Concerning the “chart of probable position,” most solvers prefer simply to keep this in mind, while others will actually set it down and make it the basis of their solution. With 176 letters of text, the average frequency of letters is about 7 (176 divided by 25). Any letter whose frequency is above that average is very likely to have been standing on the same row or in the same column of the key-square as E or T, and the two or three which lead the list are practically sure to have been substitutes for one of these two letters.
With cryptograms of the present length, or even with those of 400 to 600 letters, it is very uncertain as to whether or not the leading pair will represent th, or the leading reversal er-re. Here, in fact, we have no reversal of a definitely frequent character, and our one prominent pair, HR, might just as well represent st, at, it, on, re, se, or any other normally frequent digram capable of being used at the beginning of a sentence. Presuming, however, that it might represent th, we know that this digram is followed almost altogether by vowels, and is followed with remarkable frequency by e and a; we know also that letters have individual substitutes. Thus, we might begin solution by listing (or noting) those pairs which, in the cryptogram, have followed the supposed th: KY, DU, KY, CE, NU, assuming that their first letters, K, D, C, N, have probably represented vowels, and that, of these, D, C, and N, which rank high in the list of single-letter frequencies, are very likely to have represented e. We may attempt to identify these five pairs by working down the list of normal digrams, taking only those of v-c formation. If, in addition, it is assumed that the key-square has been filled by straight horizontals, certain assumptions can be made through possible alphabetical sequence; for instance, the U of DU and NU may have stood on the same line with R S T (U). There is a further field for suggestions to be found in patterns, such as TI BI, in which the two I’s could represent the same letter. And where the square is filled by straight horizontals, it is often possible to identify such a sequence as HZ HB as a “split double,” since the null used in these cases is often X, and Z may well represent X by alphabetical position, It is even possible to guess here a doubled L, since H and L are not far apart in the alphabet. (It may be, of course, that the two H’s represent two different letters.) The foregoing, then, has indicated the general path. If the student desires to follow out a detailed demonstration made on a cryptogram of only moderate length, a most excellent exposition can be found in the appendix to the Macbeth translation of Langie’s “Cryptography” (Dutton). It was written by Lt. Commander W. W. Smith of the U. S. Navy, and generally speaking, attacks the identifications of pairs as follows:
Having placed frequency figures beside their digrams, find those points at which two pairs of high frequency are consecutive (not necessarily a repeated sequence), and attempt to identify these tetragrams as frequent tetragrams of the language: ther, ered, ened, tion, atio, ment, beca, and so on. We have one here, provided a frequency of 3 can be considered important: HR KY. Since this happens also to be repeated, it probably represents a word, as that, this, they.
Another good demonstration, provided the student has access to it in his public library, is found in Colonel Parker Hitt’s “Manual for the Solution of Military Ciphers.” This manual is an elementary work intended for the preliminary instruction of soldiers, and the attack is made on the assumption of a key-square filled by straight horizontals. With a square of the kind we are using, most of the vowels and high-frequency letters will be standing on the upper two rows, and letters on the first two or three rows will have a much higher frequency than those of the last two or three. In fact, it can often be detected that the letters V W X Y Z were standing on the bottom row as an intact alphabetical sequence, for the simple reason that they have no frequency in the cryptogram.
Colonel Hitt’s demonstration begins with the usual pair-count, made on a chart. He selects from this chart the (approximately) ten letters having the widest variety of contact, including, if necessary, the vowel or so which would have to be present in a key-word; and these letters are assumed to have stood on the upper rows of the key-square. The remaining (approximately) fifteen letters are then set up in their alphabetical sequence and are assumed to have stood on the lower rows in about that order. They are not, of course, known to be correctly placed; the set-up merely gives a concrete idea as to where letters ought to have stood. Then, following the military case of abundant material, it is assumed that the leading pair will represent th (sure to be followed often by e), or, if th is not the leader, then he (sure to be preceded often by t). With a few obvious identifications made in the usual way, letters begin to arrange themselves on the upper rows, and a gradual adjustment takes place which corrects the few wrong assumptions of the lower rows, so that the key-square is restored far in advance of solution. When a short key-word has been used, it is not impossible, by following Colonel Hitt’s suggestions, to pick out all of the key-letters, guess the word, and decipher with the key-square. Other demonstrations, based, respectively, on French and Italian language characteristics, can be found in General Givierge’s Cours de cryptographie and in General Sacco’s Manuale di crittografia. (In the French work, the cipher is referred to simply as “orthogonal and diagonal substitution.”)
It will be seen from the foregoing that the initial difficulty lies in the correct identification of the first few pairs, and this, in a short cryptogram, is no small difficulty. By whatever means it is found possible to make these first tentative
| Figure 168 POSSIBLE RELATIONSHIPS (a) A 3-letter Equation (b) A 4-letter Equation Vertical Horizontal Vertical Horizontal Rectangular I T K I . K H T H R . I K . S Y . . R S Y . S Y (c) Impossible Equations: (d) Possible Equations: co nd em na ti on -c on de mn at io n- EO ND EC TC TI AD EO ND EC TC TI AD CT |
identifications, the operation which is to admit or disprove their correctness is step No. 2, in which we set them up as equations and then attempt to replace them into their connected relationships in the key-square. If this cannot be done, they cannot be correct; and, on the other hand, it would be an extremely rare case indeed in which we could combine as many as five or six such equations into one framework and then find them incorrectly matched. To understand “equations,” suppose we look at Fig. 168.
Assuming that the beginning pairs of our cryptogram, HR KY, represent the word this, we have two equations, HR = th, and KY = is. The first of these has only three different letters, since H is common to both members, while the second has four different letters. With the first case (a), one of the lineal encipherments must have been used, and the common letter, H, must have stood between the other two, with its plaintext partner coming first and its cryptogram partner coming last. We do not know whether these three letters stood in a column or in a row, but we do know that they were consecutive. This relationship may be expressed simply as T H R, even though, in the actual square, the letters may have been partly at the end of the row (or column) and partly at the beginning: H R * * T or R * * T H. Encipherment, remember, is cyclical, and we may come out with any one of numerous “equivalent squares.” With the second equation (b), the positions of letters are not so definite. In either of the lineal encipherments, IK must be in direct sequence and SY must be in direct sequence; either sequence may have come first, and we do not know the exact location of the fifth letter. Concerning their possible rectangular encipherment, all we know is that there must have been a parallel relationship; their distance apart, laterally or vertically, might have been anything permitted by the key-square. As to the rest of the figure, suppose that we have reason to suspect the presence in this cryptogram of the word “condemnation.” The equations of (c) are totally impossible, since, in Playfair, no letter may be its own substitute. Those of (d) are not only possible, but probable, since we find many letters from the word itself.
| Figure 169 Equations of (d),(Fig.168) Possible Combinations 1 2 3 4 5 6 7 8 O D O N O N D E D E C N O N D O D C D E N O N D E C E C D E C C 9 10 11 12 M T M . T O N D E C . M T . N C . . . . N C . N T M C 13 14 15 16 A O N D E C O N D E C T A T I . . . . I A A T I M T M I |