There are many other cases, hardly more difficult, in which our rearrangement of numbers results, not in the original order, but in an equivalent order. We could, for instance, arrive at a rearrangement in which we have taken each third number, or each fifth number, of the original cipher alphabet, so that our rearranged numbers
| Figure 135 The LINING UP of the Frequency Counts of Figure 134 ┌1st┐ ┌2d ┐ ┌3d ┐ ┌4th┐ ┌5th┐ TOTALS│ │ │ │ │ │ │ │ │ │ │ 30│11111 │ 10│111 │ 20│111 │ 21│11111111 │ 51│1111 23 * │ 40│11 │ 20│11 │ 30│11 │ 31│1 │ 61│ 7 │ 50│1 │ 30│1 │ 40│11 │ 41│11 │ 71│1 7 │ 60│11 │ 40│ │ 50│ │ 51│1111 │ 81│1 7 │ 70│11111 │ 50│111111111│ 60│111111111│ 61│111111111│ 91│111111 38 * │ 80│111 │ 60│ │ 70│1 │ 71│ │ 01│ 4 │ 90│11 │ 70│ │ 80│11 │ 81│1 │ 12│ 5 │ 00│ │ 80│1 │ 90│11111 │ 91│11111111 │ 22│1 15 * │ 11│11111 │ 90│111 │ 00│11111 │ 01│1 │ 32│1111 18 * │ 21│ │ 00│ │ 11│ │ 12│1 │ 42│ ** 1 │ 31│ │ 11│ │ 21│1 │ 22│ │ 52│ ** 1 │ 41│ │ 21│11 │ 31│1 │ 32│1 │ 62│11 6 │ 51│11 │ 31│1111 │ 41│ │ 42│1 │ 72│1 8 │ 61│1111 │ 41│11111 │ 51│11 │ 52│1 │ 82│111111 18 * │ 71│11111 │ 51│111 │ 61│11 │ 62│1 │ 92│111111111 20 * │ 81│11 │ 61│ │ 71│ │ 72│1 │ 02│ 3 │ 91│ │ 71│ │ 81│ │ 82│ │ 13│ ** 0 │ 01│1111 │ 81│111 │ 91│1 │ 92│1111 │ 23│1111 16 * │ 12│1111 │ 91│11 │ 01│111111 │ 02│11 │ 33│111111 20 * │ 22│11111 │ 01│1111111 │ 12│11111111 │ 13│1111 │ 43│11111 29 * │ 32│1 │ 12│11 │ 22│ │ 23│1 │ 53│11 6 │ 42│ │ 22│ │ 32│ │ 33│ │ 63│ (V) 0 │ 52│ │ 32│111 │ 42│11 │ 43│ │ 73│ 5 │ 62│ │ 42│ │ 52│ │ 53│ │ 83│ ** 0 │ 72│1 │ 52│111 │ 62│1 │ 63│11 │ 93│1 8 │ 82│ │ 62│ │ 72│ │ 73│ │ 03│ ** 0 265 |
are following plaintext letters in the order A D G J. . . . or A F K P. . . . ; thus, all of our frequency counts would be following one same graph, though not the graph of the normal alphabet. The problem here is to make sure that their graphs are all the same graph, and then subject them to the process called “lining up.”
To show the handling of all such cases (which would include our final autokey example), let us assume that the five frequency counts of our figure, though still following a common graph, are not following that of the normal alphabet. In this case, granting that all fifty-letter frequency counts will vary considerably from the normal, it is not quite so obvious that their pattern is the same; we shall have to cut them apart (preferably having copied numbers beside their frequencies) and place them side by side for a comparison of their graphs. Where this has been done, in Fig. 135, their similarity is plain in spite of some discrepancies, and the moving up or down of any one or more of the counts (which could be done so as to include another position, since the range of the numbers is only 25 per alphabet) does not result in greater similarity. If the alignment of this figure is correct, then all numbers found on any one row are substituting for one same original; thus, the added frequencies on any one row will be the total frequency of some one letter in a 265-letter text, and all of these totals, collectively, should resemble a frequency count taken on a simple substitution cryptogram of that length. To just what extent this is true may be seen at the right side of the figure. The nine leading letters have totalled 74%, where we normally expect 70%; but any single example can provide its surprises, and the excess 4% is not on the wrong side of the ledger. The other end of the count, as would be expected of the group J K Q X Z, is comparatively blank.
Our substitutes, remember, are assumed to be in mixed order. We do not know what letter is represented by the five numbers of the top row, or by the five numbers of any other row. To proceed with solution, we shall have to assign
| Figure 136 The NIHILIST Number-Substitution The "Checkerboard" Alphabet: 1 2 3 4 5 1 A B C D E 13 = C 2 F G H I K 34 = O 3 L M N O P 32 = M 4 Q R S T U 15 = E 5 V W X Y Z 44 = T Encipherment, with Keyword COMET: S E N D S U P P L I E S T O .... Text... 43 15 33 14 43 45 35 35 31 24 15 43 44 34 .... Key.... 13 34 32 15 44 13 34 32 15 44 13 34 32 15 .... 56 49 65 29 87 58 69 67 46 68 28 77 76 49 .... This cryptogram is usually seen without grouping: 56-49-65-29-87-58..... |
arbitrary values, calling the top row A (or 01), the second row B (or 02), the third row C (or 03), and so on; and when all of these substitutions have been made on the cryptogram, the case has been reduced to one of simple substitution. The mechanics by which the substitutions are made can be exactly those of the other case: Prepare a temporary slide, on which the numbers run in the order decided upon, and slide this against the normal alphabet (or any other); the result is a simple substitution cryptogram which can be solved by simple substitution methods. This case, first in one form and then in another, is encountered again and again; and however it may seem that its cause, in some one example, is a different one, yet the fault in all such examples is the same: The basic cipher alphabet (the primary one from which others are derived), either by its actual construction or by the method of its application, was not truly a mixed alphabet.
In some of the periodic ciphers, the basic cipher alphabet is a “checkerboard” of the kind we saw in [Chapter XI], the substitutes being two-digit numbers which will point out the columns and rows of their originals. This primary alphabet, however, seldom appears unchanged in the cryptograms, as “position 1” often does when a slide is used, or as the A-alphabet often does in the Vigenère cipher. Instead, we find a series of secondary cipher alphabets all of which have been derived from the primary one according to a mathematical process.
In view of the fact that any cipher which will necessarily double the lengths of messages is of doubtful value, it seems inadvisable here to do more than mention the infinite multiplicity of processes which would be possible; but with checkerboards, it is difficult to imagine any usable process which would not result in parallel frequency counts; that is, counts which all follow the same graph and thus are capable of being “lined up.” With most of these, in fact, the difference between any two of the (secondary) cipher alphabets will be a difference in size which is uniform from A to Z. (Often, the same result is produced with slides.) Here, then, we may content ourselves with a glance at one such cipher which is interesting rather than important. In Fig. 136, we have another of the Nihilist ciphers. Its primary alphabet is that most famous of checkerboards, the Polybius square, said to have been the invention of the ancient Greek historian, and certainly well known in his era as the basis for a signalling system — a capacity, incidentally, in which it still serves. We are showing it here in what seems to be the favorite version: The alphabet of the square is the normal one, normally arranged, with J the missing letter; and the order of reading for the two digits is row-column. It should be understood, however, that these details, in practice, are quite variable.
For the Nihilist encipherment, the message is first subjected to a simple substitution, using the checkerboard key. A key-word, treated in the same way, is repeated often enough to pair one key-number with each message-number, and the final cryptogram is formed by adding these pairs of numbers. Decipherment, of course, will be the subtraction of key-numbers from the finished cryptogram and the resubstitution of letters. We have, then, another periodic cipher, not essentially different from those already seen. Any number in the checkerboard can become a key, to be applied periodically at some given interval, and thus may govern one of the 25 possible cipher alphabets. It would be possible to lay out any one of these cipher alphabets, simply by adding a given amount to each number of the primary one; if all of them were written one below another, and if the primary alphabet were written across the top and along one side, we should have a tableau which could be used in identically the manner described for the use of the Vigenère tableau.