Elementary cryptanalysis

by Helen Fouché Gaines

Figure 54 The Columns of Figure 50, After Solution by Multiple Anagramming: 1 7 24 22 12 9 10 15 4 2 20 17 18 23 13 11 8 25 6 3 21 19 16 5 14 C A N N O T H A N D L E R A I L R O A D S T O C K D O Y O U W A N T A N Y M E X I C O L I M I T E D B U Y B L O C K A T O N E F I F T Y T O S I X T Y T R Y R A I L R O A D O N J O N E S O R S M I T H W H A T I S T E L E P H O N E S T O C K D O I N G 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 (Plaintext serial numbers, added at bottoms of columns) (Appearance of the plaintext serial numbers, if the above columns should be restored to their cryptogram order) 1 10 20 9 24 19 2 17 6 7 16 5 15 25 8 23 12 13 22 11 21 4 14 3 18

in the order 1-10-20-9-24, etc., which is a different order from that of the cryptogram numbers, and this new series is made up of plaintext serial numbers. This is the other key, having the same relationship to the first as that explained in connection with the short Nihilist key. Applied to the plaintext, it would have to be used in the writing-in manner; used on the cryptogram, it serves for taking-off. Thus we are able to recover from our reconstructed plaintext two long keys, either one of which will serve to decipher additional cryptograms, but only on condition that these new cryptograms contain exactly 25 letters.

If, then, we hope to decipher cryptograms of other lengths, which originally were enciphered with exactly the same key as our present five, it is still necessary that we take one or the other of these long Myszkowsky-type keys and reduce it to the short columnar form. The theory on which this is done should not be at all difficult to understand if it be kept in mind that both of our long keys are actually the serial numbers of letters, and that each individual serial number accompanied its letter throughout the encipherment process. This will explain any references which are made to FIRST and SECOND encipherment blocks, with their respective columns and rows. Whichever of the long keys we decide to reduce, our first objective, always, is that of determining the length of the shorter key; after that we restore its order.

The first process, summed up in Fig. 55, was originally published, so far as the writer knows, by M. E. Ohaver, and makes use of the cryptogram numbers which were the first series obtained. The discovery of the shorter key-length is made by searching the set-up for some numerical difference (between any two numbers whatever) which is repeated by corresponding pairs of numbers at some regular interval. For convenience in making the search, Ohaver suggests that the mixed cryptogram numbers be written, with uniform spacing, on two strips of paper, in one case repeated. One strip can then be moved along beside the other so as to place pairs of numbers in actual contact. It is immaterial what numerical difference is used; the

Figure 55 Finding the Original Short Key from the CRYPTOGRAM Serial Numbers - M.E.OHAVER Finding the key-length: 1 7 24 22 12 9 10 15 4 2 20 17 18 23 13 11 8 25 6 3 21 19 16 5 14. 1 7 24 22 12 9 10 15 4 2 20 17 18 23 13 11 8 25 6 3 21 19 16 5 14 1 7 x . . . . . x (Repeat series) A difference of 1 occurs again at the interval 6. The two series 9-17 and 10-18 are fragments of columns from the first encipherment block. The key-length necessary for placing either pair in a same column is 6. Replacing cryptogram numbers in first encipherment block: (As PLAINTEXT)... 1 7 24 22 12 9 10 15 4 2 20 17 18 23 13 11 8 25 6 3 21 19 16 5 14 COLUMNS of the FIRST encipherment block are converted to ROWS of the SECOND: (a) (b) (c) 1 10 18 6 14 1 10 18 6 14 22 1 - 3 - 5 - 2 - 4 - 6 22 2 11 19/ 7 15 23 3 2 11 19 7 15 23 12 20 8 16 3 12 20 8 16 24 1 10 6 24 4 13 21/ 9 17 25 5 4 13 21 9 17 25 2 . 7 5 3 8 4 9 5 At (a) the columns of the first block are (C O S M O S) arranged so as to make the cryptogram numbers At (c) the order is run consecutively in each of the new columns. shown in which the columns At (b) this block has been adjusted, so as of (b) would be taken off. to form six columns. This order is the KEY.

difference 1 pointed out in the figure seemed a little more visible than others. This difference 1 has been noted between the numbers 9 and 10, and the next difference 1 has been found six positions away between the numbers 17 and 18, but is not found again between the numbers 25 and 6, which stand at the next interval of six positions. This may be a clue, but it is not what we had hoped to find. The clue is strengthened, however, by the observation that a difference of 5 occurs just at the right of the original difference 1, and is also repeated at the lineal interval 6.

To find a good clear example, using the strips as they stand, let us go back toward the left, and look for a difference 2. We find it first between the numbers 24 and 22; exactly six positions away, we find it again between the numbers 4 and 2; another six positions, and we find it between the numbers 13 and 11; still another six positions, and we find it for the fourth time between the numbers 21 and 19. Thus we have two series of numbers, 24-4-13-21 and 22-2-11-19, which run parallel to each other with their numbers always separated by interval 6. Sequences of this kind came from the columns of a first encipherment block, and can all be placed back in these columns by re-writing the mixed cryptogram numbers in lines of six numbers each. Sometimes we find such columns broken to bits, as would be the case should we continue moving the strip until we have completely exhausted the possibilities for difference 1; and we never find them complete, since these columns of the first encipherment block were taken out in irregular order and written continuously upon the rows of a second encipherment block, and after that were sliced through in the taking out of columns from the second block. We found traces of them once before, where a difference of 8 was found throughout four consecutive pairs of numbers 12-20, 9-17, 10-18, 15-23, always at an interval of 6 positions.

The key-length, then, is 6, and the cryptogram numbers (in their plaintext order) if written into a block of that width, will reproduce the first encipherment block. From this, we wish to carry the numbers, column by column, into their second encipherment block, from which they may then be taken out, again by columns, in such a way as to bring them back to their cryptogram order 1-2-3. If this is to happen, the numbers must run consecutively in the new columns, and the number 1 must be on the top line. We select, then, from the restored plaintext block, the column which contains the number 1, then the column containing the number 2, and so on, writing these columns horizontally on the rows of the new block, in such an order as to make the numbers consecutive in every column. This may or may not require the adjustments indicated in the figure at (a) and (b). When block (b) is completely adjusted, the order in which it would be necessary to take its columns so as to produce the cryptogram numbers in their original 1-2-3 sequence, is the order of the original short key. Our key-word COSMOS, incidentally, could have been better chosen.

In Ohaver’s process, we have taken the cryptogram numbers and enciphered them. By the process of General Givierge, summed up in Fig. 56, we do the opposite: we take the plaintext numbers, in their cryptogram order, and decipher them, so as to bring them back to their correct plaintext order 1-2-3. For learning the key-length, General Givierge endeavors to find that number which, when added or subtracted throughout the series of numbers, will most often cause one of its segments to repeat another. The portions which repeat are the columns, or partial columns, not from a first encipherment block, but from a second, since the process here is to follow out a decipherment. In the figure, the left-hand block (not strictly necessary) represents the plaintext, written as a cryptogram, and the one on the right represents it in what is known to have been its first encipherment block. To develop the second block: either take columns from the right-hand block and lay them on the rows of the central one in such an order that its columns can be taken out to form the cryptogram; or, write the cryptogram arrangement into the columns of the central block in such an order that its rows will show the columns of the plaintext block on the right. The order in which columns must be taken from the right-hand block to form the central one (or that in which columns must be taken from the central block to reproduce the cryptogram arrangement) is the order of the original short key. The condensed presentation here is also drawn from the writings of M. E. Ohaver. General Givierge, who seems first to have published the method, was chiefly concerned with exposing the possibilities of analysis, as applied to numbers generally, and explains to us the reason of the increase 6 which betrays the key-length in the plaintext series of numbers. The width of the original block being 6, each number is larger by 6 than the one just above it, making every one of the columns an arithmetical progression in which the constant difference is 6. These columns, still retaining their regular increase of 6, are laid down on the rows of a second block, and, for at least a portion of their length, some two or more of them always continue parallel, with progressions of 6 running side by side. Thus the taking out of columns from the second block will, at times, select one each from two or

Figure 56 Finding the Original Short Key from the PLAINTEXT Serial tumbers - GIVIERGE To find the key-length: Try adding (or subtracting) possible key-lengths (4, 5, 6, 7, etc.) to the whole series until some one of these added numbers causes portions of the series to repeat. 1 10 20 9 24 19 2 17 6 7 16 5 15 25 8 23 12 13 22 11 21 4 14 3 18 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 16 26 15 30 25 8 23 12 13 22 11 21 31 14 29 18 19 28 17 27 10 20 9 24 The portions which repeat when the correct key-length, 6, is added, are columns, or part-columns, from the SECOND encipherment block. Plaintext Serial Numbers S E C O N D F I R S T In CRYPTOGRAM Order: Encipherment Block Enoipherment Block / 1 10 20 9 1 7 13 19 25 4 1 2 3 4 5 6 24/19 2 17 10 16 22 2 8 14 7 8 9 10 11 12 6/ 7 16 5 20 5 11 17 23 3 13 14 15 16 17 18 15/25 8 23 9 15 21 6 12 18 19 20 21 22 23 24 12/13 22 11 24 25 21/ 4 14 3 18 The ROWS of the CRYPTOGRAM BLOCK, (approximately of column-length), must be written back into the COLUMNS of the SECOND ENCIPHERMENT BLOCK in such an order that the ROWS of this SECOND encipherment block could have been taken off as a primary cryptogram from the COLUMNS of the FIRST ENCIPHERMENT BLOCK, extreme right, known to be the original order of the numbers. The original short KEY can then be found by observing (in the central block) the order in which columns have been taken from the right-hand block. That is, find the small numbers which were on the top row; these are standing in the order 1, 4, 2, 5, 3, 6 (a writing-in key), and the columns which are headed by these receive key-numbers in the order 1-3-5-2-4-6.