Elementary cryptanalysis

The Digram-Solution Method. — This method, representing another of our many debts to M. E. Ohaver, may be used either in conjunction with the vowel method, or independently, as the fundamental method of attack. For a satisfactory demonstration, however, we need more material, and Fig. 67 shows our cryptogram again, together with a suspected reply. Thus we have a length of 235 letters, so that the preparation of contact-notations, which we found sufficient in the preceding case, becomes here an irksome task.

For these longer cryptograms, it is usually best to put all of our data into the form of a digram-count, as indicated in Fig. 68. This is most easily done as follows: Using a sheet of cross-section paper, mark off the limits of a 26 x 26 square; write the normal alphabet across the top, so that each of its letters will govern a column; and write it again along one side, so that each letter will govern a row. For added convenience, these two alphabets may be repeated, as they are shown in the figure. Now, remembering that each letter in the text is the first letter of a digram (except the two which are finals), our two texts, with their total of 235 letters, are to provide a count on 233 digrams. Taking letters one by one, just as they come in the cryptograms, find each letter in the upper alphabet; find, in the side alphabet, the letter which immediately follows it in the cryptogram, and count this digram by placing a tally-mark in the cell at which the column and row governed by these two letters are found to intersect. In the figure, the tally-marks have been replaced with numbers showing their totals. It will be noted that the process described is identically the method which would have been used by Meaker in preparing the digram chart; and, just as in the case of the digram chart, the counting of the digrams has automatically counted the single letters. To obtain their frequencies, we may total either the columns or the rows, taking the larger figure in those few discrepancies caused by initial or final letters. With the chart understood, the digram-method of solution can be shown in a nutshell.

An inspection of this chart enables us to find quickly that the leading digrams are those listed: RK, VT, KV, DY, etc. These, almost certainly, are the substitutes for digrams ranking high on the normal list, and many others, having a frequency of 3, are very likely indeed to be substitutes for digrams from that same high-frequency class. Our text, of course, is still short, even with 235 letters, and we do not invariably find, in texts of this length, that the ranking digram (in this case RK, frequency 10) is the substitute for th, though the chances are, at all times, that it is. And should it prove here that RK does not represent th, then we may be quite sure that th is represented in one of the digrams VT, KV, DY, having the next frequency, 6. With the single exception of RR, each digram of the nine which are listed below the chart can be checked against three other digrams: Its own reversal; the doubling of its first letter; and the doubling of its second letter. In addition, it may be checked through the individual frequencies of its two component letters. These points of comparison, made for each of the nine leading digrams, have been tabulated in Fig. 69, so that the discussion may be easily followed.

Examining RK, assumed to represent th: Its reversal, KR, has not appeared on the chart, which is satisfactory for a digram of no greater frequency than its supposed original, ht. The doubling of its first letter, RR, has appeared four times, which is satisfactory for tt, one of the leading doubles of our language. The doubling of its second letter, KK, has not appeared, which is eminently satisfactory for a digram as rare as hh. Its first letter, R, has a frequency of 28, the highest in the cryptogram, which is not at all unusual in the case of t; and its second letter, K, has a frequency of 16, a little high for h, but not unsatisfactory. Thus, we find nothing, so far, to contradict the supposition that the digram RK is the substitute for th. But if K represents h, it should be possible to find digrams beginning with K which will check equally well as the substitutes for ke and ka. We do, in fact, find KV and KS. But which is which? Examination of Fig. 69 shows that one of these, KS, has a reversal, SK, frequency 1; but this is not informative, since it would be equally expected of eh or ah. Further examination shows that V has been doubled, which is far more characteristic of e than of a. Also, the individual frequency of V, 24, is the second highest in the cryptogram, and more likely to be that of e than that of a. Thus we may assume that KV represents he and that KS represents ha. This automatically identifies the digram SR as at. As to VT, this, apparently, involves the only reversal of any prominence in the cryptogram. Its first letter has already been identified as e, and the outstanding reversal of the language is er-re. This is not so certain as in the preceding cases, but the frequency of T is satisfactory as that of r.

Thus we have identified the letters t, h, e, a, r, which is as far as the tabulation has been carried. Having the substitute for h, we may now bring in the vowel-solution method through examination of digrams KD, KJ, KT, KZ; or continue with the digram-solution method by looking over the field for some of the other h-digrams: sh, ch, wh, ph, gh, and so on. The first of these should be easily identified by the frequency of s, and, in addition to the regular three check-digrams, we might check this against a possible st, another of our leading English digrams. With the process explained, we need not go further; the substitution of letters t, h, e, a, r, s, will surely break any simple substitution cryptogram. Possibly, enough has not been said as to the use of the trigram list, the consideration of common

affixes, common short words, and so on; but these are all points which the student can best develop for himself.

Another point, however, must not be overlooked: the long repeated sequences HVXXU, ZDYFZJX, DRKVT, GSRRVT. Repeated sequences of these lengths will usually come from repeated whole words, making it possible, to some extent, to attack the cryptogram by word-division methods. It is, in fact, the repetition of sequences, these and many others, which, in the beginning, has led us to assume that both cryptograms are using the same key. As to the recovery of this key, we need not wait until solution is complete. Even in simple substitution, it is well, during the identification of substitutes, to have before us a sort of skeleton key, in which the plaintext alphabet has been written out in normal order, so that the substitutes, as fast as their identities are discovered, can be placed below their originals.

Thus, having identified as many as twelve letters in our present cryptogram, this skeleton key, or framework, might begin to assume the appearance which is indicated in the upper tabulation of Fig. 70. Here, we are able to note a reciprocal encipherment between A and S, F and N, R and T, and U and Y, suggesting that the whole encipherment may have been reciprocal; if so, we have the identities of four additional substitutes: O, I, H, E, representing d, j, k, v, respectively. If they are present in the cryptogram, these four substitutions may be tried; but with or without their presence in the cryptogram, they can be added to the skeleton key, as in the lower tabulation of the figure. Notice that when this has been done, the cipher alphabet is beginning to show alphabetical sequences (reversed). We find H I J K, and, just before this, D F, which is an alphabetical sequence if the letter E has been taken out for use in a key-word. Between DF and HIJK of the cipher alphabet, we need only G to fill out the sequence; therefore either l or m must belong to the key-word; comparing this with what is found at the other end of the sequence, we find that either L or M would be the substitute for g. Between NO, we find V,

evidently misplaced; and, following O and preceding S, we find two positions which may be occupied by two of the letters PQR, of which R has already been placed (under t). That is, where the encipherer has used a key-word-mixed alphabet without troubling to carry it through a transposition process of any kind, we are often able to build it up again, and make it help us in the solution. This is especially true if he has used reciprocal encipherment; with the substitutes which may actually be found in our foregoing cryptograms, a little rearrangement is all that is needed in order to discover exactly what the original key was. When the cipher alphabet has been carried through a transposition block, it is not so easy to recover during the actual process of solution; afterward, however, it is not usually difficult to treat it by one of the transposition processes, just as if it were a transposition cryptogram of 26 letters. In the examples which follow, the key-word-mixed alphabets were used as they stood, though we believe that none of the encipherment was reciprocal. In one case, however, the plaintext and cipher alphabets were both mixed, according to different key-words, so that the recovery of this key may prove troublesome.

Fig. 72 sums up the entire process. At (a) we have a list of contacts taken in the order of appearance of the letters, and at (b) a rearrangement of the cryptogram letters in the order of decreasing variety of contact. Immediately above each letter is its “variety count” and directly above this is its frequency figure. In this set-up, a certain number of letters taken at the extreme right may confidently be marked off as a group of consonants. As to just where the line of demarcation may be set, recent analysis has shown that it is safe to include 20% of the total variety-count. In this case, the sum of all variety-figures is 104, and 20% of this, roughly, is 21. If we begin with P at the extreme right and add numbers backward for a count of 21, we find that the line of demarcation falls between R and C. However, we have at this point four letters, M, R, C, S, whose variety-count is uniformly 5, and any two of which might have occupied the places of C and S. To accept a vowel at this stage would mar the effectiveness of our system, and either we must discard all four of these letters, or we must find a means of differentiation other than their variety of contact. At this point, letter-frequencies come into play.

Examining the set-up just as we have it prepared at (b), note that the two figures just above M are 3-5. This is a “step-up” of 2 points. Note that just above R we have the same two figures 3-5, another step-up. Above C, we find the two figures 6-5, this time a “step-down” of 1 point; and above S we again find 3-5, a step-up. According to years of observation, confirmed by investigation of special cases, a vowel nearly always shows a tendency to step up, while consonants are

prone to step down. Thus among our four doubtful letters, there are three, M, R, and S, of which one will probably be a vowel. But the remaining letter, C, has the step-down peculiar to consonants; and while a step-down of only 1 point would not be definitely informative when found at the left end of the set-up, it almost certainly indicates in the present position that C is a consonant. Thus, we are able to include seven letters, C T N E O D P, in our group of “sure-fire” consonants, but have to dispense with several points of our 21-count.

At (c) we have the beginning stage of the actual investigation, while (d) and (e) are amplifications of the first stage. In these the original consonants are used to determine other consonants, progressing from stage (c) to stage (e). First, the original group of consonants is set down on a sheet of paper, and the space below it is bisected by the consonant-line. Consulting (a), we then find the letters of this group one by one, and all contacting letters which precede any one of them we set down on the left side of the consonant-line, and all those which follow any one of them we set down on the right side, always once for each time that the contacting letter appears. Thus we have stage (c). While we do not often encounter doubled letters in this form of cryptogram, it may be well to say here that while a doubled letter would be counted among the frequencies of letter appearance, its contacts with itself would not be entered on the consonant-line. That is, a doubled L would add a frequency of 2 to the general count, but contacts L-L would be ignored.

At (d) we have the first step in amplification, for which we are indebted to Chester A. Griffin. If there is any letter in the cryptogram which does not appear at all in (c), such a letter is practically sure to be another consonant. In this case we find A and U, and in (d) these two letters have been added to the consonant group and their contacts placed on the consonant-line. From this point onward, the work becomes more tentative, and, as a detail of operation, Mr. Griffin suggests that further additions to the consonant-line be made in another color of lead; if it then becomes a matter of necessity to erase, only the new letters will be included in the erasing.

Further work includes the application of the “force method.” That is, we turn our attention to the cryptogram itself, marking with a small cross, or otherwise, all letters determined as consonants, and placing a dot, or other indication, over all letters determined as vowels. Some vowels become evident as early as stage (c), as here we find both X and Y freely contacting our preliminary group of consonants, and if confirmation is needed, a glance back at set-up (b) will show that both of these are step-up letters. They may be labeled vowels without hesitation.

As to consonants, there are two clear text letters, h and n, which, owing to their many contacts with other consonants, and particularly with the low-frequency consonants (as in the digrams CH, GH, PH, WH, NG, NC, NK, NQ, etc.), will often show up clearly on the consonant-line. Of these two, n will appear largely upon the left side of the line, and h, the more reliable of the two, upon the right side. Examining (d) we find W and H appearing exclusively on the right side of the line, and since, under the rules of the game, no letter may be its own substitute, we may assume here that W represents h.

Further concerning h: An examination of the cryptogram shows that W has occurred twice as the second letter of a word, and the second-position is particularly characteristic of h. Then, assuming that W actually does represent h, we have in the seventh word of the cryptogram an intimation that the letter Z is also a consonant (since formations like AHEAD, AHA, are very rare, and seldom, if ever, occur in long words). Thus, we have two new consonants, W and Z, to be added to the consonant-line, with their contacts below, extending operations to stage (e). If desired, the spotted consonants may now be crossed off on the line itself, or merely indicated as in (e). It seems evident from (e), confirmed by (b), that B is our third vowel, and the supposition can be strengthened by inspecting the third cryptogram word, which, at this time, will have appeared as in (f). It also appears from (e) and (b), confirmed by the aspect of the tenth cryptogram word, that R is a consonant. Similarly, M, which, on three appearances, has twice contacted a vowel, may be placed as a consonant. These two new consonants, R and M, are added to the group of known consonants, and all of their contacts are placed on the consonant-line. Our next victim is S, evidently a vowel in spite of (e) because of its position in the seventh cryptogram word (g), which, otherwise, must begin with five consonants in succession. Presuming that a fifth vowel is to be found, the same word suggests either K or H as the candidate. The choice falls on H, according to the fourth cryptogram word (h); and thus continuing the force method with one eye on the consonant-line and the other on the cryptogram, the v-c formation of the cryptogram is finally established as in Fig. 73. Actual identification and solution follow

the usual path of patterns, cross-comparison of words, and inspiration, where all systems are subordinate to the solver’s own perspicacity . . . or “cipher brains.” Chapter IX has given some methods for identifying letters from their characteristics, and also mentioned the preparation and use of pattern-word lists.

At this point, it might be well to mention the “vowel-line” method, whose appearance was antecedent to that of the consonant-line. This earlier method was conceived by Erik Boden, and in principle works in reverse to the consonant-line method. Its set-up is like that of the consonant-line except that vowels, instead of consonants, are placed at the top. The contacts made by the determined vowels are listed fore and aft as is done with the consonants in the consonant-line method. The vowel-line shows several letters by certain characteristics . . . a letter appearing exclusively on the left might represent h, and one appearing solely on the right can be taken for n. The liquids, l and r, straddle the line about equally. On the supposition that you have located three vowels, the list of contacts on the vowel-line will not include, or only rarely, any other vowels as yet unidentified. A good suggestion is to use the consonant-line as specified, and then follow up with the vowel-line, using the vowels you have definitely identified as such. The result will be thus: The letter appearing exclusively to the right of the consonant-line will appear solely on the left of the vowel-line, and vice versa. If such appearances are noted, then you have spotted h and n . . . . . identified as suggested in another part of this chapter.

The workability of the consonant-line system in unravelling the mysteries of the “Dizzy” crypt is best judged by making a series of preliminary sheets from clear text. In Fig. 74, for instance, we have the solution to one of the most skillfully manipulated cryptograms in the collection of 130. This was prepared by Rufus T. Strohm as No. 17 Aristocrat of the April, 1932, Cryptogram. The total variety-count is 104, 20% of which is about 21. The line of demarcation thus falls in the group H L T U, each with a variety-count of 5 and no step-down. H, with figures 5-5, could be grouped with the remaining letters, giving us H Y N W B D as consonants, with P and M to be added at step (d). We thus include Y among our sure-fire consonants, and, in fact, it often is a consonant, but this is a problem no solver

has yet been able to overcome. However, the letter Y can usually be spotted by position. Note that except for Y, every vowel here is a step-up letter.

The above two examples have represented the “tough” case. In Fig. 75, where the text is the solution to a crypt by J. Lloyd Hood published as No. 9 in the February, 1932, Cryptogram, we have the average comparatively simple case. The total variety-count is 118, making the isolation count about 23 and throwing the line of demarcation into the group D H Z, where D is the only step-down letter. Every letter in the isolated group is actually a consonant, and step (d) adds the letters T G H Z. On the consonant-line, A, O, and to a lesser extent U, stand out clearly as vowels. E might be mistaken for H until we apply the force method, while I shows a step-up of three points, in addition to whatever shows up on the cryptogram. So up and at ’em! Edgar Allan Poe spoke truly when he suggested that whatever the human mind can devise, the human mind can also untangle.

Note: For additional methods of analytical attack on this kind of cryptogram, the student is referred to the booklet “Cryptogram Solving,” by M. E. Ohaver. This can be purchased directly from the author (Columbus, Ohio) for twenty-five cents, or may be purchased from the Frank A. Munsey Company. The textbooks of the National Puzzlers League also include chapters devoted to the solving of cryptograms; further information concerning these may be had by writing to R. T. Strohm, 1328 E. Gibson Street, Scranton, Pa.

"IDFURSF   UJBDOC   UJY   NEGNXDNOWN   IDFU   FUN   CXJKGDOC
JOY   RGGXNKKDBN   AJO   IN   WJOORF   JGGXNWDJFN  FUN
JZZJHDMDFL   RZ   FUN   URONKF   JOY   CNONXRSK."  --RXDNOFJM
JYJCN.

50.  M, MK, KB, KOH, BAH, MA, PUKO.
51.  FUN, IDFU, IN, IDFURSF.
52.  B, MPJ, MPBM, PJ, RA-, -RAN.
53.  AL-, -SANL, ZNL-, SKD.
54.  UZ, FAUZ, FU, -FAH, -UFZA, FV, ZAJ.
55.  BP, BP-, -NBDP.
56.  IT, TO;  IEAR, IEAP, AUTNME.
57.  A, AV, -AVU;  BR, RBF, RFBQ.
58.  DH, CD, CYV, HSV; identify W through its frequency.
59.  Note Pattern group;   DRDSLW-DRDS.
60.  SPL; O; EFE, word 5, last word.
61.  Use of J, LJHJ, JKJC, -ZCE, ZA.
62.  WGF, QHWWQF, WTF.
63.  AXF, KXY, AKY;  FIFBZ, YB.
64.  -GHJ, KHL.