For the beginner, however, who might like to have it in a more visible form, another chart, of a kind which we believe has never before been published, appears on [page 220]. This is F. R. Carter’s contact chart, on which every letter of the alphabet has been listed in the center of the page, with its favorite contact-letters beside it. The arrangement here is from the center outward; the letters shown on the left of any given letter are those which most often precede it, with percentages as found in Ohaver’s digram chart; letters shown on the right are those which most often follow, with percentages from the same digram chart. This information was not completed to the end for every letter, since the only information wanted is the actual preferences of each letter, or the fact that it has none. However, the outermost columns will show the complete percentages of vowel and consonant contacts for all letters as these were found in one 10,000-letter text. With such a chart before us, it becomes very easy, in the absence of Q, and other particularly vulnerable letters, to make good use of whatever letters we happen to have; and it is hoped that this new “contact chart” will prove sufficiently valuable to justify Carter’s labor in having compiled it for us. As to the other data in the [appendix], the student will do well to look it over. The list of trigrams is that of the Parker Hitt Manual, where THE was shown as having been found 89 times in 10,000 letters, the others graduating downward to MEN, found 20 times. Now let us return to our columnar transposition.
| Figure 36 Key-length Columns, 75 Letters Key 5: (Impossible under system) Key 6: 3 columns of 13 3 columns of 12 Key 7: 5 columns of 11 2 columns of 10 Key 8: 3 columns of 10 5 columns of 9 Key 9: 3 columns of 9 6 columns of 8 Key 10: (Improbable under system) Key 11: 9 columns of 7 2 columns of 6 (Etc.) |
When a digram QU is actually present in a text, or when it is fairly certain that some other digram may be present, such as the YP of CRYPTOGRAM (that is, one composed of two infrequent letters), it is possible to discover (or limit) the key-length by observing the distance apart of these two letters in the cryptogram. To approximate such a case, using the foregoing cryptogram (Fig. 30), we will make use of the digram VI, and, in order to be brief, we will assume that the letter V, position 5, is the only one in the cryptogram, and that the only I’s present are those at positions 46 and 61. In one case the interval which separates V from I is 41, and, in the other, 56. As a preliminary step, we may discard all key-lengths which are factors of 75: 3, 5, 15, 25. In addition, we may discard, for the time being, the key-lengths 10, 20, etc., which are multiples of 5. Of those left, any very short length, as 2 or 4, is very improbable. We may consider, then, possible key-lengths of 6, 7, 8, 9, 11, etc., as far as we care to take them.
To make ready for the investigation, we first prepare a sheet of the kind shown as Fig. 36, where each possible key-length has been used as a divisor in order to learn the column-lengths for each one in a 75-letter cryptogram.
| Figure 37 Q U I T E A F * E W F A N S W I L L B E D E L I G H T E D ** T O S E E *) From F in a short column to E in a long column is an interval of 5; but from E to F is an interval of 4. **) From T to E is an interval of 5; from E to T is an interval of 4. |
Now let us picture any text written into any block, as in Fig. 37, where long columns have five letters and short columns have four. Considering any digram in the text, as QU at the beginning, its two letters are separated by exactly one column of length, provided the letters are counted straight down the columns and columns are taken in one straight direction, or provided the counting is done strictly upward with columns always taken in one direction. In the case of QU, this column of separation is a long one (five letters), while, in the case of AF, on the right-hand side of the block, it is a short one (four letters), but in both cases it is a full column. This is true, also, of the digram FE, which is on two different lines, presuming that, having counted all the way to the end of the last column, we start again with the first. If both letters are in short columns, the interval which separates them is that of a short column, and if both are in long columns, this interval is that of a long column. But if one letter is in a long column and the other in a short column, the separating interval may be long or short, according to whether the columns are taken in straight order or in reverse order.
If the columns of Fig. 37 should be cut apart, and placed in some other order, then other columns might be placed between the one containing Q and its neighbor containing U, but these would be full columns, never fractional columns, so that the interval from Q to U would always be an exact number of full columns.
This is what happens in columnar transposition. If the digram VI, which we intend to consider, was actually present in the original encipherment block, then, in the cryptogram, its letters V and I are separated by an exact number of columns, long or short or mixed. Also, if the column containing V was taken off first, the distance from V to I may include the full number of long columns permitted by the key-length, but must fall one short of including all of the short columns; but if the I comes first, the opposite is true. Now, assuming that the only V’s and I’s in our cryptogram are those appearing at positions 5, 46, and 61, we find that if the first of the I’s is considered, the distance from V to I is 41, while, if the other is the one considered, then this distance is 56. We will investigate, first, the interval 41.
If V and the first I stood in sequence in the encipherment block, either as VI or as IV, then the interval 41 represents a certain number of complete columns, and if the digram was VI (since the V-column was evidently taken off first), this interval 41 must not include the full number of short columns, but may include the full number of long ones.
Consulting Fig. 36, we find that key-length 6 calls for columns having 12 and 13 letters, and it is impossible to divide an interval 41 into columns of such lengths. The key-length 7 calls for columns having 10 or 11 letters, of which only two columns may have the shorter length; an interval 41 can be divided into the right lengths, but only if three of the columns are short. Thus, if the first I is the correct one, the key-lengths 6 and 7 are totally impossible, as is also key-length 8. The key-length 9, however, calls for columns having 8 and 9 letters, of which six have the shorter length. An interval 41 can be divided to produce four short columns and one long column. Again, the key-length 11 calls for columns having 6 and 7 letters, of which two columns may be short; and an interval 41 will provide five long columns and one short column. These two key-lengths, then, 9 and 11, are possible, presuming that the first I is the one which actually followed V. When the other I, interval 56, is investigated in the same way, it is found that the only key-lengths possible are 8 and 11.