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 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
| 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 |
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.
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.