The make-up of this new cryptogram is not hard to understand if it is noticed that what we have done is to carry out simultaneously the seven A-decipherments of seven tableaux like that of Fig. 122. We saw there that the odd-numbered letters of a series react as Vigenère encipherment and the even-numbered letters as variant Beaufort. With seven A-decipherments made at once, the same will apply to odd-numbered and even-numbered groups. Thus, our new cryptogram has seven columns enciphered in Vigenère and another seven enciphered in variant Beaufort. The original seven-letter initial key-word will decipher both sets of columns; for the first seven, it must be applied in the Vigenère manner, and, for the other seven, in the variant Beaufort manner.

Figure 123
a a a a a a a L C N D M E K A A A L M C J B G X K B B X R M K M Z V M
L C N D M E K L C N O Y G T B G X V N D G S S H W A W J Q E V L H O W
L C N D M E K A A A L M C J B G X K B B X R M K M Z V M Z S L Z I T K
Z S L Z I T K Z Q Y X D L N B Z X H Z A O G E T X L U I Q C Y K M P X
Y I J W L E X A P V E C L B H D Q E K U W W G R H X J F B D Y P I P K
Z Q Y X D L N B Z X H Z A O G E T X L U I Q C Y K M P X L B A F W A N
L B A F W A N F C W V V G H R P P L S C J B W G A M Q G G I O T L L I
Q D W A R G U W R L G N I Q S L V L E S P H E U T X B O N D H V X D C
F C W V V G H R P P L S C J B W G A M Q G G I O T L L I H V T C M S U
H V T C M S U H Z K Q X B L C O K A Q P A Y M Y L W K U Z W G L S R E
O U D S J T F J N U Q N Q L A A I L M Z U X I E W O B Y I W E H P D Q
H Z K Q X B L C O K A Q P A Y M Y L W K U Z W G L S R E J A Y W X M M
New Cryptogram: L C N D M E K A A A L M C J - B G X K B B X...........(Etc.)

As to why this encipherment reduces to alternate Vigenère and variant Beaufort groups, this is best understood by resorting once more to the “mathematical” aspects of the Vigenère cipher. In a previous discussion, we have said that Vigenère encipherment consists in the “addition” of key to message, and that variant Beaufort encipherment (which, in Vigenère, would be decipherment), consists in the “subtraction” of key from message. In the beginning, our plaintext is a series of groups, as A, B, C, D, E, etc. and the first encipherment operation consists in the addition of a key, as X, but only to the first group, A. To encipher group B, we add A; to encipher group C, we add B, and so on, so that when the auto-encipherment is complete, we have a cryptogram in which the groups are made up as follows:

1st:2d:3d:4th:5th:
A plus XB plus AC plus BD plus CE plus D. . . . . . (etc.)
Figure 124
The New Cryptogram from Figure 123
L C N D M E K A A A L M C J B G X K B B X R M K M Z V M
Z S L Z I T K Z Q Y X D L N
B Z X H Z A O G E T X L U I
Q C Y K M P X L B A F W A N
F C W V V G H R P P L S C J B W G A M Q G G I O T L L I
H V T C M S U H Z K Q X B L
C O K A Q P A Y M Y L W K U
Z W G L S R E J A Y W X M M
1 2 3 4 5 6 7 1 2 3 4 5 6 7(Vigenère) (Variant Beaufort)

Now, remembering what the mathematical valves were for key-letters, the trial key, made up entirely of A’s, is made up entirely of zeros. When we subtract zero from the first group, we leave it unchanged, that is, the first cryptogram group is still A plus X (plaintext plus key, or Vigenère). When we subtract A plus X from the second group, this cancels the A of both, and leaves B minus X (plaintext minus key, or variant). When we subtract this from the third group, we cancel the two B’s, leaving C plus X, again Vigenère. When we subtract this from the fourth group, we cancel the two C’s, leaving D minus X, again variant Beaufort. And so to the end. Always we come out with the original plaintext group plus or minus X, the key. Those groups which are plus X are Vigenère, and those which are minus X are variant. And X, in all, is the same: the original preliminary key. A comparison

Figure 125
Tables of High-Frequency Co-Efficients PHILLIP D. HURST

VIGENÈRE(Cipher Letters)
ABCDEFGHIJKLMNOPQRSTUVWXYZ
Eaehinorst
)Thinorstae
KAaehinorst
eOnorstaehi
yNnorstaehi
sIstaehinor
)Sinorstaeh
Htaehinors
Rnorstaehi
641-4444423432321442264224
BEAUFORT(Cipher Letters)
True Beaufort:ABCDEFGHIJKLMNOPQRSTUVWXYZ
...VARIANT...AZYXWVUTSRQPONMLKJIHGFEDCB
Eeatsronih
)Ttsronihea
KAatsronihe
eOoniheatsr
yNniheatsro
sIiheatsron
)Ssroniheat
Hheatsroni
Rroniheats
94124333233334333323333214
PORTA(Cipher Letters)
ABCDEFGHIJKLMNOPQRSTUVWXYZ
Erstnoaehi
)Tnorstehia
KAnorstaehi
eOnorsthiae
yNtnorshiae
sIrstnoaehi
)Snorstehia
Hrstnoaehi
Rnorsthiae
333244323465342344-2333332

of the same kind applied to the two Beauforts (or a few trials made on actual groups, if the student is not mathematically disposed) will show whether or not the auto-enciphered Beauforts can also be reduced to periodic form, and, if so, what their period is likely to be. In the case of the true Beaufort, it may be necessary to straighten out a quirk as to the application of the trial key.

While the foregoing methods are intensely interesting as an example of what can be learned by analyzing the structure of a cipher, most members of the American Cryptogram Association, in practical work, prefer methods of their own which are quicker in giving results. These methods, for the most part, have subordinated other considerations to certain original observations concerning the use of the purely plaintext key. Where message and key, as in the case of the autokey and “running key” encipherment, are each made up of normal text, with both members including the normal 70% of high-frequency letters, it becomes inevitable that high-frequency letters in the key and high-frequency letters in the message will be paired again and again as the co-efficients of cryptogram-letters, so that cryptograms enciphered with this kind of key must contain a great many letters caused by this kind of co-incidence. For convenience in making use of this fact, each member has his own ideas. Phillip D. Hurst, for instance, prepared a set of tables of about the kind shown in Fig. 125, one table for each multiple-alphabet cipher with which he expected to