| Figure 12 Nihilist Plan (a) Transposal of Columns (b) Transposal of Rows S C O T I A S-5 E U J W T O 5 2 4 6 3 1 C-2 R A F O R E O-4 A N E B C O S E U H T L (Let us h) T-6 X L X X S E R A F O R E I-3 A Y U T O M A Y U T O M A-1 S E U H T L (Let us h) A N E B C O E U J W T O (c) Cryptogram: E U J W T O R A F O R E A N E X L X X S E B C O X L X X S E A Y U T O M S E U H T L. |
unit at one continuous writing. The decipherer, too, having restored his cryptogram unit to its block and written his two series of numbers, may read, or copy, continuously. The decipherer, in fact, uses the exact method which would produce a Nihilist cryptogram if a key were used in the “taking out” manner. What we have described is the encipherment of a single major unit; and all cryptograms must contain an exact number of these major units.
The second operation, that of taking off the cryptogram, is not always done by straight horizontals as we have shown this under (c) of Fig. 12. This, of course, is the expected way; but the Nihilist square is quite frequently taken off by some other one of the forty-odd routes possible to rectangular transpositions. The decipherer, knowing this route, merely writes his units back into their blocks; but the decryptor is often faced with a preliminary problem of discovering how they were taken off. Sometimes he must also discover how many units a cryptogram contains.
To understand how such problems are solved, it is necessary to pause and consider the make-up of ordinary written plaintext. English vowel-percentage, as mentioned, is about 40%, and practically never varies out of its limits 35%-45%. Each 40 vowels are fairly evenly distributed throughout their 100 letters. Take any English text whatever, not composed of initials or otherwise distorted, and, beginning where you please, mark it off into ten-letter segments and count the vowels in each of the segments. You will find that the majority of these have exactly the normal number of vowels, which is 4. Others will have 3 or 5, which, though outside of the limits 35%-45%, are the closest variations possible. It will be a rare segment indeed which contains fewer than 3 vowels or a greater number than 5.
But suppose, having marked off such a text into ten-letter units, or segments, we take each of these segments individually and mix up the order of its letters, though still allowing it to stand where it is. And suppose, having done this, we erase the original division-marks and, beginning at some point in the midst of a former segment, we again mark off a series of ten-letter units, and count the vowels of these new segments. This time, we are just as likely as not to find seven or eight vowels in one segment and none at all in the next, depending on just what we did to the old units, and still we have not actually mixed the units; we simply have our division marks in the wrong places. Imagine, then, how the vowel distribution can vary when a transposition is one so planned as to break up units and scramble their letters.
This fact of uniformity in vowel distribution is of enormous assistance in dealing with the simpler transpositions. For instance, it may be that what we want to know is the length of the units, and that what we have is a cryptogram of 144 letters, which could be a single square, or a series of 36-letter squares, or even a series of
| Figure 13 5 2 4 6 3 1 5 2 4 6 3 1 5 2 4 6 3 1 5 5 5 2 2 . A . . . E 2 R A F O R E 4 4 4 6 6 6 3 3 3 . Y . . . M 1 . E . . . L 1 S E U H T L 1 S E U H T L |
16-letter or 9-letter squares. We may start at the beginning of this cryptogram and mark it off into equal segments of any length we like, afterward counting the vowels per segment. If every segment shows approximately a 40% vowel count, the chances are that we have a series of intact units, each one merely transposed within itself; but if one segment shows 50%, another 30%, another 28%, and so on, we may be quite sure that our division marks are in the wrong places.
Returning, now, to the Nihilist cipher, suppose we consider the make-up of its major unit, that is, of any one block. This major unit is a series of minor units, and each of these minor units, at the time of encipherment, was written by itself on its own line. In the beginning, it was a small fragment of plaintext, presumably conforming closely to a 40% vowel count. It is true that we placed it on the line in transposed order, but we did not remove any of its letters or add any new letters. Even in the transposal of the lines themselves, we merely removed a number of intact units from one place to another. There has never been a time, throughout the entire encipherment, when we took any letter out of its original minor unit and put it with some other unit. Thus, as we first see our completed Nihilist square, we still have, on each horizontal line, a small fragment of an English sentence in which all of the original vowels are still present. If such a block is now taken off by straight horizontals, it is no more than a series of intact units. To break up these units, we must at least take it out by verticals; and they will, of course, be much more thoroughly mixed when taken out by diagonals or spirals.
The decryptor, hoping for the best, writes his cryptogram into a square (or series of squares) by straight horizontals and counts the vowels per horizontal line. If his block is wide, he may estimate the actual number of vowels represented by 40%; if it is narrow, he may only roughly approximate the number; but in either case what he hopes to see is evenness of distribution. More than half of his units must be exactly normal, and any which are not exactly normal must show the smallest variation possible. If he finds that this is the case, he assumes that his block arrangement is the encipherer’s original square, with only the minor possibility that half of his lines may be written in the wrong direction. If his distribution is not uniform, he counts the vowels per column so as to find out what kind of distribution he would get from a vertical arrangement (ascending or descending). If this, too, fails to show him a uniform vowel distribution, he writes out a new block by the route of alternating verticals (or gets this count from his first block; this is possible, though a little confusing). Afterward, he may go on to the diagonals and spirals until finally he reaches the arrangement in which more than half of his horizontal lines show a 40% vowel count, and the rest a minimum variation.