In applying the variant encipherment, in which key-letters are found first, he need find a given key-letter but once, then lay a ruler along the row (or column) indicated by that key-letter, and encipher at a single writing all plaintext letters which are going to have that particular key. But if, as previously recommended, he has familiarized himself with the use of the Vigenère tableau, he will see instantly that the operation which, in the variant Beaufort he is calling encipherment, is identical, in every particular, with the operation which, in Vigenère, he would have called decipherment, and that, in order to decipher the variant, he must perform the operation which, in Vigenère, is called encipherment. Neither of these operations
| Figure 99 How to find the C-alphabet of each Beaufort TRUE BEAUFORT VARIANT BEAUFORT Key: C C C C C C C C Key: C C C C C C C C Plaintext: A B C D E F G H... Plaintext: A B C D E F G H... Cipher ALPHABET: C B A Z Y X W V... Cipher ALPHABET: Y Z A B C D E F... |
provides a reciprocal substitution; instead, they are reciprocal to each other. Once it is seen that this is true, it becomes equally plain that the Saint-Cyr slide serves just as well for the variant as for the Vigenère. To make use of it in applying the variant encipherment, set key-letters below index-letter A, exactly as if making ready to encipher in Vigenère, but reverse the functions of the two alphabets; that is, find all plaintext letters in the lower one, and take their substitutes from the upper one.
Now, consider the true Beaufort cipher: Here, plaintext letters are found first, and keys are found by tracing into the square, so that encipherment is more or less a letter-by-letter process, and hardly so convenient as in the other two ciphers. It is true that every ascending diagonal in the tableau is made up of only one key-letter, so that a ruler, laid diagonally across this tableau, will point out a whole line of C’s, or O’s, or M’s. But practically every one of these diagonals is broken into two portions, so that in attempting to encipher by one key-letter at a time, we find it rather confusing to make the necessary adjustments. Is there not, then, a more convenient method for applying the Beaufort? Every cipher of this family, remember, provides a certain number of individual simple substitution cipher-alphabets. For every key (whether it is a letter or a number) there is some kind of cipher alphabet showing a substitute for A, a substitute for B, a substitute for C, and so on. To isolate one of these cipher alphabets, and find out what it is like, we have merely to take some one key-letter (or some one key-number) and discover what these substitutes are, and what their order is; that is, we need merely encipher the normal alphabet, using this one key. This is true of every cipher of the multiple-alphabet type. The process can be seen in Fig. 99, where the C-alphabet (that is, the alphabet governed by key-letter C) is being isolated for each of the Beaufort ciphers.
In the Beaufort proper, we find that the C-alphabet will begin with C and come out in the order C B A Z Y X. . . . , which is merely the normal alphabet reversed. Should we investigate the D-alphabet, we should find that this begins at D and comes out in the order D C B A Z Y. . . . , again the normal alphabet reversed; or, investigating the E-alphabet, we should find E D C B A Z. . . . , always the normal alphabet written backward, and always beginning with whatever letter is called the key. This being the case, it becomes quite evident that a slide is possible, and the formation of this slide is clearly indicated in the left-hand tabulation of the figure: Its upper alphabet must run in one direction and its lower alphabet in the other; if one of the two is made of double length, it becomes possible to place any one of the 26 key-letters in juxtaposition with index A, thus bringing into position any one of the 26 cipher-alphabets which are governed by these keys. Nor does it make a particle of difference which of the two A’s, the upper or the lower, is regarded as the index-letter; when C is standing below A, then A is also standing below C. We saw, in the tableau itself, that the true Beaufort encipherment gives reciprocal substitution. This, however, was not our first meeting with one of the Beaufort alphabets; in [Chapter IX], we met the Z-alphabet. We saw there that whenever a cipher alphabet is merely the plaintext alphabet written backward, it makes no difference which of the two is called a cipher alphabet; we may see here that this fact is not
| Figure 100 A Pair of COMPLEMENTARY Alphabets: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z A Z Y X W V U T S R Q P O N M L K J I H G F E D C B By doubling the length of one or the other of these two alphabets, we may use them to form a slide which will encipher and decipher the true BEAUFORT. |
altered by shifting one of the alphabets. Since a slide is possible, it follows that a disk is also possible. This particular cipher disk, on which one alphabet runs forward and the other backward, was used long ago in our own army, and is widely known in this country as “The United States Army Cipher Disk.” Most persons, apparently, prefer the slides, on which the letters are always right-side up, and the preparation of which does not involve the division of a circle into 26 equal arcs. Of those who prefer the disks, practically all will make the smaller disk reversible, with the normal alphabet on one side and the reversed alphabet on the other.
Now, returning to our Fig. 99, and examining its right-hand tabulation: We find that, in isolating the C-alphabet of the variant Beaufort, we have merely reproduced the Y-alphabet of the Vigenère. Should we now isolate its Y-alphabet, we should find that we have obtained the C-alphabet of the Vigenère. Further investigation will show that the D-alphabet of one is the X-alphabet of the other, that the E-alphabet of one is the W-alphabet of the other; and so on. Only their A-alphabets and their N-alphabets are keyed alike. Thus we seem to have here a case of “reciprocal” key-letters. These particular pairs of corresponding letters, B and Z, C and Y, D and X, and so on, are called complements, one letter of each pair being complementary to the other. Since the letters A and N have no complements (or serve as their own complements), the normal alphabet will furnish only twelve such pairs, and these are shown complete in Fig. 100. In this same set-up, it can be seen that the A-alphabet of the Beaufort cipher is the complement of the normal alphabet. Thus, having provided ourselves with a Beaufort slide (or disk), we have always at hand a means for finding out the complements of letters. Once it is clearly understood that the chief difference between a Vigenère cryptogram and a variant cryptogram lies in the names of their respective cipher alphabets, it becomes evident that we might decrypt a variant, believing it to be a Vigenère, and have no trouble whatever in reading its message, though finding that it has an incoherent key. Vigenère keys, of course, can be incoherent; occasionally they are based in some way on numbers, following the Gronsfeld scheme. But usually, this is not true; the incoherency is only apparent, and a little investigation will discover what the trouble is. In the case just mentioned, the variant key-word COMET will come out in Vigenére as Y M O W H, or vice versa; all that is necessary, in order to discover the original key-word, is to set the Beaufort slide at the A-alphabet, and perform a bit of simple substitution. Another cause for the apparently incoherent key lies in the
| Figure 101 Applying a PROBABLE WORD to BEAUFORT (a) Cryptogram, TRUE BEAUFORT: K K Z B B I Z X T L Y W T Q Probable word.......... S C C R T T A R P L D Q O L I U E T V V C T R N F S Q N K P O Q Q X O M I A N L I F P Q . . . M I A N . I . L E I T E C S O Use the word SUPPLIES as a trial key, exactly as in Vigenère, but make use of the VARIANT method, and not tho TRUE BEAUFORT. *** *** (b) Cryptogram, VARIANT BEAUFORT: Q Q B Z Z S B D H P C E H K Probable word............. S Y Y J H H A J L P X K M P S U W H F F Y H J N V I K N Q P M K K D M O S A N P S V P . . . . O S . N . P . L W I H E Y S M This was deciphered as a Vigenère, and showed the repeating of a scrambled key: Y M O W H. Had it been deciphered with the BEAUFORT SLIDE, suggested in Figure 100, it would have reproduced the plaintext keyword, C O M E T. |
use of some other index-letter than the stationary A. Say, for instance, that the encipherer has used the key-word COMET, but has placed his key-letters beneath index D. The key recovered by the decryptor is Z L J B Q; to find the original key-word, he need merely “run it down the alphabet.”