Now, putting aside the fact of the mixed plaintext alphabet (since we do not intend to recover the letters) we have here a cipher which, to all intents and purposes, is the Vigenère autokey initiated with a single letter. In place of the letters A to Z we have numbers 1 to 26, and the encipherment is a series of additions. In the corresponding Vigenère case, the group-length 1 will usually show up plainly in the number of doubled letters — “letters repeated at interval 1.” And with the group-length determined as 1, it is possible to begin with some given initial key, as A, and either reproduce the plaintext or convert the autokeyed cryptogram to a periodic one in which the period is 2 (twice the group-length). Considering the analogy between the two cases, it should be possible to do the same thing here. That is, it should be possible to take the autokeyed cryptogram of (b), initiate its decipherment with some number chosen between 1 and 26, and either reproduce the primary cryptogram or convert the autokeyed cryptogram to a periodic one in which the alternate numbers will belong to two cipher alphabets. Where this reduction has been carried out in Fig. 128, the initial decipherment was made with key 9 in order to avoid a discussion of negative numbers. Also, the fact of numbers will usually limit the range of the trial keys: here, the first number, 29, was not enciphered by adding any number smaller than 3.
Now, looking at Fig. 129, let us compare the new cryptogram of Fig. 128 with the primary cryptogram of Fig. 127(b), and see whether or not it has the expected formation. Between the two cryptograms (the supposedly periodic one obtained from the trial decipherment and the one we hope to recover), there is a constant numerical difference in the pairs of corresponding substitutes, and this difference, throughout, is alternately plus and minus. Further comparisons can be made, if the student so desires, by initiating other partial decipherments with trial-keys 10, 11, 12, etc. Always, the constant numerical difference persists, and always it is alternately plus and minus. Moreover, for every time that the initial key-number increases in size, there is a corresponding decrease in all numbers occupying the odd serial positions and a corresponding increase in all numbers occupying the even positions.
| Figure 129 Comparison of TRIAL DECIPHERMENT with TRUE DECIPHERMENT True decipherment - (See Figure 127): 21 17 11 10 21 5 13 13 9 22... Trial decipherment of Figure 128: 20 18 10 11 20 6 12 14 8 23... CONSTANT DIFFERENCE: 1 -1 1 -1 1 -1 1 -1 1 -1 |
We have, then, a periodic cryptogram whose period is 2, and two cipher alphabets, consisting of numbers, in which the only difference is one of size. But these substitutes, unlike those of the Vigenère, will not be placed in normal alphabetical order; to complete the solution by one of the general methods, it may become necessary to take a number of frequency counts. For instance, considering the first of the two Bassières processes, it would be possible to set up the same tableau (Fig. 122), causing numbers to run alternately backward and forward (and beginning again at 1 whenever the number 26 is reached). In this way, one of the columns would contain the primary cryptogram, and a frequency count taken on the numbers of that column should resemble a simple substitution frequency count.
Considering the second of the Bassières processes, the autokeyed cryptogram is already reduced to a period of 2; the subsequent solution of the periodic cryptogram belongs to the general case of the [next chapter]; that is, a case in which the cipher alphabets are in mixed order but parallel. But we have, here, a special method, and a short-cut. The only difference between our two cipher alphabets is a matter of size in all corresponding substitutes. If we can find out what this numerical difference is, we have only to increase or decrease the size of the numbers in one of the cipher alphabets and bring it to the level of the other. Our short-cut, as pointed out by Ohaver, lies in repeated sequences (or even repeated single letters) in the autokeyed cryptogram. A glance back at the plaintext of the foregoing example will show that two repetitions were pointed out: STO and TOMOR, and that these were still present in the primary cryptogram as 21-23-15 and 23-15-7-15-24. In the autokeyed version, they were still repeated sequences, but shorter in length: 44-38 and 38-22-22-39.
Had we initiated our trial decipherment with the correct number, 8, these two repetitions would, of course, have worked back to their original length. But where this trial decipherment was made with a different initial key-number (Fig. 128) we find that only one of the sequences, TOMOR, has done this; the other, STO, has disappeared. The explanation for this has been summed up in Fig. 130. One sequence was repeated at interval 8, which is even. When the autokeyed cryptogram is converted to one having period 2, any interval which is divisible by 2 will contain a certain number of periods; thus, any repeated sequence at interval 8, will appear in the periodic cryptogram as one of the ordinary periodic repetitions. The other sequence, STO, was repeated at interval 17, which is odd, and thus cannot show up as a repetition in any cryptogram whose period is 2.
It is this repeated sequence found at the odd interval which is to give us our short-cut. We have only two cipher alphabets, each one having a substitute for S, a substitute for T, and a substitute for O. When the repetition occurs at the odd interval, we obtain both substitutes for S, both substitutes for T, and both substitutes for O. By subtracting one sequence from the other, we may learn the numerical difference between the two cipher alphabets. Notice that the difference is constant, is alternately plus and minus, and is divisible by 2. (One alphabet is larger than the original, and the other is smaller by the same amount.) Our special
| Figure 130 Respective Behavior of the Cryptogram's Two Repeated Sequences Sequence 38-22-22-39 Sequence 44-38 Repeated at interval 8 Repeated at interval 17 Trial Decipherments: 1st occurrence: 22-16- 6-16-23 1st occurrence: 22-22-16 2d occurrence: 22-16- 6-16-23 2d occurrence: 20-24-14 2 -2 2 (This interval was EVEN) (This interval was ODD) |
method, then, for a cryptogram known to have been enciphered in this way, is as follows: First, underscore all repeated sequences which occur at odd intervals, or, in their absence, the repeated single letters. Those which are long will almost surely represent repetitions in the plaintext. Then, selecting a suitable number, make a trial decipherment and examine the resulting sequences. If, by any chance, those repetitions found at the odd intervals have worked back to longer repeated sequences, then the trial key and the original initial key must have been the same. If not, try subtracting one result from the other. If both have represented the same plaintext sequence, the result of the subtraction will be a constant difference, alternately plus and minus, and divisible by 2. To restore the primary cryptogram, split this uniform difference, adding half of it to the numbers of one alphabet, and subtracting half of it from the numbers of the other alphabet. This, as mentioned in the beginning, will leave a simple substitution cryptogram still to be investigated.
Our explanation, perhaps, has been a little rapid, but the student who has read carefully will be able to discover the “germ” originally referred to, and to make his own laboratory tests. Also, there may be an interesting answer to the following question: When the cipher is one of the Beauforts (using letters), and the auto-encipherment is initiated with a single letter, does a trial decipherment, initiated with some other single letter, result in a period of 2?