| Figure 166 The PLAYFAIR Cipher KEY Encipherment Some Equivalent Keysquares C U L P E (1) bl = IB L P E C U N O Q S T Y Z V W X R A B D F ez = FE B D F R A V W X Y Z P E C U L G H I K M (2) cl = UP I K M G H C U L P E D F R A B N O Q S T ce = UC Q S T N O R A B D F K M G H I V W X Y Z (3) th = OM X Y Z V W G H I K M S T N O Q ht = MO Example: SE ND DI AM ON DS TO AM ST ER DA MX MO ND AY WE EK WI TH OU TF AI LX. TP SR BK FH QO KY NQ FH TN CF FB IZ HT SR DW ZU PM XH OM WA ZM BH BL. |
granting that some are practical and not too unwieldy for use, are entirely beyond our present scope, and we will spend our few remaining pages on a cipher of a less cumbersome nature and presenting far more points of interest.
The Playfair cipher, which may be examined in Fig. 166, requires no apparatus other than pencil and paper. Its key is the usual 5 x 5 square, based on a key-word, and filled in by any agreed plan (preferably not by straight horizontals). For encipherment, the plaintext is marked off into pairs, and these pairs are enciphered according to three very simple rules:
1. If the two letters of the pair are found in the same column in the key-square, replace each letter with the one directly beneath it; and if one letter stands at the bottom of the column, use the one standing at the top of the same column. With the key of the figure, ha becomes OH; wa becomes UH.
2. If the two letters of the pair are found in the same row in the key-square, replace each letter with the one immediately to its right; and if one letter stands at the extreme right end of the row, use the one standing at the extreme left end of the same row (os becomes QT; st becomes TN).
3. If the two letters of the pair have a diagonal relationship in the key-square (and these are usually in the majority), consider them to be standing at the diagonally opposite corners of an imaginary small rectangle, and substitute for each letter that letter of the other diagonal which stands on the same row with itself (bu becomes AL, not LA). The decipherment rules, as usual, are the same rules in reverse.
Notice that this encipherment is cyclical. So long as the order 1-2-3-4-5 is maintained in both columns and rows, it makes no difference whatever how many columns are transferred from one side to the other, or how many rows are transferred from top to bottom. This may be investigated in the three equivalent squares of the figure. Notice, too, that our three rules do not make any provision for the case in which the two letters of a pair are the same. If, in marking off the plaintext into pairs, we encounter a pair which is a double, it becomes necessary to dispose of this, usually by inserting a null which will throw the second letter into the next pair. Occasionally we find a sequence such as LESS SEVEN, in which it is necessary to do this twice in succession: LE Sx Sx SE VE Nx. An unpaired final letter also requires a null (unless left unenciphered), and when five-letter groups are to be used, it often becomes necessary to complete a final group by adding nulls.
The foregoing description and rules are those of the original Playfair cipher. Many encipherers, however, will vary the rules, especially the one concerning doubles; perhaps one letter will be omitted or replaced with a null; sometimes one double is replaced with another; occasionally an encipherer will separate every doubled letter in the message whether or not this is necessary. We meet, too, with variant forms. A 24-letter alphabet will be used in a 4 x 6 rectangle, or a 27-letter alphabet (with character &) will be used in a 3 x 9 rectangle. One variation, attributed to W. W. Rouse-Ball, uses the standard key-square with the standard rule 3, but varies the two rules for lineal encipherment. Rule 1: If the two letters of the pair stand in a column, use the two letters immediately to their right. Rule 2: If they stand in a row, use the two letters immediately beneath them. In all of these cases, presuming the method to be known, the degree of difficulty would be the same as if the standard system had been used; otherwise, it is only necessary to keep in mind the fact that variations occasionally occur. We will give our attention, then, to the standard encipherment. But before entering into the subject of decryptment, let us look carefully at the system itself.
Primarily, we have a fixed substitution. No plaintext pair ever has more than one substitute pair; and no substitute pair ever changes its original. We might say that the Playfair is, in effect, a “simple substitution” based on an “alphabet” of 600 pairs; and, just as in simple substitution proper, the Playfair cryptograms will very often contain long repeated sequences which represent whole words. Again, the reversal of a plaintext pair means the reversal of its substitute pair, and vice versa, so that the discovery of any one equation (as th = OM) always means the discovery of another (as ht = MO); and if, in addition, the encipherment was a rectangular one (rule 3), we obtain also the two reciprocal equations (as om = TH, and mo = HT). The two lineal encipherments, however (rules 1 and 2), are not reciprocal. But notice particularly that, in spite of the polygram theory, each letter has its individual substitutes. No letter in the key-square may have more than five of these; the four which are standing on its own line, and the one which stands directly beneath it. It may be learned, too, by writing out the 24 (or 48) possible pairs for any one given letter, that the letter standing immediately to its right in the key-square is twice as likely as any one of the other four to act as its substitute; and, further than this, that any letter which is paired with it will be limited to eight possible substitutes, all of which must be found either in the column or on the row of the letter itself. To clarify this important point, let us assume that the letter in question is E, and that the key-square is that of Fig. 166. The letter E may have only the substitutes C, U, L, P, and F, with C twice as likely to be used as any one of the other four. Any letter which is paired with E must take one of the following substitutes: U L P E F M T Z.
Naturally, then, those letters which, in the key-square, are standing on the same row or in the same column with the normally frequent letters will have high frequencies in the cryptograms; in fact, the two or three which predominate in a given cryptogram will practically always be letters which, in the key-square, were standing in the same row or column with E or T (in English). Moreover, if any letter has been identified once as the substitute for E, there is a most excellent chance that it can be identified again as the substitute for E. Say, for instance, that CF has been identified, or assumed, as the substitute for er. This means that C is individually the substitute for e, and when another pair CT is found to be of some frequency, it can be tried as the substitute for en, es, et, and so on. Single-letter frequencies, then, will play an important part in the decryptment of the Playfair. But the process will rest fundamentally upon the frequencies of digrams, and will follow, in general, three steps repeated over and over in the same rotation: