Alphabets of the kind we saw in No. 2 can be much more satisfactorily identified by means of a graph. This graph, when the cipher is Vigenère, is no more than a picture of the normal frequency table. Ordinarily, it will be a strip of paper on which the normal alphabet has been written twice in succession, with a straight line standing at right angles to each letter, this line being long or short according to the normal frequency of its accompanying letter.
A description of one such graph, suggested by L. H. Patty, will serve to explain them all: Assuming that the several frequency counts are standing in a vertical position, as we see them in Fig. 105, and that the work has been done on quadrille paper, the graph will also be prepared vertically, and the strip of paper will be quadrille paper with squares of the same size, so that the spacing, vertically, will be the same for graph and frequency counts. The graph, however, will be twice the length of the frequency counts, and will carry the normal alphabet written twice in succession (except that the final Z can be omitted). The basis for frequencies can be 200, 100, or any other basis desired. If the basis is 200, each small square might represent a frequency of 5, so that a horizontal line placed beside E (frequency 24), would have a length of nearly five of the small squares. Or, if the basis is 100, each small square might represent a frequency of 2, and the horizontal line placed beside E (frequency 12), would have a length of six of the small squares. Or, if this same graph is being made on a typewriter, we might dispense with the horizontal lines and use a series of diagonals (or 1’s, or asterisks), after the manner of tally-marks, using whatever number of these is the actual frequency of the letter per 200, or per 100; this will give a good clear picture of the normal frequency count. It is understood that the upper and lower halves of the graph are to be prepared exactly alike, and that there is to be no skipping of extra spaces between them. Thus the graph, being twice the length of the frequency counts, and spaced to match them, can be moved up and down beside each one of these until some point is found at which the pattern of the given frequency count bears some resemblance to a pattern found somewhere on the graph. If no such pattern can be found, the conclusion is that the frequency count was not made on one of the simple shifted alphabets; however, due allowance must be made, as in the case of our alphabet 2, for the difference in length and for the fact that frequency counts of this kind have been made on columns. Patty’s graph, so far, is representing only the shifted normal alphabet; that is, the cipher alphabets belonging to Vigenère, variant, and Gronsfeld ciphers. If its horizontal lines be made very heavy, and retraced on the opposite side of the strip, and if the letters be written on that side, opposite exactly the same horizontal lines as before, the reverse side of the strip will furnish another graph for identifying the reversed alphabets of the Beaufort cipher. Other graphs can be prepared for other kinds of alphabets. For instance, a graph suitable for examining a series of Porta frequency counts could be made in two halves, each of double length;
| Figure 106 Another Tabulation for Finding the Period EDWIN LINDQUIST Repeated Interval List of all PRIME Factors.....Sequence 2 3 5 7 11....(Etc.) J C V 24 111 1 C V 13 D D V 36 11 11 D S 12 11 1 S S 8 111 D T J 60 11 1 1 T J 48 1111 1 This was based on a cryptogram whose period was 12. The PRIME FACTOR 2 is obviously included twice, and the PRIME FACTOR 3 once. |
the A-to-M half would serve for comparison with the N-to-Z halves of the frequency counts, and vice versa.
The Vigenère cipher, and, in particular, the Kasiski method of solution, have given rise to much research among members of the American Cryptogram Association. We doubt that any of this research has ever resulted in any new or valuable discovery. Yet it is interesting in that it shows a body of amateurs arriving at devices which are fully as effective or convenient as those proposed by seasoned cryptanalysts. Carter’s “discovery,” for instance, which we saw in Fig. 90, was purely his own device; at that time, he had never heard of the “probable word method” proposed by Commandant Bazeries, one of the greatest of modern cryptanalysts. A great many of the first suggestions were directed at methods for making the trigram-search less tedious; these were largely duplications of a same idea, involving the use either of a tableau or of a slide; one example will be shown in the [next chapter]. The use of graphs, also, was a sort of simultaneous “invention.” As to Kasiski processes, while Ohaver’s tabulation had been published, it had been out of print and was not available for several years. The only information to be had was the fact that a period could be discovered by factoring intervals between repetitions, and Edwin Lindquist, finding this rather vague, devised for his own use the tabulation which is shown as Fig. 106. This tabulation was made from a cryptogram in which the period was 12. Lindquist, instead of preparing columns for all possible factors, prepared them only for prime factors, the repeated sequences and their separating intervals being listed in about the same way as in Ohaver’s tabulation. Now, taking one of the intervals, as 24: Tally in column 2, and the interval is reduced to 12. Tally again in column 2, and the interval is reduced to 6. Tally again in column 2, and the interval is reduced to 3, which is itself a prime factor. Tally a final time in column 3, and the interval 24 has been reduced to its prime factors. This process is almost entirely mental, and very rapid. Examining the results: Columns 2 and 3 are very full, indicating that prime factors 2 and 3 are both included in the period. But in column 3, the tallies are largely single, indicating that this factor is included only once in the period; while, in column 2, the tallies are largely in pairs, indicating that this factor is probably included twice in the period; had it been included three times, it would have shown up oftener in threes. Conclusion: The period is 2 x 2 x 3, which is 12. This tabulation will be found fully as convenient as Ohaver’s, and its results fully as accurate.
Mr. Lindquist also developed his own method for identifying alphabets. This method, which, in theory, is graphic, is not particularly applicable to the kind of
| Figure 107 The "SHIFT" Method for Identifying Alphabet 4 EDWIN LINDQUIST Letters apparently of the high-frequency class: I O P U Their possible originals................E T A O N I R S H Amount of SHIFT if I represents.. 4 15 8 20 21 0 17 16 1* " " " " O " .. 10 21 14 0 1* 6 23 22 7 " " " " P " .. 11 22 15 1* 2 7 24 23 8 " " " " U " .. 18 1* 20 6 7 12 3 2 13 A SHIFT of 1 (the B-alphabet) makes all four of these letters the substitutes for high-frequency originals. It almost certainly the shift which was made. |
alphabets we have been considering; that is, it would not be needed when there is so much material. But for shorter examples, where alphabets contain only ten or fifteen letters each, it comes close to being that magical thing referred to by Lamb, a “mechanical crypt-solver.” This method can be examined in Fig. 107, where it is being applied to our so-far unidentified alphabet 4. An examination of this alphabet 4 (of Fig. 105) shows that it has four letters of more prominence than the rest: I, O, P, U. These letters, or most of them, should represent high-frequency originals; and our method consists in examining them collectively in order to find out what amount of “shift” must have taken place in order that some four of the letters E T A O N I R S H would have resulted in these four particular substitutes. The word “shift” is best understood by picturing the movement of the lower alphabet on a Saint-Cyr slide. If the two A’s are together, this is the starting position, and the “amount of shift” is zero. If the B-alphabet be moved into position, we have a shift of 1; if the C-alphabet be moved into position, we have a shift of 2; and so on. These “shift-numbers,” 0 to 25, can be written below the letters of the sliding alphabet.
Now, considering only one of our letters, I: If this is the substitute for e, the normal alphabet was shifted 4 positions; if it is the substitute for t, the amount of shift was 15; if it is the substitute for a, the amount of shift was 8; and so on through the rest of the nine letters belonging to the high-frequency group. Finally, having considered our letter I as the substitute for all nine of these possibilities, we arrive at a series of nine shift-numbers: 4-15-8-20-21-0-17-16-1. And unless one of these is the correct shift, the cryptogram-letter I does not represent a high-frequency letter at all. In the figure, this examination has been made for all four of the letters I, O, P, and U, and opposite each of these we have the resulting series of nine shift-numbers. A comparison of the four series of numbers will show that each one includes a shift of 1. A shift of 1, then, that is, the B-alphabet, would have caused all four of our cryptogram-letters to become substitutes for high-frequency originals. This is almost certainly the shift which was made; but should the assumption prove incorrect, then a shift of 7 has appeared in three of the lines, and the H-alphabet would be the next choice. Lindquist’s method was found so effective for cases of scant material, that two members of the Association, M. R.
| Figure 108 Tableau Showing SHIFTS for Each Letter of the Alphabet - (MORRIS R. COLLINS) For VIGENÈRE For BEAUFORT E T A O N I R S H E T A O N I R S H 22 7 0 12 13 18 9 8 19 A 4 19 0 14 13 8 17 18 7 23 8 1 13 14 19 10 9 20 B 5 20 1 15 14 9 18 19 8 24 9 2 14 15 20 11 10 21 C 6 21 2 16 15 10 19 20 9 25 10 3 15 16 21 12 11 22 D 7 22 3 17 16 11 20 21 10 0 11 4 16 17 22 13 12 23 E 8 23 4 18 17 12 21 22 11 1 12 5 17 18 23 14 13 24 F 9 24 5 19 18 13 22 23 12 2 13 6 18 19 24 15 14 25 G 10 25 6 20 19 14 23 24 13 3 14 7 19 20 25 16 15 0 H 11 0 7 21 20 15 24 25 14 4 15 8 20 21 0 17 16 1 I 12 1 8 22 21 16 25 0 15 5 16 9 21 22 1 18 17 2 J 13 2 9 23 22 17 0 1 16 6 17 10 22 23 2 19 18 3 K 14 3 10 24 23 18 1 2 17 7 18 11 23 24 3 20 19 4 L 15 4 11 25 24 19 2 3 18 8 19 12 24 25 4 21 20 5 M 16 5 12 0 25 20 3 4 19 9 20 13 25 0 5 22 21 6 N 17 6 13 1 0 21 4 5 20 10 21 14 0 1 6 23 22 7 O 18 7 14 2 1 22 5 6 21 11 22 15 1 2 7 24 23 8 P 19 8 15 3 2 23 6 7 22 12 23 16 2 3 8 25 24 9 Q 20 9 16 4 3 24 7 8 23 13 24 17 3 4 9 0 25 10 R 21 10 17 5 4 25 8 9 24 14 25 18 4 5 10 1 0 11 S 22 11 18 6 5 0 9 10 25 15 0 19 5 6 11 2 1 12 T 23 12 19 7 6 1 10 11 0 16 1 20 6 7 12 3 2 13 U 24 13 20 8 7 2 11 12 1 17 2 21 7 8 13 4 3 14 V 25 14 21 9 8 3 12 13 2 18 3 22 8 9 14 5 4 15 W 0 15 22 10 9 4 13 14 3 19 4 23 9 10 15 6 5 16 X 1 16 23 11 10 5 14 15 4 20 5 24 10 11 16 7 6 17 Y 2 17 24 12 11 6 15 16 5 21 6 25 11 12 17 8 7 18 Z 3 18 25 13 12 7 16 17 6 |