately speaking, this transfers the cipher to the “irregular” class). Units, once transposed in this way, may continue to stand intact, one after another; or they may remain intact, merely exchanging places with one another; or the cipher may be so planned that they do not remain intact, as was the case with our cryptogram (b) of Fig. 5.

Often, two ciphers will differ from each other only in the method by which their cryptograms are produced; oftener, there will be an actual difference, but one which is purely superficial. For instance, we have just mentioned a plaintext unit _A B C D E F_ as having been transposed with a key 6 2 1 4 3 5 to result in the order _F B A D C E_. Identically the same numerical key, used in another way, will transpose this unit in the order _C B E D F A_. The two resulting cryptograms would be different, but the _kind_ of cryptogram would not.

An extremely common form of complete-unit transposition is that indicated in Fig. 7, where a short message, LET US HEAR FROM YOU AT ONCE CONCERNING JEWELS QQ (38 letters plus 2 _nulls_), has been written into an oblong, or _block_, in one order and taken off in another. Both the writing in and the taking off follow a _route_, rather than a key and, for that reason, the cipher is often spoken of as _route transposition_, rather than _rectangular transposition_.

Three of the many possible _routes_ are shown in the three (partial) cryptograms of the figure. In this connection, the American popular terminology seems to favor _horizontals_ and _verticals_, rather than “rows” and “columns.” The writing in or the taking out of a text is said to be done by _straight horizontals_, or by _reversed horizontals_ (backward), or by _alternate_ (or _alternating_) _horizontals_ (written alternately in both directions). Similarly, we find ascending, or descending, or _alternate verticals_; and again the _diagonal_ routes will be described as _ascending_, _descending_, or _alternate_. The route may also be a _spiral_ one, and in this case it is said to be _clockwise_ or _counter-clockwise_.

Figure 7

L E T U S Cryptograms: H E A R F R O M Y O (a) By descending verticals, from the left: L H R U C U A T O N C E C O N C N E E E O A E E G L T A M T C R J S U, etc. C E R N I N G J E W (b) By alternating verticals from the right, top: E L S Q Q S F O N N I W Q Q E N O O Y R U T A M T, etc.

For all of these routes, the point of beginning is nearly always one of the four corners, except in the case of the two _spiral_ routes, which are just as likely to begin with a central letter, particularly when the rectangle is a square. Colonel Parker Hitt, in his _Manual for the Solution of Military Ciphers_, shows the same series of letters written into forty different blocks, always beginning at one of the four corners.

Rectangular transposition, when used as cipher and not simply as a puzzle, requires that one dimension of the oblong be fixed, the other dimension being entirely dependent on the length of the message to be conveyed. In the figure, the pre-arranged width of the block, called its _key-length_, was 5, and the filling of the block required 8 complete units. These were written one by one as simple bits of plaintext, and were then broken up in the method of taking off. Occasionally it will be the vertical dimension of the block which is fixed, and the plaintext will be written in by columns, beginning at the left or at the right. But there is so little difference in the results of the two procedures that a decryptor may solve and read a cryptogram without learning which of the two was actually followed. Ordinarily, it is the simple operation which comes first, the writing in of intact units one after another. Sometimes the opposite is true, the operation of writing in being made very complex, so that the whole block is the unit, the taking off being done by simple rows or columns. Frequently both operations are complex. This kind of transposition belongs rather to the category of puzzles than to cipher; any reasonably intelligent person can decrypt it, knowing what it is. However, it has not infrequently been applied to serious purposes, and a decryptor, encountering an unknown transposition, would not overlook the possibility of simple rectangular encipherment.

Decryptment, here, is merely a matter of trying out the known routes, and it would never be actually necessary to write out the entire forty-plus blocks, or even half of these, for any one rectangle. The decryptor begins by counting the letters of his cryptogram and factoring the number of these, to find out what oblongs are possible. A 36-letter cryptogram, for instance, might mean dimensions 6 x 6, or dimensions 4 x 9. It could, conceivably, represent dimensions 3 x 12, or 2 x 18. But key-lengths are hardly ever shorter than 5, or as long as 18. He would seize upon the square as the object of his first investigation, writing the cryptogram into that block by various known routes, and also _reading_ by various known routes, diagonally, horizontally, vertically, backward, or upside down, until he begins to find words. As a rule, this does not take him very long; often the very efforts of an encipherer to achieve complexity will result in an easier task for the decryptor. However, a spiral will sometimes give trouble.

Figure 8

A E I B D F H J C G K........etc.

Taken off: A E I & B D F H J & C G K...

The examples appended to this chapter are all of the complete-unit type, and require little knowledge of cryptanalysis for their solution.

Passing on to irregular types, we find these in all degrees of difficulty, from the very simple “rail fence” to the formidable “U. S. Army” double transposition.

The “rail fence” family is outlined sketchily in Fig. 8. The writing in of the plaintext follows a zig-zag route, downward by so many letters, then upward to the line of beginning, as indicated by the series _A B C_ . . . . . , and the taking off of the cryptogram is done by straight lines. In explanation of the character _&_, this has been used here as a signal to show the ends of the straight lines. No such signal is needed if a proper understanding exists between correspondents as to the construction of the “fence” and the length of it which may occupy one line of writing; and in some cases the straight lines are all equal in length.

In Fig. 9, we have a suggested _grille-transposition_, of a kind described by Mario Zanotti as “indefinite.” This kind of grille, we believe, is the invention of General Sacco. To picture it complete, we may imagine a flat surface, such as a piece of cardboard, marked off into squares, having dimensions 12 x 6, and turned sidewise. Assuming this to be shown in full, we are looking at 12 _columns_, and each column has 6 of the small squares, or cells. To convert this piece of cardboard into an encipherment grille, we clip out three squares from each one of its 12 columns, always in the most haphazard manner possible. The resulting grille will thus have 36 openings, and, if placed over a sheet of paper (preferably also marked into cells), enables us to transpose the first 36 letters of a message by writing them one at a time into the 36 apertures in some one order and taking them off in another. The original plan was the reverse of the usual: write the letters by columns and take them off by rows.

Figure 9

[ (N) ... [(S) (O) ( ) ... [(T)(I) ( )( ) ... [ (K) ... [(R)(E) ( ) ... [ (W)( )( ) ...

Cryptogram: N S O T I K R E W.

In the figure, a 9-letter message, STRIKE NOW, has been written into the first three columns of such a grille, and, taken off by rows, comes out in the order _N_, _SO_, _TI_, _K_, _RE_, _W_. While the figure shows this cryptogram regrouped in the usual fives, the original method, as prescribed with the device, would have grouped it in threes, that is, to correspond with the number of apertures per column. This would facilitate the operation of decipherment, which is as follows: Count the number of letters in the cryptogram _and divide this number by 3_, in order to find how many columns were used. Cover (or ignore) the unused portion of the grille, write the cryptogram by straight horizontals into the uncovered portion, then read, or copy, by descending verticals. The recipient of the present cryptogram, for instance, finds nine letters, divides this number by 3, thus ascertaining that three columns were used, covers up the other nine columns, then, proceeding by straight horizontals, places one cryptogram-letter wherever he sees a hole. Having thus restored all letters to their proper columns, he has the plaintext message before him. It will be noticed that an encipherer uses only the number of columns that he needs. His last column does not have to be completed with nulls, as in the case of complete-unit ciphers.

As this grille has just been described, its full capacity is 36 letters, and it has a repeating cycle of that length, presuming that, after the transposition of the first 36 letters, another 36-letter unit is to be transposed by the same grille standing in the same position. But this grille, reversed, provides a new pattern; and the opposite side of the grille provides two additional patterns. These positions may be numbered, thus providing for the encipherment of 144 letters, even assuming that the positions are to be used in 1, 2, 3, 4 order and without varying the method of use. Add to this that the cryptographic offices may have provided half-a-dozen different grilles to be used interchangeably and not always in exactly the same way, and it becomes plain that such an encipherment, in the hands of an operator who knows his business, could be made to furnish a very effective form of transposition.

Figure 10

2 1 L E T U S H E A R

5 4 3 2 1 F R O M Y O

0 9 8 7 6 5 4 3 2 1 U

7 6 5 4 3 2 1 A T O N

3 2 1 C E C O N C E R

4 3 2 1 N I N G J E W

7 6 5 4 3 2 1 E L S Q

Zanotti, and others, have also described mechanical devices of a patentable type for accomplishing very involved transpositions. The principle on which most of these operate can be seen in Fig. 10. A certain number of pointers, or narrow sliding rulers, all carrying the same progression of numbers, are so attached to a framework that they can be set, by means of a numerical key, to project at irregular lengths over a sheet of quadrille paper cut to fit into the frame. Thus, each pointer indicates a certain number of empty cells, as nine on the first line, six on the next, and so on. In the example of the figure, presuming that each pointer carries only ten numbers, and that the full number of these pointers is seven, the numerical key would be the column of numbers at the extreme left: 2-5-0-7-3-4-7. The message here is written in the usual horizontals, with a null (not strictly necessary) completing the last line. It could be taken off by columns: _L_, _EC_, _TEN_, _UFCI_, etc. The decipherer, having a duplicate apparatus, would set this according to the pre-arranged key, copy the cryptogram by columns, and read it by rows. The exact method, of course, can be varied.

Some attempt has been made, too, to evolve cipher machines which will produce effective transpositions, but our understanding is that these have never been accepted as worthwhile. The accomplishment of transposition by mechanical means is far from new. In fact, the oldest transposition cipher of which we have any record was accomplished by means of the Lacedaemonian _scytale_. The Spartan general, departing for foreign conquests, carried with him a rod, or scytale, of exactly the same diameter as one retained by the administration. When it was desired to communicate matter of a confidential nature, the sender, using a narrow strip of parchment, wound this carefully around his scytale with edges meeting uniformly at all points, and wrote his message lengthwise of the rod. When the strip was unrolled, the message appeared as a series of short disconnected fragments, one letter, or two letters, or portions of one or two letters. It was presumed that no person would be able to read the message without being possessed of a duplicate scytale on which to rewind the strip. We are left to suppose that this presumption was justified by fact, though the decryptor of today would make short work of such a system. The scytale, we believe, is the oldest known cipher of any kind, and is still serving today as the emblem of the _American Cryptogram Association_.

Before leaving types, it should be mentioned that any of the transpositions ordinarily used for disarranging single letters can also be used for the transposal of entire words. The popular name for this is “Route Cipher” — possibly because it is rather cumbersome to accomplish by any other than a “route” transposition.

We have said little concerning _decipherment_. This, in practically all cases, is a mere matter of performing inversely the two encipherment operations. For either process, the operator begins by setting down his key or design, or adjusting his mechanical device in the agreed manner. The encipherer “writes in” a plaintext, and “takes off” a cryptogram; the decipherer “writes in” a cryptogram, and “takes off” (or reads) a plaintext. If the encipherer, by agreement, has written the text in rows and taken it off by columns, then the decipherer must do the reverse: write his text by columns and take it off by rows.

Before entering into the subject of _decryptment_, the student should acquaint himself with the significance of the various tables appended to this text, in order that he may consult these or similar tables for information as to _frequencies_, and _sequence_. Every written language has its individual characteristics in these two respects, and, to learn just what these are for each language, various cryptologists have, from time to time, counted the letters, the short words, the combinations, and so forth, often on extremely long texts, afterward arranging these data in the form of charts, or tables, or lists. Two such counts are never duplicates, and there may be a noticeable difference, say, between results obtained from literary text and those obtained from military or telegraphic text; yet results for any one language are surprisingly uniform. Finding, for instance, an unexplained cryptogram in which a count of the letters shows that about 40% of these are vowels (with or without _Y_), we may classify it, not only as a transposition, but as one enciphered in English or German, since one of the Latin languages can hardly be written with so low a vowel percentage. Then, if we note the occurrences of the letter _E_, and find that this makes up about 12% of the total number of letters, we may discard the possibility of German, in which the letter _E_ is far more likely to represent 18% of the text. Or, if the vowel percentage is high enough to point to one of the Latin languages, French would be distinguished from the others by the outstanding frequency of its letter _E_, sometimes as great as that of the German _E_, while the Spanish, Portuguese, or Italian language will not always show it as the leading letter, its place having been taken by _A_. In the Serb-Croat language, the letter _A_ always predominates, and in Russian the letter _O_.

As to sequence, and considering English combinations only, certain digrams, such as _TH_, _HE_, _AN_, etc., very consistently predominate over all others. These almost never show identical percentages in any two digram counts (as the single letters sometimes will), and seldom, if ever, are ranked in exactly the same order, aside from the fact that _TH_ invariably comes first. But in all counts, the same fifty to sixty digrams (out of 676) are always found at the top of the list. Thus the Meaker digram chart differs from similar charts made by many others; yet _any_ digram chart is the most valuable weapon we have for attacking a cipher. The Carter contact chart contains the same general information expressed in another way for special use in transpositions. (This was not figured from the Meaker chart, but from an earlier one by Ohaver, made on the same kind of text.)

One very useful phase of frequency data is seen in the group percentages. Single letters, especially in short texts, may vary greatly from their normal percentages, while certain classes, taken as a whole, maintain a fairly constant percentage no matter how short the text. Such classes, or groups, listed under the general heading of _English Frequency and Sequence Data_, can be memorized as having roughly approximate percentages: Vowels, 40%; selected high-frequency consonants, 30%; extreme low-frequency group, 2%; the five most frequent letters, mixed, 45%; the nine most frequent letters, 70%. This final group of nine letters, _E T A O N I S R H_, hardly ever varies appreciably; the shorter groups will sometimes vary as much as 5% one way or the other.

Very useful in _code_ decryptment is a list of the commonest words. Trigrams have also been investigated, the favorite positions of individual letters in their own words, average word-length, _patterns_, and endless other information, some of which is indispensable, and some merely convenient. It will not be possible, in the space at our disposal, to point out all of the uses to which this kind of information can be put; the student is urged to take his cue from the occasional short references made in connection with examples.

* * *

All ciphers are decrypted by the _general methods_ suitable to their type, and a transposition cryptogram may involve _factoring_, _examination of the vowel distribution_, and _anagramming_, either singly or in combination. These are best explained in connection with examples, which may themselves have _special methods_, and we have selected for general discussion four ciphers, two belonging to the complete-unit type and two to the irregular. A careful study of the methods used in individual cases should furnish the student with a basis for analyzing other ciphers and evolving other special methods to suit particular cases.

Concerning the paper work, which, admittedly, is onerous in most forms of cipher investigation, much reference may be found, in the matter which follows, to “paper strips.” These are old stand-bys. Most decryptors prefer to do all of their work on cross-section (quadrille) paper, since the writing of the letters into cells enables them to obtain an accurate spacing both laterally and vertically, and this paper is easily cut apart along the separating lines. But for the kind of cryptograms we are likely to see here, many persons prefer to work with a set of anagram blocks. These can be prepared at home from cardboard squares, or may be bought in sets with frequent letters represented in approximately the correct proportions.

7. By TITOGI.

T S S N I H A Y S T I N T P I S E R O O I A A S N. Also this: S H C V I E O L E A E W E R M.

8. By G. A. SLIGHT. (Something found in every school-book - IF found!)

T G H M R R I A Y E X N U E E S D E X S H M T I D E Q U O A Y R O A U N P U E T G T I T E S Y S N O A Q N X A T U A D S I S H X.

9. By PICCOLA.

W I N T A H D A E S W H L E T Y L W A I L H O Q L A S S S A S Q.

10. By NEMO. (Magic Square).

L E A S U L T S G M S L O E I E O I M E A R N S A S R C D E K I U S U H E M A Q L Y S P R M E O A.

11. By THE ADMIRAL.

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

12. By THE ADMIRAL.

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

13. By DAN SURR. (Received from General Headquarters following a skirmish).

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

14. By PICCOLA. (This is serious advice!)

F F L T A A R N I E U O R N T O T D L A N R W S O I A T T E Y B A N T M E H S K O G R Z E P S R E I O A O A M S S S M A L P I L Y S.

15. BY FRA-GRANT. (This might have been a little easier. Still - ?)

Q Y T E Y O F U B U Q E H I H T E C H T H S A U A O N S I T I T T T I E T T E L L S E A P L T N T.