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
| Figure 137 Cryptogram by EDWIN LINDQUIST: Final Investigation of Supposed Period 424-66-35-77-37-77-55-59-55-45-55-88-28-66-46- 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) |
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.
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.