Elementary cryptanalysis

by Helen Fouché Gaines

At stage (c), we are practically in possession of the numerical key, and to show this, the cryptogram segments have been numbered. The first one, containing V C C O T X, has been set up in the partial block as column 3; thus the third column of (b) should have key-number 1. The second segment, containing S O E U Y E, has been set up as column 5, showing that the fifth column of (b) should have key-number 2. And so on with the rest, until the eight key-numbers are standing in the order 4-6-1-7-2-3-5-8. This is shown at (d), and directly below this, at (e) is the encipherer’s original key. It can be seen that we are now in very much the same position as the legitimate decipherer; by making a few trials, each time shifting one key-number from the left side to the right, we need do little more than decipher. Usually, however, it is quicker simply to go on and rough out the block we have already started, and then make the necessary adjustments, approximately as shown in Fig. 35. Having noted, in the cryptogram, that there are some unused letters,

Figure 34 Investigating the Key-length 8 (a) (b) 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 4 6 1 7 2 3 5 8 A D V I S A B . I N C R E A S E E A C C O U N . D V O L U M E ? I N C R E A S E D V O L U M E . E N T B Y A I . (c) X X X X E H R . (1) (2) (3) E N T H/V C C O T X/ ¦R E M U/S O E U Y E/¦ I S G/A U A M A H/Y ¦ (4) (5) (6) T/A E I D E X/T N¦B H/B N S E I R/A¦S T /D A N V N X/G¦G E O (7) (8) /I C R L B X║C S U T L T E S R x (d) Apparent key-numbers: 4 6 1 7 2 3 5 8 (e) True key-numbers: 6 1 7 2 3 5 8 4

E N T H, on the left side of segment 1, we assume temporarily that all other unused letters belong to the segment which follows them, and add them all, indiscriminately, at the top of the block. Where this is shown, at the left side of Fig. 35, the true key-numbers, as found in the cryptogram, have been added above the original reference numbers, and similarly with the adjusted block on the right.

With the block roughed out, and knowing that a cryptogram of 75 letters using key-length 8 cannot have columns of any other length than 9 and 10, the first obvious maladjustment is seen in column 1 (key 4), which has only 8 letters. Since this is the 4th segment of the cryptogram, its remaining letter (or its remaining two letters) will have to be found at the end of the third segment or at the beginning

Figure 35 Forming and Adjusting a Tentative Block 4 6 1 7 2 3 5 8 4 6 1 7 2 3 5 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 E G R T C R E G R E T C A N G E I N S H A N G E I N S Y S T E M S B U Y S T E M S B U T T H O U G H T T T H O U G H T A D V I S A B L A D V I S A B L E A C C O U N T E A C C O U N T I N C R E A S E I N C R E A S E D V O L U M E S D V O L U M E S E N T B Y A I R E N T B Y A I R X X X X E H R X X X X Column 1 must have another letter, top. (Found at bottom of Column 6). (4) (3)Column 6 must then have another letter, top. (Found a bottom of Column 5). (3) (2)Column 7, bottom, then shows an extra letter, which must be transferred to (5) Column 2, top. When these transfers have been made, as shown on (6) the right, all that remains is to transfer the short column (1) (4) to the right-hand side of its block, raising it by one position.

of the fifth (keys 3 and 5), that is, at the bottom, or at the top, respectively, of the columns originally set up as columns 6 and 7. The selection of H from the bottom of column 6 leaves this column too short, while the top row of the block shows a gap in sequence, and evidently needs the E at the end of the second segment. The lone R which remains at the bottom of column 7 is then erased and written at the top of column 2, and thus we arrive at the adjustment shown on the right side of the figure, where the only remaining operation will be that of transferring the misplaced nine-letter column to its own side of the block. This final adjustment shows us the segments of the cryptogram in their key order: 6-1-7-2-3-5-8-4.

Having seen the ideal case, the student will understand how the less perfect example would be handled, or the case in which the probable word is not long enough to overlap at all. For the latter, he would attempt to find some word like CRYPTOGRAM, in which there are letters such as C, Y, P, G, M, not likely to appear more than once or twice in a short text. We need not discuss this latter case, since we are to see something very much like it before the present chapter ends.

Now, as a preliminary to those cases in which we are unable to find a probable word, suppose we turn to the back of the book, and make an inspection of the tool chest. First in importance, and valuable in ciphers of all kinds, is the digram chart which O. Phelps Meaker has been kind enough to prepare especially for this text. To learn how often he encountered any given digram in his 10,000-letter count, note its first letter in the horizontal alphabet, at the top of a column, then note its second letter in the vertical alphabet, at the beginning of a row, and observe the figure which occupies the cell at the intersection of this column and row. If the digram is TH, its frequency was 315; if the digram is JN, the cell is blank. This does not mean that the digram TH will appear exactly 315 times in any other 10,000-letter text, or that JN will never be found (occurring, say, as initials). It merely shows that the digram TH is of remarkably high frequency, while a digram JN is so rare that it practically never appears. The most commonly occurring digrams of this chart have been listed on another page in the order of decreasing frequencies. A list of the principal reversals is also given, with other data which will be found useful in the majority of ciphers. Meaker’s digram chart shows also the frequencies found for single letters in the same text. These are shown at the extreme right, and were obtained by adding the figures found on the 26 rows of the chart proper. When such counts are made, every letter in the text is considered to be the first letter of a digram, and no attention is paid to the separations between words. Thus the single-letter frequencies can be found by totalling either the columns or the rows, which, except for minor discrepancies, will check against each other.

So much for frequencies. Now let us take a closer look at sequence. Certain letters, ordinarily those of lowest frequency, are peculiar in their contacts with other letters. The shining example, in most languages, is the letter Q, followed, almost 100% of the time, by U plus another vowel; and if it seems, in the present text, that the significance of QU is being overlooked, this is simply because the individuality of this digram, like that of the German CH (CK), is so well advertised that even the novice encipherer finds a way to avoid using it. It is impossible, however, to avoid all letters having individual preferences. We still have J and V, practically sure to be followed by vowels, and Z, almost as sure. We have X, nearly always preceded by a vowel, but more often followed by a consonant. If these are missing from the cryptogram, we may have letters like K, B, and P, which confine an enormous percentage of their contacts to vowels; or to vowels and liquids; or to letters from the high-frequency group E T A O N I R S H. Even among the high-frequency letters themselves we find that H is followed about 75% of the time by either E or A, and that it is preceded largely by T, with S, C, and W as the next favorites; or we find that N is inordinately fond of vowels on its left, though with some preference for consonants on its right. All information of this kind is present in the digram chart, and usually is known to the decryptor without recourse to a chart.