CHAPTER XVIII

Periodic Ciphers with Mixed Alphabets

Periodic cryptograms in which the cipher alphabets are mixed are nearly always produced by means of slides. Before discussing these ciphers, it may be well to clarify a few terms which otherwise could leave room for uncertainty. We have, for instance, two, and sometimes three, key-words. There is a primary one (sometimes two) used in the preparation of the slide, and a secondary one, often called the “specific” key, which is used, as in Vigenère, for the encipherment of cryptograms. Since we shall have practically no occasion to mention the primary key-word (or words), any references which are made here to a key-word, unless clearly seen to refer to the preparation of a mixed alphabet, can be understood as meaning the secondary one, that is, the specific key which selects the cipher alphabets. Perhaps it is also advisable to call attention once more to the existence of a primary cipher alphabet (the basic one which is written twice in succession on the slide) and of the 26 secondary cipher alphabets which can be derived from it by placing it in its 26 possible positions. These are usually referred to simply as “the alphabets,” while the basic one is more commonly called “the sliding alphabet.” All, of course, are the same alphabet except for their points of beginning.

To see clearly what is meant by an “equivalent slide,” the student may make an experiment: First, form a temporary slide, using any two 26-letter alphabets, and use the slide to encipher a short message. Now form another temporary slide on which the two alphabets of the preceding slide (both treated by exactly the same plan) have been rearranged so that their letters are taken at every interval 3 (or at every interval 5, or 7, or 9 — any interval whatever that is not divisible by 2 or 13), and with care taken always to maintain this constant interval even when the 26th letter is reached and the 1st reappears. Then, using this new slide in the same way as before, encipher the same message with the same key, and compare this new cryptogram with the first. Finally, an _alphabetical_ interval (or distance) between two letters will mean their distance apart in the normal alphabet, while a _lineal_ interval (or distance) will mean their distance apart in any alphabet whatever. That is, the _alphabetical_ distance from _A_ to _B_ is invariably 1 (position), while their _lineal_ distance apart on a slide, or in the rows or columns of a tableau, could be anything from 1 to 25. Where these intervals must be mentioned often, the distance from _A_ to _B_ will be referred to more briefly as “the distance _AB_.”

* * *

Now let us consider the four slides of Fig. 138, which are being designated (arbitrarily) as belonging to _Types I_, _II_, _III_, and _IV_, in what would seem to be the order of their potential resistance to decryptment. Their actual resistance, however, might depend largely upon the manner of their use, and we are assuming throughout the chapter that the encipherment process is identically that described for the Saint-Cyr cipher: The upper alphabet, in all cases, is to be the plaintext one; the index-letter is always the initial one of this plaintext alphabet; and, for the encipherment of cryptograms, the letters of the chosen key-word are to be found in the lower alphabet and brought one by one to stand below the index-letter in order to set up their cipher alphabets. Also, for our immediate purposes, we are neglecting certain precautions the advisability of which will be seen later: First, the mixed alphabets have all been left undisturbed with their primary key-words (CULPEPER, DAMASCUS) and their alphabetical sequences in plain view; in practice, such alphabets ought to be carried through a transposition block, or otherwise made to appear incoherent. Second, the index-letter should never be _A_ (or any other frequent letter) unless the details of encipherment are varied. (We might, for instance, consider that the index-letter is in the sliding alphabet and that keys are in the upper.)

Figure 138

SLIDE - TYPE I.

Plaintext: C U L P E R Z Y X W V T S Q O N M K J I H G F D B A CIPHER: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z A B C.....

══════════════════════════════════════════════════ a b c d e f g h i j k l m n o p .......

Key A: Z Y A X E W V U T S R C Q P O D ....... Key B: A Z B Y F X W V U T S D R Q P E ....... Key C: B A C Z G Y X W V U T E S R Q F ....... ══════════════════════════════════════════════════

SLIDE - TYPE II.

Plaintext: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z CIPHER: C U L P E R Z Y X W V T S Q O N M K J I H G F D B A C U L.....

══════════════════════════════════════════════════ a b c d e f g h i j k l m n o p .......

Key A: A C U L P E R Z Y X W V T S Q O ....... Key B: B A C U L P E R Z Y X W V T S Q ....... Key C: C U L P E R Z Y X W V T S Q O N ....... ══════════════════════════════════════════════════

SLIDE - TYPE III.

Plaintext: C U L P E R Z Y X W V T S Q O N M K J I H G F D B A CIPHER: C U L P E R Z Y X W V T S Q O N M K J I H G F D B A C U L.....

══════════════════════════════════════════════════ a b c d e f g h i j k l m n o p .......

Key A: B D A F P G H I J K M U N O Q L ....... Key B: D F B G L H I J K M N C O Q S U ....... Key C: A B C D E F G H I J K L M N O P ....... ══════════════════════════════════════════════════

SLIDE - TYPE IV.

Plaintext: D A M S C U B E F G H I J K L N O P Q R T V W X Y Z CIPHER: C U L P E R Z Y X W V T S Q O N M K J I H G F D B A C U L.....

══════════════════════════════════════════════════ a b c d e f g h i j k l m n o p .......

Key A: C R P A Z Y X W V T S Q U O N M ....... Key B: A E L B R Z Y X W V T S C Q O N ....... Key C: U Z E C Y X W V T S Q O L N M K ....... ══════════════════════════════════════════════════

In the _Type I_ slide, the cipher alphabet is in normal order, and “slides against” a mixed plaintext alphabet. In _Type II_, we find a mixed cipher alphabet “sliding against” the normal one; in _Type III_, we find this mixed cipher alphabet “sliding against” itself; and in _Type IV_, we find it “sliding against” another, and different, mixed alphabet. Every slide, used in any manner, has an equivalent _tableau_ and while tableaux are seldom used, it is very important that we carry in mind a clear picture of their appearance; otherwise we shall find it difficult to understand how slides can be restored with only partial information. The imaginary tableau which is to serve this purpose, using any one of the four slides in the manner specified, is formed as follows: The plaintext alphabet, _with letters in exactly the order of the slide_, appears at the top. The 26 cipher alphabets, standing below and parallel to the plaintext one, are all seen in exactly the order of the slide, _and are shifted, one letter at a time, exactly as the normal alphabet is shifted in the Vigenère tableau_. Thus, exactly as in Vigenère, _the columns of this imaginary tableau are duplicates of the rows_. Keys, if considered, would repeat the first column of such a tableau. This tableau, as mentioned, is imaginary. Should the encipherer or the decipherer actually desire to make use of a tableau in preference to the slide, he would probably prefer one in which both his plaintext alphabet and his key alphabet are running in normal order, so that letters are easier to find. To form this tableau, he would begin by laying out, in normal order, his plaintext alphabet and his key-alphabet, and then lay out his 26 cipher alphabets in the manner explained in connection with the Beaufort alphabets. Each of the four slides of the figure is accompanied by a partial tableau of this kind, and it will be noticed there that we have only one case in which the (secondary) cipher alphabets bear any resemblance to the primary one. This tableau, too, should be well understood, since the cipher alphabets recovered from cryptograms will be like those of the figure.

Figure 139

5 10 15 20 25 30 Y V N G K Y E G D P Z E A Y K H S M D Q K K W S J I Q V I O P E I T E A v c I c c c v

35 40 45 50 55 60 K C F K Q J P M L B J X G K C Z D B G N G Q B D Q M E O N K I T c v I T H T H E P E c E c E

65 70 75 80 85 90 X T Y A D D D G J R X R X F W G D A Y T Q S G G C G P B Y O H H I E H A I P H E E v

95 100 105 110 115 120 C L W K C B I C F E Z D G J W K U F K C B U I Z Y B K E K C T H A T H I I c T H A c E A c T H

125 130 135 140 145 150 G K T A O Q C B Y Q U U F Z G G Z Y F N F M J V Z B L Q J U E c H v E c c E E v A

155 160 165 170 175 180 V M M J T A E F V S M E N K Q J E I Z Y A L Q Y R X R X F R v W E v E T c E E W

185 190 195 200 205 210 O U F V S V V V V P K T B K C G O M I K B Q V Z N B I N A O c v c v I E T H E v A c c E A H v

215 220 225 230 235 240 C E V V J F V U Z S B K M K C G P M D T K K Y A D D D Y Z C E c v E A c v T H E v I c H H E H

245 250 255 260 265 270 B T K V S G Q W I T Z D A K P G W B I O N D G R C H P B H U A M v E c T H A T E E v H I H E

275 280 285 290 295 300 G K T Q H G U V Z N Y X M L H F S M D Q K K W Z Q U D A M T E c E c c E v v c I c E c H A

305 310 315 320 325 330 Z D B J O P E U L R Y U G K U Z E U S J Z D B O D R E S I O T H E v E c I T T E T H E E v

335 340 345 R L A B L J R S Z Q Y Q V F L A L E c c c

* * *

Of our four slides, only the _Type I_ is radically different from the rest. Since its basic cipher alphabet is not a mixed one, it makes little difference what has been done to its plaintext alphabet. Notice, in the partial tableau which accompanies it, that the difference between one cipher alphabet and another is purely a matter of alphabetical shift (or of “size,” if we wish to replace all of these letters with numbers). Properly speaking, this cipher belongs to the case of the preceding chapter; it is presented here largely as a warning of what could happen through misuse of the _Type I_ slide. In the remaining three cases, the sliding alphabet is a mixed one; a series of frequency counts taken from cryptograms cannot be “lined up” unless letters can be placed in the right order before these frequency counts are taken. The “right” order may be the original one of the cipher alphabet, or an equivalent order in which the original letters are taken at a constant interval. In these cases, as with any other periodic cipher, the period is found in the usual way. Individual frequency counts are then taken on the several cipher alphabets, and these are examined in the hope of finding a known alphabetical graph; that is, the graph of some mixed alphabet recovered from previous decryptments — (but notice also the _C_-alphabet under the _Type III_ slide!). It can also be ascertained whether or not the frequency counts have followed one common graph, whether any two or more have followed one graph, and so on. But when it is found that the frequency counts are those of unknown mixed alphabets, then each alphabet is to be treated by simple substitution methods. Here, the principles will still be those of Chapter IX, and we will examine, as briefly as possible, the mechanical phases of their application.

Our cryptogram, shown in Fig. 139, is already written into its correct period, 5, with a few substitutions already made, and a few letters noted as vowels or consonants (_v_-_c_). With the period determined as 5, and alphabets found to be in an unknown mixed order, our next step is the preparation of a contact sheet (contact chart, contact count) for each one of the five alphabets, the usual form being that shown in Fig. 140. The necessary number of sheets is prepared in advance by writing the normal alphabet through the center, and each is numbered to show what alphabet it represents. It may also carry the numbers of the two contacting alphabets (those in parentheses in the figure). Then, if the cryptogram is properly grouped, so that all first letters of groups belong to alphabet 1, all second letters to alphabet 2, and so on, the putting down of contact letters is very rapid.

Illustrating with alphabet 2: Start with its first letter, _V_; find _V_ in the prepared alphabet numbered 2; place on its left side the _Y_ of alphabet 1; place on its right side the _N_ of alphabet 3. Pass on to the next letter, _E_: contacts are _Y_-_G_. Pass on to the third letter, another _E_: contacts are _Z_-_A_. And so on to the end of alphabet 2. Each contact chart, of course, will serve also as a frequency count and as a graph. The five graphs should now be compared with one another in the hope that some two or more may represent the same alphabet. Such a key-word as DENSE, for instance, makes use twice of the _E_-alphabet, thus doubling the amount of material in one of the alphabets. In our present case, it is found that the five alphabets are all different. Now, just as in simple substitution, we wish to determine, for each of the five alphabets, what letters are apparently representing vowels, and what letters are more likely to be consonants. For this purpose, some of our “pointers” are still available, and are just as valid as in