As pointed out more than once, those characters having the highest frequencies in periodic cryptograms will nearly always have derived these high frequencies because of their occurrence in more than one of the cipher alphabets; while
| Figure 132 Cryptogram Enciphered with the Slide of Figure 131 32 41 31 61 33 12 32 60 91 91 30 81 70 92 92 51 52 61 23 43 71 01 90 61 71 71 41 12 92 51 01 52 12 91 91 80 50 30 92 53 30 81 62 72 62 30 41 00 02 43 71 20 60 41 51 01 81 00 61 81 71 12 12 31 93 61 50 00 32 33 70 41 00 52 33 22 50 20 51 92 80 31 61 92 23 11 91 01 13 92 81 51 12 91 91 01 30 90 21 82 90 50 01 21 23 70 20 60 01 82 90 31 20 51 91 22 51 12 91 32 12 50 51 51 33 71 10 01 13 92 40 50 91 61 51 60 52 42 91 91 61 01 90 61 43 11 31 60 41 92 51 50 01 02 92 81 21 60 21 33 70 21 60 13 72 70 80 60 21 23 01 90 80 91 43 30 32 20 63 32 80 01 90 61 23 70 90 01 21 82 72 51 30 12 91 50 01 00 62 82 40 50 40 21 53 12 50 12 91 32 12 90 01 81 92 11 41 80 13 92 22 10 21 61 43 11 31 60 21 82 60 32 60 51 92 61 01 42 21 82 22 10 51 63 22 11 01 40 91 51 22 01 90 61 62 30 91 12 42 32 61 12 12 61 33. |
those having the lower frequencies will more often represent repetitions in some one cipher alphabet. Thus, when we find, in the present cryptogram, that the numbers 02, 53, and 63, have each a frequency of 2, it seems reasonable to suppose, for each number, that its two occurrences were in a single cipher alphabet; that is, that each one is a periodic repetition. Now, considering Fig. 133, and confining our observations, for the moment, to the number 02, we find that this number, in the cryptogram, occupies serial positions 49 and 154. Having first laid out a series of columns headed by the various possible periods, 2, 3, 4, 5. . . . . , we use each possible period in turn as a divisor, first applying them all to the serial number 49, and then applying them all to the serial number 154, each time setting down, in the proper column, the remainder from the division. This remainder tells us, each time, into what cipher alphabet the number 02 would fall, should the cryptogram be rewritten into the period indicated at the top of a given column. Still confining our observations to the number 02: It is seen, under possible period 2, that if this were the period, then the two occurrences of the number 02 would be in different alphabets. The same can be seen under possible periods 4, 6, 8, 9, 10. But if the period were 3, both occurrences of our number would fall into alphabet 1; if it were 5, both occurrences would fall into alphabet 4; if it were 7, both occurrences would fall into alphabet 7 (remainder zero indicates the final alphabet of the given period). Here, then, it would appear that possible periods 3, 5, and 7, are more likely than the rest, as far as the tabulation goes. When exactly the same observations are made for the number 53, it appears that the most likely periods are 3 and 5. And when these observations are made again for the number 63, only the period 5 is indicated as a likely one. Since the period 5 has been indicated oftener than any other, this is probably the correct period, as we happen to know that it is.
When the same method is applied to repeated sequences, the serial numbers can be those of the repeated first number. And it may, of course, be applied to letters, just as the Kasiski method might have been applied here. As to why Ohaver might have preferred this method in dealing with numbers, let us examine, in the figure, the entire column under possible period 5. The Ohaver method, unlike the Kasiski, not only indicates the period, but, in addition, shows the exact alphabet of that period into which a repeated number will fall. The number 02 is shown as belonging to alphabet 4 ; the number 53 as belonging to alphabet 5; and the number 63 as
| Figure 133 An OHAVER Method for Finding Period P O S S I B L E P E R I O D S Substitute Serial Position 2 3 4 5 6 7 8 9 10 ... 02 49 1 1 1 4 1 0 1 4 9 ... 154 0 1 2 4 4 0 2 1 4 ... 53 40 0 1 0 0 4 5 0 4 0 ... 205 1 1 1 0 1 2 5 7 5 ... 63 179 1 2 3 4 5 4 3 8 9 ... 244 0 1 0 4 4 6 4 1 4 ... X X |
belonging to alphabet 4. It is thus possible to see that the very small number 02 and the very large number 63 belong to a same cipher alphabet; and since a range of over sixty numbers cannot correspond to only twenty-six letters, we may conclude at once that the numbers on the slide were not in consecutive order. Often, our information is exactly the opposite.
Returning, now, to our cryptogram: In the beginning, we probably made a general frequency count; if not, we now have the five individual counts to be taken. And, as previously mentioned, a frequency count made on numbers is much more conveniently accomplished on a 10 x 10 chart than by sorting and listing the numbers. The moment our five frequency counts are made, in the present case, two facts become evident: Each count includes only fifteen or twenty different numbers, with about the frequency-distribution of simple substitution; and, while the tens-digits have included a full series, the units have never run beyond 3. The cipher, then, is a simple periodic; had multiple substitutes been used, the frequency counts would have included more different numbers, and with frequencies more uniformly distributed. As to the series of numbers, two probabilities are suggested, and these, in effect, are the same thing: The numbers may have run in straight order into the thirties, and with each number reversed; or: the numbers may have been grouped by tens. It is further possible that the whole series runs backward, or that the tens do, or the units, or sections of a certain length; and some uncertainty may arise as to the rank, in the series, of the digit zero; this digit is ordinarily last, but occasionally is ranked first. It is, of course, possible that the series of numbers is well mixed, but the chances are that it is merely methodized; the person who uses numbers in a simple periodic cipher is not usually one who knows the dangers of regularity in a cipher alphabet.
We may try, then, to restore his original arrangement (or an equivalent one), placing beside it the five frequency counts in their five columns, as shown in Fig. 134. The probable arrangements are very few, and the placing of tally-marks opposite their numbers is very rapid, since this, at each trial, is a mere matter of
| Figure 134 A Series of PARALLEL Frequency Counts Which Can Be LINED UP By PATTERN |
copying them from their charts. Once the correct rearrangement is reached, notice, in the figure, the appearance of the five frequency counts. Insofar as is ever likely to happen with columnar counts, all five have followed the same graph. This, of course, is the simplest case; the finding of the encipherer’s original order, so that every frequency count has followed the graph of the normal alphabet. Any substitute can be identified, as in Vigenère, by its serial position in its own alphabet; and where numbers are used, there is seldom any doubt as to what number comes first in its alphabet. The shortest road to solution would be as follows: Prepare a temporary slide exactly like the one which was used (except that we have no way of knowing what the key-letters were), mark the points at which the five alphabets begin, and decipher with the slide.