Elementary cryptanalysis

by Helen Fouché Gaines

Collins and Helen S. Pearson, decided, independently of each other, to set it up in permanent form, so as to avoid fresh computations for each new cryptogram.

Collins’ device took the form of a tableau, as shown in Fig. 108. In this figure, the vertical alphabet running through the center is a list of possible cryptogram-letters. On the side marked “Vigenère,” the four lines of numbers standing beside the letters I, O, P, and U, are the same as those included in Fig. 107. It will be noticed that only the first line of numbers (opposite A) need be found from the slide; after that, each column is a series 0 to 25. The same is true with reference to the Beaufort shifts. These, incidentally, were computed on the assumption that the Beaufort keys, A, B, C, D. . . . . . . are passing in their normal alphabetical order beneath the stationary A (as most of us prepare the Beaufort slide, this is backward). Fig. 109 shows a similar tableau prepared for the Porta shifts. The zero-position here is the AB-alphabet, a shift of 1 is the CD-alphabet, and so on. Collins, however, did not use the shift-numbers. He increased these by 1, using numbers 1 to 26, which represent the 26 positions of the slide, or, better, the serial positions in the normal alphabet of the 26 key-letters. Others who have since prepared similar tableaux have dispensed altogether with numbers, and have used the key-letters themselves. Doing this, the first row of the Vigenère portion will show the nine key-letters W H A M N S J I T, the second row will show key-letters, X I B N O T K J U, and so on. If the letters appearing on the slide have been numbered, one method is fully as convenient as the other, though in dealing with a plaintext key one would probably prefer the letters. In any case, where some four letters, such as our I O P U of the foregoing alphabet, have been found more than once in a given frequency count, it is merely necessary to find these four letters one by one in the vertical alphabet and copy their accompanying numbers. It is even possible, having these three tableaux, to decide whether the frequency counts taken

Figure 109 Tableau Showing SHIFTS for PORTA E A I H T O N R S 9 0 5 6 N 6 1 0 4 5 A 10 1 6 7 O 5 0 12 3 4 B 11 2 7 8 P 4 12 11 2 3 C 12 3 8 9 Q 3 11 10 1 2 D 0 4 9 10 R 2 10 9 0 1 E 1 5 10 11 S 1 9 8 12 0 F 2 6 11 12 T 0 8 7 11 12 G 3 7 12 0 U 12 7 6 10 11 H 4 8 0 1 V 11 6 5 9 10 I 5 9 1 2 W 10 5 4 8 9 J 6 10 2 3 X 9 4 3 7 8 K 7 11 3 4 Y 8 3 2 6 7 L 8 12 4 5 Z 7 2 1 5 6 M

from a periodic cryptogram represent the alphabets of the Vigenère, the Beaufort, or the Porta.

Miss Pearson’s device took the form of strips, a set of 26 for each of the three ciphers. Fig. 110 shows the first five of her Vigenère set as she originally prepared them, using the “position-numbers,” which are all larger by 1 than those of the tableau. Aside from this, each strip represents one row from the Vigenère half of Collins’ tableau. But where Collins had arranged his numbers according to the frequencies of the nine possible originals (so that possibilities found on the left might have more significance than others found on the right), Miss Pearson arranged hers in straight numerical order, and spaced them in such a way that No. 1 is always in the first column, No. 2 is always in the second column, and so on. Had she used key-letters, all A’s would have been in the first column, all B’s in the second column, and so on. As to the use of these strips: Presuming that the four leading cryptogram-letters are the same as before, simply pick out the four strips which are headed by the letters I, O, P, and U, and set them together. If any of the numbers are duplicated, you will find them standing in the same column. These, remember, are the devices of amateurs, and both will be found very effective. It will be noticed that the basis is the finding of key-letters (or numbers) and not the identification of cipher alphabets.

Now compare these devices with a method proposed by an expert, in which the basis is the identification of cipher alphabets, and not their keys: With this method, a tableau is prepared (which could be arranged like the one of Fig. 85) in which the only letters shown on any one line are the substitutes for the nine high-frequency letters. If, for instance, the tableau is intended for the Vigenère cipher, the top row will contain only the letters A E H I N O R S T, and the other 17 positions will be left blank. The second row will contain only the letters B F I J O P S T U, the third will contain only the letters C G J K P Q T U V, and so on. Or, if the tableau is intended for the Beaufort cipher, the top row will contain only the letters A W T S N M J I H, the second row only the letters B X U T O N K J I, and so on. Thus, after having taken a series of frequency counts, we may find out, in each of these frequency counts, which are its leading letters, then consult the prepared tableau to find out which of its alphabets will show these same leading letters. An added suggestion is as follows: Prepare the tableau, as described, using black ink. Then, using red ink, add to each alphabet the substitutes for J K Q X Z (perhaps, also, for

Figure 110 Strips for Determining SHIFTS HELEN S. PEARSON SET FOR VIGENÈRE _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ A 1 _ _ _ _ _ _ 8 9 10 _ _ 13 14 _ _ _ _ 19 20 _ _ 23 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ B _ 2 _ _ _ _ _ _ 9 10 11 _ _ 14 15 _ _ _ _ 20 21 _ _ 24 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ C _ _ 3 _ _ _ _ _ _ 10 11 12 _ _ 15 16 _ _ _ _ 21 22 _ _ 25 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ D _ _ _ 4 _ _ _ _ _ _ 11 12 13 _ _ 16 17 _ _ _ _ 23 24 _ _ 26 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ E 1 _ _ _ 5 _ _ _ _ _ _ 12 13 14 _ _ 17 18 _ _ _ _ 24 25 _ _ NOTE: The numbers here are POSITION-numbers, instead of SHIFT-numbers. A shift of zero is "position 1" of a slide. This is also the numerical, or serial, position of A in the normal alphabet. Most members of the American Cryptogram Association prefer to dispense with numbers, and use key-letters.

B P V W); that is, the substitutes for those letters which ought to be largely absent. This makes it much easier to decide between two alphabets in which the more frequent letters have made it seem that one is as likely as the other. It will be found that letters of low or moderate frequency are ordinarily as helpful in these ciphers as those of high-frequency; an instance has been pointed out in which those of the cryptogram can be more so: Where the question is one of deciding between two possible periods, a new tabulation can be made using only the sequences found in connection with those letters which are less frequent in the cryptogram than others, and thus not so sure to belong to more than one alphabet.

We have seen, then, what can be done in place of the trigram-search in the case of those longer cryptograms. Having one of only 170 letters, we first found out its period, and then (presuming that we accepted the B-alphabet in the case of alphabet 4), found out all five of its key-letters J A C B H, and even the type of encipherment (obviously Gronsfeld), without having deciphered a single letter of its message. We are now in a position to go back and investigate any which are still unsolved. With Vigenère methods and principles thoroughly understood, the student is fully in possession of methods for dealing with any periodic cipher whatever in which he knows what the cipher alphabets are. All that remains, then, is to pick up a few loose ends, and observe a few variations from the strictly periodic encipherment, after which we may consider the case of the unknown cipher alphabets.