Chapter 18 of 25 · 6324 words · ~32 min read

CHAPTER XVII

Some Periodic Number-Ciphers

The use of numbers in a periodic cipher does not, in itself, create a problem essentially different from that of the letter-ciphers. Numbers may, in themselves, cause weakness; we saw such a case in the last of our autokey examples, where a complete disarrangement in their order did nothing to conceal their size. But oftener than not, the weakness lies in the construction of the cipher or in the manner of its application, and while this is fully as true of letter ciphers, the numbers, for some reason, appear to be more inviting for certain kinds of misuse.

In order to observe a weakness which need not have existed, let us consider the slide partially shown in Fig. 131. The use of two-digit numbers will furnish a hundred substitutes; but a strip of that length is awkward to handle, and the constructor of the present slide has confined its length to forty numbers. Then, since he has only fifteen different cipher alphabets, and wishes to make use of word-keys, he has adjusted his 26 key-letters to fit the number of alphabets. Now if the alphabet of the slide (that is, the _cipher alphabet_, or series of numbers) is written in straight 1-2-3 order, and if it is considered that letters may have two or more values, so that _A_ has the values 1, 27, 53, etc., _B_ the values 2, 28, etc., _C_ the values 3, 29, and so on, a slide of this kind is exactly the equivalent of the Saint-Cyr slide, since any cryptogram accomplished with it could be promptly converted to a Vigenère cryptogram by substituting letters for numbers. The keys, of course, might differ. The constructor of the slide has desired something more difficult. But instead of carrying his forty numbers through a transposition block, and really mixing them, he has been content to group them, in regular order, by their tens. We shall see in a moment what happens to his cryptograms. But he neglects also an opportunity: Presuming that his circumstances are such as to make the use of numbers practical at all, why waste the opportunity to use the full one hundred substitutes? The remaining sixty numbers might have been placed on the next two rows, and thus, in every position of the slide, he could have had two or three optional substitutes for every letter — a much more difficult case than the simple periodic.

Figure 131

A Slide Carrying a NUMBER-Alphabet (and Keys)

( Plaintext Alphabet - Stationary ( ( A B C D E F G H I J K L M ..... ( 10 20 30 40 50 60 70 80 90 00 11 21 31 41 51 61 71 ..... Sliding Cipher-Alphabet: Key-letters may be added: A B C D E F G H I J K L M N O) ( * P Q R • S T U * V W X Y Z *)

The addition of key-letters makes it possible to employ a keyword. For the present forty-number slide, it was necessary to double them up as in Porta. A slide having fifty-one numbers would have accommodated all twenty-six of the key-letters.

The cryptogram of Fig. 132 was enciphered with the slide of the preceding figure, using the key-word CABLO (equivalent to the numerical key 30-10-20-21-51), and its period, 5, can be determined in the usual way. However, we have already seen the Kasiski method; suppose, here, we look at another, originated by Ohaver; and, since Ohaver himself, explaining his method in connection with a number-cipher of much the kind we have here, illustrated with single numbers instead of with sequences, it seems fitting that it be illustrated again in the same way.

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.

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 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 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.

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

* * *

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.

Figure 134

A Series of PARALLEL Frequency Counts Which Can Be LINED UP By PATTERN

10 . 111 . . . 20 . 11 111 . . 30 11111 1 11 . . 40 11 . 11 . . 50 1 11111 1111 . . . 60 11 . 11111 1111 . . 70 11111 . 1 . . 80 111 1 11 . . 90 11 111 11111 . . 00 . . 11111 . . 11 11111 . . . . 21 . 11 1 11111 111 . 31 . 1111 1 1 . 41 . 11111 . 11 . 51 11 111 11 1111 1111 61 1111 . 11 11111 1111 . 71 11111 . . . 1 81 11 111 . 1 1 91 . 11 1 11111 111 11111 1 01 1111 11111 11 11111 1 1 . 12 1111 11 11111 111 1 . 22 11111 . . . 1 32 1 111 . 1 1111 42 . . 11 1 . 52 . 111 . 1 . 62 . . 1 1 11 72 1 . . 1 1 82 . . . . 11111 1 92 . . . 1111 11111 1111 02 . . . 11 . 13 . . . 1111 . 23 . . . 1 1111 33 . . . . 11111 1 43 . . . . 11111 53 . . . . 11 63 . . . 11 . 73 . . . . . 83 . . . . . 93 . . . . 1 03 . . . . .

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 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.

There are many other cases, hardly more difficult, in which our rearrangement of numbers results, not in the original order, but in an _equivalent order_. We could, for instance, arrive at a rearrangement in which we have taken each third number, or each fifth number, of the original cipher alphabet, so that our rearranged numbers are following plaintext letters in the order _A D G J_. . . . or _A F K P_. . . . ; thus, all of our frequency counts would be following one same graph, though not the graph of the normal alphabet. The problem here is to make sure that their graphs are all the same graph, and then subject them to the process called “lining up.”

Figure 135

The LINING UP of the Frequency Counts of Figure 134

┌1st┐ ┌2d ┐ │ │ │ │ │ 30│11111 │ 10│111 │ 40│11 │ 20│11 │ 50│1 │ 30│1 │ 60│11 │ 40│ │ 70│11111 │ │ 80│111 │ 60│ │ 90│11 │ 70│ │ 00│ │ 80│1 │ 11│11111 │ 90│111 │ 21│ │ 00│ │ 31│ │ 11│ │ 41│ │ 21│11 │ 51│11 │ 31│1111 │ 61│1111 │ 41│11111 │ 71│11111 │ 51│111 │ 81│11 │ 61│ │ 91│ │ 71│ │ 01│1111 │ 81│111 │ 12│1111 │ 91│11 │ 22│11111 │ 01│1111111 │ 32│1 │ 12│11 │ 42│ │ 22│ │ 52│ │ 32│111 │ 62│ │ 42│ │ 72│1 │ 52│111 │ 82│ │ 62│ ┌3d ┐ ┌4th┐ ┌5th┐ TOTALS │ │ │ │ │ │ │ 20│111 │ 21│11111111 │ 51│1111 23 * │ 30│11 │ 31│1 │ 61│ 7 │ 40│11 │ 41│11 │ 71│1 7 │ 50│ │ 51│1111 │ 81│1 7 50│111111111│ 60│111111111│ 61│111111111│ 91│111111 38 * │ 70│1 │ 71│ │ 01│ 4 │ 80│11 │ 81│1 │ 12│ 5 │ 90│11111 │ 91│11111111 │ 22│1 15 * │ 00│11111 │ 01│1 │ 32│1111 18 * │ 11│ │ 12│1 │ 42│ ** 1 │ 21│1 │ 22│ │ 52│ ** 1 │ 31│1 │ 32│1 │ 62│11 6 │ 41│ │ 42│1 │ 72│1 8 │ 51│11 │ 52│1 │ 82│111111 18 * │ 61│11 │ 62│1 │ 92│111111111 20 * │ 71│ │ 72│1 │ 02│ 3 │ 81│ │ 82│ │ 13│ ** 0 │ 91│1 │ 92│1111 │ 23│1111 16 * │ 01│111111 │ 02│11 │ 33│111111 20 * │ 12│11111111 │ 13│1111 │ 43│11111 29 * │ 22│ │ 23│1 │ 53│11 6 │ 32│ │ 33│ │ 63│ (V) 0 │ 42│11 │ 43│ │ 73│ 5 │ 52│ │ 53│ │ 83│ ** 0 │ 62│1 │ 63│11 │ 93│1 8 │ 72│ │ 73│ │ 03│ ** 0 265

To show the handling of all such cases (which would include our final autokey example), let us assume that the five frequency counts of our figure, though still following a common graph, are not following that of the normal alphabet. In this case, granting that all fifty-letter frequency counts will vary considerably from the normal, it is not quite so obvious that their pattern is the same; we shall have to cut them apart (preferably having copied numbers beside their frequencies) and place them side by side for a comparison of their graphs. Where this has been done, in Fig. 135, their similarity is plain in spite of some discrepancies, and the moving up or down of any one or more of the counts (which could be done so as to include another position, since the range of the numbers is only 25 per alphabet) does not result in greater similarity. If the alignment of this figure is correct, then all numbers found on any one row are substituting for one same original; thus, the added frequencies on any one row will be the total frequency of some one letter in a 265-letter text, and all of these totals, collectively, should resemble a frequency count taken on a simple substitution cryptogram of that length. To just what extent this is true may be seen at the right side of the figure. The nine leading letters have totalled 74%, where we normally expect 70%; but any single example can provide its surprises, and the excess 4% is not on the wrong side of the ledger. The other end of the count, as would be expected of the group _J K Q X Z_, is comparatively blank.

Figure 136

The NIHILIST Number-Substitution

The "Checkerboard" Alphabet:

1 2 3 4 5

1 A B C D E 13 = C 2 F G H I K 34 = O 3 L M N O P 32 = M 4 Q R S T U 15 = E 5 V W X Y Z 44 = T

Encipherment, with Keyword COMET:

S E N D S U P P L I E S T O .... Text... 43 15 33 14 43 45 35 35 31 24 15 43 44 34 .... Key.... 13 34 32 15 44 13 34 32 15 44 13 34 32 15 .... 56 49 65 29 87 58 69 67 46 68 28 77 76 49 ....

This cryptogram is usually seen without grouping: 56-49-65-29-87-58.....

Our substitutes, remember, are assumed to be in mixed order. We do not know what letter is represented by the five numbers of the top row, or by the five numbers of any other row. To proceed with solution, we shall have to assign arbitrary values, calling the top row _A_ (or 01), the second row _B_ (or 02), the third row _C_ (or 03), and so on; and when all of these substitutions have been made on the cryptogram, _the case has been reduced to one of simple substitution_. The mechanics by which the substitutions are made can be exactly those of the other case: Prepare a temporary slide, on which the numbers run in the order decided upon, and slide this against the normal alphabet (or any other); the result is a simple substitution cryptogram which can be solved by simple substitution methods. This case, first in one form and then in another, is encountered again and again; and however it may seem that its cause, in some one example, is a different one, yet the fault in all such examples is the same: The basic cipher alphabet (the primary one from which others are derived), either by its actual construction or by the method of its application, was not truly a mixed alphabet.

* * *

In some of the periodic ciphers, the basic cipher alphabet is a “checkerboard” of the kind we saw in Chapter XI, the substitutes being two-digit numbers which will point out the columns and rows of their originals. This primary alphabet, however, seldom appears unchanged in the cryptograms, as “position 1” often does when a slide is used, or as the _A_-alphabet often does in the Vigenère cipher. Instead, we find a series of secondary cipher alphabets all of which have been derived from the primary one according to a mathematical process.

In view of the fact that any cipher which will necessarily double the lengths of messages is of doubtful value, it seems inadvisable here to do more than mention the infinite multiplicity of processes which would be possible; but with checkerboards, it is difficult to imagine any usable process which would not result in parallel frequency counts; that is, counts which all follow the same graph and thus are capable of being “lined up.” With most of these, in fact, the difference between any two of the (secondary) cipher alphabets will be a difference in _size_ which is uniform from _A_ to _Z_. (Often, the same result is produced with slides.) Here, then, we may content ourselves with a glance at one such cipher which is interesting rather than important. In Fig. 136, we have another of the Nihilist ciphers. Its primary alphabet is that most famous of checkerboards, the _Polybius square_, said to have been the invention of the ancient Greek historian, and certainly well known in his era as the basis for a signalling system — a capacity, incidentally, in which it still serves. We are showing it here in what seems to be the favorite version: The alphabet of the square is the normal one, normally arranged, with _J_ the missing letter; and the order of reading for the two digits is row-column. It should be understood, however, that these details, in practice, are quite variable.

For the Nihilist encipherment, the message is first subjected to a simple substitution, using the checkerboard key. A key-word, treated in the same way, is repeated often enough to pair one key-number with each message-number, and the final cryptogram is formed by adding these pairs of numbers. Decipherment, of course, will be the subtraction of key-numbers from the finished cryptogram and the resubstitution of letters. We have, then, another periodic cipher, not essentially different from those already seen. Any number in the checkerboard can become a key, to be applied periodically at some given interval, and thus may govern one of the 25 possible cipher alphabets. It would be possible to lay out any one of these cipher alphabets, simply by adding a given amount to each number of the primary one; if all of them were written one below another, and if the primary alphabet were written across the top and along one side, we should have a tableau which could be used in identically the manner described for the use of the Vigenère tableau.

Decryptment, too, can parallel that of the Vigenère: The period of a cryptogram can be found through repeated sequences, or, in their absence, through repeated single numbers, and individual frequency counts can be taken on the several alphabets of the period. If the arrangement of letters in the checkerboard is that of the figure, or any other strictly alphabetical one to which the order of the numbers can be adjusted, these frequency counts will all follow the graph of the normal alphabet, with allowance made for the missing letter. Or, if the arrangement of letters in the checkerboard is not strictly alphabetical, then the several frequency counts, no matter how badly mixed, will still be parallel; they will all follow one graph, and thus can be “lined up.” Very often, however, given the opportunity to examine and analyze a cipher, it becomes possible to formulate for it a special method which is much more rapid than the general one; Ohaver, who first published a special method for the Nihilist, has compared this cipher to a leaky boat in the open ocean.

Notice that its primary alphabet contains only the digits 1-2-3-4-5. The maximum difference among these is 4; and the addition of any same number to all of them does not change this fact; the maximum difference between any two of the sums would still be 4. But the number which is added during encipherment is also a number containing no digit other than 1-2-3-4-5; thus any number found in a cryptogram can be considered as carrying two separate additions, one for tens and one for units. Even when two 5’s are added together, the result is an all-revealing zero; the “carried” digit 1 can be mentally “borrowed” back, causing the zero to become 10, and decreasing by 1 the size of the digit which precedes the zero. Specifically: Finding in a cryptogram the number 40, we may regard this as having only 3 tens, with 10 units; or finding the number 110, we may regard it as having 10 tens and 10 units. Thus, there is never a time when it is impossible to see the tens and units as having been separately added; if we find, in a Nihilist cryptogram, the two numbers 29 and 87, with a difference greater than 4 in their respective tens-digits, we may say promptly that they were not enciphered with the same key; no digit whatever added to any two digits of the original square can produce a difference greater than 4. But if the two cryptogram numbers are 30 and 77, where the difference in the tens-digits appears, at first glance, to be only 4, the presence of the zero must be taken into account; thus, the number 30 has only 2 tens, and the difference between 2 and 7 is greater than 4; therefore, the numbers 30 and 77 could not have been enciphered with the same key. It is interesting, also, to note that the digit 2, found in a cryptogram, can have been produced in only one way: the addition of 1 and 1; and that the digit 0, found in a cryptogram, can only have been produced by the addition of 5 and 5. Either one of these digits gives away its key; but, further than this, the cipher provides four “give-away” numbers, 22, 30, 102, and 110, the presence of any one of which in a cryptogram will give away the key to a whole cipher alphabet.

Figure 137

Cryptogram by EDWIN LINDQUIST: Final Investigation of Supposed Period 4 24-66-35-77-37-77-55-59-55-45-55-88-28-66-46- 24 66 35 77 88-37-67-33-59-58-65-45-66-67-58-44-55-34-79- 37 77 55 59 55 45 55 88 44-59-55-45-42-87-28-76-43-78-46-86-26-67-24- 28 66 46 88 37 67 33 59 85-26-67-28-76-26-78-46-65-65-88-36-49-54-67- 58 65 45 66 67 58 44 55 28-65-42-88-36-49-44-89-57-58-54-66-47-67-26. 34 79 44 59 55 45 42 87 28 76 43 78 46 86 26 67 24 85 26 67 28 76 26 78 46 65 65 88 36 49 54 67 28 65 42 88 36 49 44 89 57 58 54 66 47 67 26

(Acceptable throughout)

Now, to look at Ohaver’s special method, let us consider the cryptogram of Fig. 137, prepared by another “inventor” of exactly the same method. It can be noted, first, that this cryptogram has not resulted from the addition of a single number throughout, since it contains pairs of numbers like 24-88, 42-87, and so on, which have a greater difference than 4 in either their tens or their units. Now, using a bit of scratch-paper, we may, if we like, scribble down a series of possible periods, 2, 3, 4, 5, 6, and so on, to be crossed off as fast as we eliminate them. Considering these, one by one:

_Period 2_: With a thumbnail on the first number, 24, and another on the third number, 35, we may run quickly through the cryptogram comparing numbers found at interval 2; that is, the first and third numbers, the second and fourth, the third and fifth, and so on, until stopped by the two numbers 33 and 38, whose difference, in the units, is greater than 4, showing that their key was not the same. Period 2, then, is eliminated.

_Period 3_: Here we are stopped short at the very first comparison. The numbers 24 and 77, found at the first interval 3, have a difference greater than 4 in their tens, and thus cannot have been enciphered with the same key. Period 3 is also eliminated.

_Period 4_: Starting again, and comparing numbers taken at interval 4, we are able to go all the way to the end of the cryptogram without finding any two numbers whose difference, either in tens or in units, is greater than 4. The numbers compared, however, included only those which would have been adjacent in their columns. To make sure that period 4 is possible, we must see numbers collectively in each of the four columns, and this is best done by recopying the cryptogram into its apparently possible period 4. Further examination, made individually on each column, still shows no two numbers in any one column whose difference, either in tens or in units, is greater than 4. It is possible, then, that each of the four columns was enciphered with a single key; and while this is not absolute proof that the period 4 is correct, those cases are extremely rare in which a period found in this manner is not the correct one. With period 4 accepted, and given as much material as we have, perhaps we can also discover just what key-number was added to primary numbers in order to produce each of the four alphabets. Considering alphabets one at a time, and examining separately the tens and the units:

_Alphabet 1_: The tens-half of the first column contains a digit 2; and since this can only have been produced by the addition of 1 and 1, the only possible key-digit here is 1. (We have already ascertained that all digits in this column could have had a same key.) The units-half has a range of 4-5-6-7-8, the maximum range possible. The smallest digit which can result in 8 is 3, and the largest which can result in 4 is 3; that is, the only digit which can result in all of the digits 4-5-6-7-8 is 3, so that the only possible key-digit here is 3. Conclusion: The key which produced the first cipher alphabet must have been 13, _since it cannot possibly be anything else_.

_Alphabet 2_: The tens-half of the second column ranges over the full five digits 4-5-6-7-8 (key 3), and the units-half ranges over the digits 5-6-7-8-9 (key 4). The key which produced the second cipher alphabet is 34.

_Alphabet 3_: The tens-half of the third column contains the “giveaway” digit 2, and the units-half contains this digit also. The key which produced the third cipher alphabet is 11.

_Alphabet 4_: The tens-half of the fourth column ranges only over the digits 5-6-7-8, with nothing to indicate whether the missing one is 4 or 9. Thus, the key to the tens might have been either 3 or 4, though it could not have been anything else. The units have the full range of digits, 5-6-7-8-9, key 4. In the fourth cipher alphabet, then, we cannot tell immediately whether the key is 34 or 44. Granting, however, that the arrangement of letters in Lindquist’s key-square was the same as that of Fig. 136, the substitution of letters for numbers may suggest which of the two numbers, 34 or 44, is the correct key. With one of these we obtain letters _C O A O_, and, with the other, _C O A T_, a word (The student might find it of interest to decipher this cryptogram and learn what the minister had to say).

* * *

Any sufficiently long cryptogram, then, will reveal both its period and its key, and this regardless of how the letters were arranged in the encipherer’s checkerboard. It may then be deciphered with its own key, and the case, at worst, becomes one of simple substitution. With shorter cryptograms, we often find, as here, that some one or more of the cipher alphabets could have had two or more possible keys. This happening, presuming that the alphabetical arrangement of the square is a known one, or one easily reconstructed, presents no real problem; a little experimentation on the cryptogram will show which keys bring out a message. When the alphabet of the square is an unknown mixed one, the problem may vary according to length, and the number of key-combinations which are found to be possible. If, for instance, the case resembles that of our preceding cryptogram, where only one alphabet out of four was in doubt, then, remembering that the Nihilist cipher alphabets are of a kind whose frequency counts can be “lined up,” we might take frequency counts on the several alphabets, and supply the missing numbers of the doubtful one by making its pattern match that of the rest. With several alphabets in doubt, which could only happen when frequency counts are too scant to betray their graph, it might become necessary to decipher the periodic cryptogram with each possible combination of key-numbers, each time obtaining a new cryptogram, and accept, among these new cryptograms, the one whose general frequency count seems most likely to be that of a simple substitution. The correct cryptogram, in this case, should also contain some fresh repetitions; that is, repetitions which were not present in the periodic one. As to the three examples which follow, there should be little difficulty in deciding whether or not the Nihilist cipher is represented.

131. By B. NATURAL.

45 68 48 46 60 78 45 78 24 59 35 67 50 75 38 58 53 60 65 26 54 46 68 55 38 67 42 69 56 59 24 59 70 54 30 85 32 90 44 46 45 56 79 54 30 86 22 78 27 26 44 49 78 75 38 54 55 78 47 27 45 49 89 44 49 88 42 59 56 49 42 86 50 52 26 55 42 60 47 36 22 50 78 65 50 76 35 78 28 59 26 50 68 54 60 76 25 87 28 29 55 58 59 73 59 97 54 69 66 57 26 46 78 65 48 76 45 57 47 29 65 79 77 55 30 57 35 89 45 49 53 46 66 75 57 97 55 68 28 47 22 66 66.

132. By PICCOLA. (Keyword, CRYPT. Fifth alphabet contains Q. But: Can you rearrange the numbers on the strip before taking frequencies?

15 20 23 18 03 15 26 12 26 25 03 30 40 14 20 09 20 25 11 15 17 25 16 02 29 30 25 21 18 03 11 16 27 30 26 10 02 21 17 01 06 25 13 01 25 03 30 23 26 23 06 27 12 11 20 12 22 16 18 03 29 20 19 01 19 17 19 12 12 20 02 11 14 18 19 13 20 38 11 23 19 01 19 01 27 30 16 21 01 23 17 24 22 25 03 19 26 21 11 28 11 17 16 21 03 13 20 28 05 20 06 26 13 11 26 11 16 27 26 16 02 26 18 05 25 06 03 16 03 03 30 26 16 27 28 10 02 16 02 29 06 26 27 11 24 15 20 23 13 15 11 25 13 05 24 28 20 40 27 19 19 30 27 19 19 13 02 23 21 28 11 30 14 28 03 18 26.

133. By PICCOLA. (If you recognize this gem of literature, you are beyond the draft age. It got around the censor in 1918).

20 08 17 29 15 09 01 05 08 29 24 11 06 05 10 26 13 22 06 01 18 19 05 03 16 24 13 16 04 08 07 19 12 18 24 11 17 09 07 27 26 22 01 15 21 21 10 03 06 22 03 18 04 22 20 06 07 24 12 19 10 19 10 30 10 19 16 24 13 16 04 08 23 01 10 10 23 10 09 05 08 17 21 22 09 15 21 21 10 03 06 06 21 20 12 22 21 08 18 19 23 05 02 01 11 34 19 27 12 06 02 15 10 22 03 03 02 11 12 19 10 11 19 27 13 12 18 24 19 13 24 15 07 16 16 16 26 20 04 05 11 29 26 20 03 10 19 10 23 11 16 19 13 16 04 08 25 17 05 24 20 20 23 09 10 25 20 25 02 05 07 16 26 20 04 05 11.

134. By DAN SURR. (Should you be worried at finding this in Daughter's boudour?)

A B C D E F G E H D G E F J E K H D L J D G J M M J D G J M E E F J E O J E L F A C B D G. - P G M G.