CHAPTER XIV

The Kasiski Method for Periodic Ciphers

Prior to the 1860’s, the ciphers of the past two chapters had been regarded as entirely safe. A radical change of opinion took place in 1863, when Major F. W. Kasiski, a German cryptanalyst, was so indiscreet as to publish certain of his observations. The student will surely have noticed, among the examples of Chapters XII and XIII, the happening which is suggested in Fig. 102. Some sequence, usually a digram, is repeated in the plaintext, and happens to be enciphered more than once by exactly the same few key-letters; the result is a _repeated sequence in the cryptogram_. What he may have failed to notice is the _periodicity_ of such repeated sequences. In order that the same few key-letters be used again, _the key-word must have been repeated an exact number of times_, so that, in these cases of repeated cryptogram-sequences, the distance from first-letter to first-letter is _evenly divisible by the key-length_ — or period (in the figure, the distance from _V_ to _V_ is 10, which is twice the key-length, 5). This does not mean that all repeated sequences found in Vigenère cryptograms are periodic. Often, they are purely accidental; oftener still, they will be due to the repetition of alphabets in the key itself, especialiy if it is such a word as CORCORAN or DESDEMONA. But a distinct majority of them, according to Kasiski, are caused by periodicity; and if all of the repeated sequences found in a given cryptogram be examined to find what the separating interval is in each case, and if all of these intervals be factored, _the factor which predominates will betray the period of the cryptogram_.

Figure 102

A common happening in all PERIODIC ciphers:

Vigenère key: C O M E T C O M E T C O M E T C O M Plaintext: T H E R E I S A N O T H E R Q U E S REPEATED SEQUENCE: V V Q V . . . . . . V V Q V . . . .

In order to have a look at the Kasiski method, we will consider the cryptogram shown in Fig. 103; and, to approximate a more troublesome case, we will assume that no repeated sequences can be found except those few which have been underscored in the figure. With the cryptogram-letters serially numbered, in the manner shown, the distance apart of any two of them is readily learned by subtraction. The digram _CH_ is found beginning at the 1st letter and again at the 46th letter; 46 minus 1 equals 45, their distance apart. Thus, if _CH_ is one of the periodic repetitions, the period could be 15, 9, 5, or even 3. The trigram _UBF_, 8th and 63d letters, shows an interval 55; here, the period could be 11 or 5. Notice that a period 5 has been indicated by both.

Now let us look at Fig. 104, where the method of presentation is once more a debt to M. E. Ohaver. In this figure, the repeated sequences have been listed, and each one is accompanied by the two serial numbers of its two first letters, together with the interval which was obtained by subtraction. Ohaver’s process provides a column for each possible factor, beginning with 2 and carried as far as desired. Opposite each interval, its various possible factors may then be noted in their correct columns. In the average case, the correct period will be pointed out by _the column showing the largest number of entries_. But in this connection, it must be taken into consideration that _small_ factors 2 and 3, and even factors 4 and 5, are usually present in considerable numbers, partly as accidentals, but also because they are factors of the period itself; that is, if the period is 6, there will surely be factors 2 and 3 for every factor 6, and there will usually be a few extra appearances due to accidental repetition.

Figure 103

5 10 15 20 25 30 C H G S L F A U B F X U P H S J D A G Y X M N Z U W W J P D

35 40 45 50 55 60 J S U P L G C G F K R N I M F C H K O A Q A V X O N N U I L

65 70 75 80 85 90 N S U B F N D V P K A I P L S N M Q O H M E U I L B L K Q W

95 100 105 110 115 120 N D V I Y X U I I A Q E U U Y J W C O K O E N M P W W J J J

125 130 135 140 145 150 Q I U O V C M W D O X F C O L F S K U L V B W U N R V G T B

155 160 165 170 B S Q N L U E P H A Q T Q X V A K Q O E

Now, considering our tabulation, and ignoring the fact that short periods like 2 and 3 are seldom encountered, we find that factors 3 and 5 are present in equal numbers. Often, we are faced with exactly this problem. Here are two factors which have appeared in approximately equal numbers. Which one of these actually represents the period? Ohaver’s recommendations include these: Where two factors seem almost equally prominent, select the larger if it is a multiple of the smaller. If one factor is not a multiple of the other, try to select a period which is a multiple of both (as 15 here, includes both 3 and 5). He points out also that the factor which is the correct key-length will usually be accompanied, in the tabulation, by quite a number of its own multiples, growing gradually fewer and fewer as their size increases. In this respect, our factors 3 and 5 are both disappointing. If we consider factor 5, we find factors 10 and 15, but not growing fewer; instead the number increases. We find no factor 20, but we do find a factor 25; another increase. Or, if we consider factor 3, we find factors 6 and 9, but no factor 12, and then a sudden increase in the number of factors 15. This is a case in which the decryptor would play safe by selecting the period 15.

Figure 104

Tabulation for Finding Period M.E.OHAVER

Repeated Positions - POSSIBLE FACTORS of INTERVALS Sequence Intervals

C H 46 - 1 = 45 3 5 9 15 U B F 63 - 8 = 55 5 11 U P 33 - 12 = 21 3 7 21 S U 62 - 32 = 30 2 3 5 6 10 15 P L 73 - 34 = 39 3 13 W W J 116 - 26 = 90 2 3 5 6 9 10 15 18 N D V 91 - 66 = 25 5 25

The factors found in the largest number of DIFFERENT intervals are 3 and 5.

The student who cares to examine this matter more closely may do so by preparing for himself a less haphazard listing of the cryptogram’s repeated sequences. Perhaps the most satisfactory way of doing this is to begin by making a general frequency count. Then, in order to have the more reliable information at once, start the tabulations by examining those letters whose frequency is only 2; follow this with an examination of those having a frequency of 3, and so on. The theory is that letters of these frequencies are much more likely to belong to only one alphabet, while the letters of higher frequency have probably been enciphered in several different alphabets, so that their repeated sequences are not so sure to be periodic. For other cases in which there may be some doubt, the writer’s advice is to select large factors in preference to small factors. Or, if the decision must be made between two factors such as 6 and 7, where a period of 42 would be necessary in order to include both, simply select the handiest and give it a trial. With the longer cryptograms, as we shall see in a moment, an error in the choice is very speedily discovered; as to the shorter cryptograms, there is one rule which invariably holds good: _If you meet with any resistance at all_ in dealing with the kind of ciphers which were shown in the past two chapters, you have probably selected the wrong period.

Figure 105

Individual Frequency Counts - PERIOD 5

Alphabet 1 Alphabet 2 Alphabet 3 Alphabet 4 Alphabet 5

A 11 11 1 111 B 11 1 11 1 C 111 1 11 D 111 1 1 E 1111 1 F 11 1 1 111 G 1 111 1 H 11 11 1 I 11 11 1111 J 111 11 1 1 K 1 111 111 L 1 1 11111 11 M 1 111 11 N 11111 11 11 1 1 O 1 11111 1 11 P 111 111 1 Q 1111 1111 1 R 11 S 1111 1 11 T 1 1 U 1 11 11111 11 111 1 V 1 1 111 11 W 11 111 11 1 X 1111 11 Y 111 Z 1

Often, however, where one clue is missing, there will be another present to take its place. Repeated _trigrams_ are less likely than repeated digrams to be accidental, and longer repeated sequences are still less likely to be so. In the present tabulation, we find that three of the repetitions are trigrams; in all three cases the period 5 is suggested, while only one suggests also a period 3. That is, if we use a period of 15, two of these trigrams will have to be considered accidental.

If the period here is 5, then we are dealing with _five simple substitution alphabets_. These five alphabets have been used over and over again, always in a given rotation; therefore, if the cryptogram be rewritten into _five columns_ (it is already conveniently grouped), the letters in each column will belong to one same alphabet, and it becomes possible to _take a separate frequency count on each one of these five alphabets_. These individual frequency counts may be seen in Fig. 105. Originally, we had a length of 170 letters, and, if the student desires to take a frequency count on the complete cryptogram, he will find that he has no truly predominant letters which could represent some of the letters _E T A O N I R S H_. Instead, he has a series of frequencies which are all fairly close to 4% of the text (6 or 7), and which, should he rearrange them in decreasing order, would have somewhat the following appearance: 10-10-10-9-9-9-8-8-8-7-7-7-7. . . . . .3-2-2-1. He will probably find, also, that every letter of the alphabet has been used at least once, something which would be very rare indeed in any normal English text of 170 letters. But in these five individual frequency counts, each belonging to a separate alphabet, matters are different. Here, the alphabets represented have a length of only 34 letters each, and yet, in the third, fourth, and fifth alphabets, there is one predominant letter, which could represent _E_, or some other letter which has taken the place of _E_, while, in the first and second alphabets, there are some few letters distinctly more prominent than others. Also, each alphabet has shown some gaps in sequence, where letters of the class _J K Q X Z_, and possibly also some letters like _B P V W_, would surely be missing in a normal text of only 34 letters.

A frequency count made on columns is not, of course, normal. We saw this in dealing with transpositions, when we considered vowel-distribution. Yet, as length increases, we find that the letters present in columns begin to approach more and more the proportions found in normal text; here, with only 34 letters, it would be possible, in any one of these frequency counts, to assign the letters to groups of high, moderate, and low frequencies. _Whenever our frequency counts do not have this general aspect, the period cannot be correct_. (There are, of course, the very short cryptograms, in which the actual frequencies are not apparent.) So far, we are dealing with any cipher whatever of the periodic type, and many of these ciphers do not make use of simple shifted alphabets, or even of alphabets which are in any way related to one another.

Now let us consider the one case in which the alphabets are all “Caesars.” In this case, whether the cipher is Vigenère, Beaufort, or Porta, we have only to identify one letter in order to identify a whole alphabet. Suppose we examine, first, alphabet 5, in which the one outstanding letter, _L_, has appeared 7 times. Does this letter represent _e_? If _L_ of alphabet 5 represents _e_, then, counting backward (that is, upward), we find that the letter _a_ will have to be represented by _H_; this alphabet, then, will be the _H_-alphabet if the cipher is Vigenère. The letter _H_ has a frequency of only 1, which, in normal text, is not particularly satisfactory as the frequency of _a_, but this frequency count has not been taken from normal continuous text; suppose we examine the rest of the alphabet, and find out what the frequencies would be for other letters. Beginning at _H_, and calling letters in the order _a_, _b_, _c_, we find that this fifth alihabet, provided it is the _H_-alphabet, will contain: 3 _d_’s, 7 _e_’s, 2 _h_’s, 2 _l_’s, 2 _o_’s, 3 _r_’s, 3 _t_’s, and 3 _y_’s. That is, each letter present which shows a frequency greater than 1 will represent some plaintext original which, normally, is of some frequency, the only exception being _y_, which is a vowel. This is the best we can expect of any columnar frequency count made on only 34 letters; but more convincing still, and more reliable, is the fact that out of the entire group _j k q x z_ we find only _x_, represented once. Alphabet 5, then, is entirely acceptable as the _H_-alphabet of the Vigenère cipher.

Let us see what we can find out about alphabet 3. Here, the strongly predominant letter is _U_. But when we attempt to identify this as _e_, we find that we should have to accept an alphabet containing 3 _q_’s, 2 _x_’s, and 3 _z_’s, all occurring in only thirty-odd letters of text. We meet with similar trouble when we attempt to identify _U_ as _t_, as _a_, as _o_, and so on. It is not until we try it as _s_ that we have good luck, finding only a series of blanks to represent the letters _b_, _j_, _k_, _q_, _v_, _w_, _x_, and _z_. And if _U_ represents _s_, this alphabet begins at _C_. Alphabet 3, then, is entirely acceptable as the _C_-alphabet of the Vigenère cipher, and we have two of the key-letters: * * _C_ * _H_.

In alphabet 1, the leading letter, _N_, is not so strongly predominant, and yet, when we assume it as the substitute for _e_, we find that the rest of the count is satisfactory. Alphabet 1, then, is acceptable as the _J_-alphabet of the Vigenère cipher and we have three of the key-letters: _J_ * _C_ * _H_.

In alphabet 2, we find no one leading letter, but the two most prominent frequencies are standing opposite _E_ and _S_, as if this count might represent the normal alphabet itself. The absence of _O_ and the presence of only one _T_ is hardly significant in a columnar count; but further examination shows an excess of _M_’s and _W_’s, and this is more disturbing. However, a single _K_ has appeared as the only representative of the group _J K Q X Z_; the low-frequency letters _B_ and _V_ have appeared but once each; and there is an absence of _Y_’s to counterbalance those which were too numerous in one of our other alphabets. So that a detailed examination, and the failure to identify this as any other alphabet, will lead to its tentative acceptance as the _A_-alphabet of the Vigenère cipher. (We can know definitely when we attempt to decipher with it.) With alphabet 2 accepted as the _A_-alphabet, we now have four of the key-letters: _J A C_ * _H_. We shall return in a moment to consider the one which is still missing; but according to those present, it does not look as if our key is going to develop into a recognizable word.

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; the _A_-to-_M_ half would serve for comparison with the _N_-to-_Z_ halves of the frequency counts, and vice versa.

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 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.

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.

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 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. Collins and Helen S. Pearson, decided, independently of each other, to set it up in permanent form, so as to avoid fresh computations for each new cryptogram.

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

Collins’ device took the form of a tableau, as shown in Fig. 108. In this figure, the vertical alphabet running through the center is a list of possible cryptogram-letters. On the side marked “Vigenère,” the four lines of numbers standing beside the letters _I_, _O_, _P_, and _U_, are the same as those included in Fig. 107. It will be noticed that only the first line of numbers (opposite _A_) need be found from the slide; after that, each column is a series 0 to 25. The same is true with reference to the Beaufort shifts. These, incidentally, were computed on the assumption that the Beaufort keys, _A_, _B_, _C_, _D_. . . . . . . are passing in their normal alphabetical order beneath the stationary _A_ (as most of us prepare the Beaufort slide, this is backward). Fig. 109 shows a similar tableau prepared for the Porta shifts. The zero-position here is the _AB_-alphabet, a shift of 1 is the _CD_-alphabet, and so on. Collins, however, did not use the shift-numbers. _He increased these by 1_, using numbers 1 to 26, which represent the 26 positions of the slide, or, better, the serial positions in the normal alphabet of the 26 key-letters. Others who have since prepared similar tableaux have dispensed altogether with numbers, and have used the key-letters themselves. Doing this, the first row of the Vigenère portion will show the nine key-letters _W H A M N S J I T_, the second row will show key-letters, _X I B N O T K J U_, and so on. If the letters appearing on the slide have been numbered, one method is fully as convenient as the other, though in dealing with a plaintext key one would probably prefer the letters. In any case, where some four letters, such as our _I O P U_ of the foregoing alphabet, have been found more than once in a given frequency count, it is merely necessary to find these four letters one by one in the vertical alphabet and _copy_ their accompanying numbers. It is even possible, having these three tableaux, to decide whether the frequency counts taken from a periodic cryptogram represent the alphabets of the Vigenère, the Beaufort, or the Porta.

Figure 109

Tableau Showing SHIFTS for PORTA

E A I H T O N R S

9 0 5 6 N 6 1 0 4 5 A 10 1 6 7 O 5 0 12 3 4 B 11 2 7 8 P 4 12 11 2 3 C 12 3 8 9 Q 3 11 10 1 2 D 0 4 9 10 R 2 10 9 0 1 E 1 5 10 11 S 1 9 8 12 0 F 2 6 11 12 T 0 8 7 11 12 G 3 7 12 0 U 12 7 6 10 11 H 4 8 0 1 V 11 6 5 9 10 I 5 9 1 2 W 10 5 4 8 9 J 6 10 2 3 X 9 4 3 7 8 K 7 11 3 4 Y 8 3 2 6 7 L 8 12 4 5 Z 7 2 1 5 6 M

Miss Pearson’s device took the form of _strips_, a set of 26 for each of the three ciphers. Fig. 110 shows the first five of her Vigenère set as she originally prepared them, using the “position-numbers,” which are all larger by 1 than those of the tableau. Aside from this, each strip represents one row from the Vigenère half of Collins’ tableau. But where Collins had arranged his numbers according to the frequencies of the nine possible originals (so that possibilities found on the left might have more significance than others found on the right), Miss Pearson arranged hers in straight numerical order, and spaced them in such a way that No. 1 is always in the first column, No. 2 is always in the second column, and so on. Had she used key-letters, all _A_’s would have been in the first column, all _B_’s in the second column, and so on. As to the use of these strips: Presuming that the four leading cryptogram-letters are the same as before, simply pick out the four strips which are headed by the letters _I_, _O_, _P_, and _U_, and set them together. If any of the numbers are duplicated, _you will find them standing in the same column_. These, remember, are the devices of amateurs, and both will be found very effective. It will be noticed that the basis is the finding of key-letters (or numbers) and not the identification of cipher alphabets.

Now compare these devices with a method proposed by an expert, in which the basis is the identification of cipher alphabets, and not their keys: With this method, a tableau is prepared (which could be arranged like the one of Fig. 85) in which the only letters shown on any one line are the substitutes for the nine high-frequency letters. If, for instance, the tableau is intended for the Vigenère cipher, the top row will contain only the letters _A E H I N O R S T_, _and the other 17 positions will be left blank_. The second row will contain only the letters _B F I J O P S T U_, the third will contain only the letters _C G J K P Q T U V_, and so on. Or, if the tableau is intended for the Beaufort cipher, the top row will contain only the letters _A W T S N M J I H_, the second row only the letters _B X U T O N K J I_, and so on. Thus, after having taken a series of frequency counts, we may find out, in each of these frequency counts, which are its leading letters, then consult the prepared tableau to find out which of its alphabets will show these same leading letters. An added suggestion is as follows: Prepare the tableau, as described, using black ink. Then, using red ink, add to each alphabet the substitutes for _J K Q X Z_ (perhaps, also, for _B P V W_); that is, the substitutes for those letters which ought to be largely _absent_. This makes it much easier to decide between two alphabets in which the more frequent letters have made it seem that one is as likely as the other. It will be found that letters of low or moderate frequency are ordinarily as helpful in these ciphers as those of high-frequency; an instance has been pointed out in which those of the cryptogram can be more so: Where the question is one of deciding between two possible periods, a new tabulation can be made using only the sequences found in connection with those letters which are less frequent in the cryptogram than others, and thus not so sure to belong to more than one alphabet.

Figure 110

Strips for Determining SHIFTS HELEN S. PEARSON

SET FOR VIGENÈRE --------------------------------------------------------------------------------- _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ A 1 _ _ _ _ _ _ 8 9 10 _ _ 13 14 _ _ _ _ 19 20 _ _ 23 _ _ _ ================================================================================= _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ B _ 2 _ _ _ _ _ _ 9 10 11 _ _ 14 15 _ _ _ _ 20 21 _ _ 24 _ _ ================================================================================= _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ C _ _ 3 _ _ _ _ _ _ 10 11 12 _ _ 15 16 _ _ _ _ 21 22 _ _ 25 _ ================================================================================= _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ D _ _ _ 4 _ _ _ _ _ _ 11 12 13 _ _ 16 17 _ _ _ _ 23 24 _ _ 26 ================================================================================= _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ E 1 _ _ _ 5 _ _ _ _ _ _ 12 13 14 _ _ 17 18 _ _ _ _ 24 25 _ _ ---------------------------------------------------------------------------------

NOTE: The numbers here are POSITION-numbers, instead of SHIFT-numbers. A shift of zero is "position 1" of a slide. This is also the numerical, or serial, position of A in the normal alphabet. Most members of the American Cryptogram Association prefer to dispense with numbers, and use key-letters.

We have seen, then, what can be done in place of the trigram-search in the case of those longer cryptograms. Having one of only 170 letters, we first found out its period, and then (presuming that we accepted the _B_-alphabet in the case of alphabet 4), found out all five of its key-letters _J A C B H_, and even the type of encipherment (obviously Gronsfeld), _without having deciphered a single letter of its message_. We are now in a position to go back and investigate any which are still unsolved. With Vigenère methods and principles thoroughly understood, the student is fully in possession of methods for dealing with any periodic cipher whatever _in which he knows what the cipher alphabets are_. All that remains, then, is to pick up a few loose ends, and observe a few variations from the strictly periodic encipherment, after which we may consider the case of the unknown cipher alphabets.

116. By NEMO. (A Vigenère? Or a snare and a delusion?)

W L P C V M O G K E E I F M U R W W F H V M F F W E Y X A V U B I C Z O J M L C H V X Y F K S C U S X I L M G B Q I D B W I F G B I Q Z G Z H F J Y P M K I G V P T W Y K W Z H W M Z H W I F A P S D N W F H E D S C X A V O E B Y Y O K C O Y U I H U J L H U D X P P W V V H P F W Y L G F B V E J M A A G B P I E B A V U V Q L Z N L P W A J W.

117. By NEON. (U.S.Army Cipher Disk. Surely not an advertisement?)

D J T X J M H L M K O M F D T F N E U I G D D N A A U S N S A C F G Y M Z Y A Q A N M W U W S R B R F J J Q S K A Y B A N B L T O J E R K S N W X A G T J L Z Y S T V A R B X L K N R L V D U U F O F A K Z L W Y T E E W.

118. By TITOGI. (What! Another Vigenère? Some collusion here!)

D W P W Z T C G H H Z B B V W F B H I F W Q B L L J D Z R G U M M E S W B D W L J K X I F Y Z D G K Y I O I K D W P M F H C M S F Q G C E L J I I H W A M I W L J Z I W S W K V W E.

119. By THE ADMIRAL. (Vigenère).

N S R V K D K S I W J W Y C E C E G K C E B D K N Q Y S J U L X Z O L X P S U V U T F B S O I N P C R R E U Y O N U F K H K Z D D O J P Q Z C K J I E N A F J D W B U S J U R C L C J C E P C O K T V F A F P Y X G K K Y Z V.

120. By THE ADMIRAL. (Beaufort).

Z N J L N Y H C Z D A U D D Z I N H R C Z Y Z K H G B P E C L M L W Y R O I J Q D T L Q O Z H Q S N D V E S E P E J O Y L S Z O J U P G T K J F K C U W N S H G W F D T M G K K D W E H L Z R N S B G V E S R A U K K U M J Z M T K N K F Q L G K C U P Z U S D L W D E Z U B D Y F O D.

121. By DOR. (Another "Aristocrat." - Not hard. No keyword).

A B C D E F G C H G I J A K G F D J F B L M E D M M I M G B A N F L C L O G J P N F D R F C L N. O G P I M S D A N T D L I F U. F C B G N B P J E G J F C L E F, K C G A I E D V B F.