CHAPTER XX
Vigenère with Key-Progression
Before leaving the study of multiple-alphabet ciphers, we will consider briefly the process which, in its simplest form, would be that shown in Fig. 154. The initial key, in each of three examples, is _A_, and a long key has been formed by causing the initial one to progress in the normal alphabet according to an agreed index. In the first example, the progression index is 1, in the second, it is 2, and in the third, it is 25 (or minus 1). The resulting long key will govern a period, which is 26 or 13, according to whether the progression index is odd or even. This encipherment, logically, would be applied with a cipher disk. The initial key, as _A_ of the examples, would indicate the starting position of the revolving disk, the first letter being enciphered with the disk in this initial position, after which the disk is made to revolve, so many angles at a time, without further reference to key-letters. For this kind of cryptogram, the solution is purely mechanical. A series of alphabets may be extended, with each cryptogram letter as a beginning, and the message can be found following a diagonal path in the resulting set-up.
Figure 154
Forms of Key-Progression
Keys: A B C D E F G... A C E G I K M... A Z Y X W V U... S E N D... S E N D... S E N D... S F P G... S G R J... S D L A...
This type of key-progression can be decrypted by "running down the alphabet," and watching the diagonals for plaintext.
A much commoner scheme, when using a cipher disk, is that of following a series of irregular shifts in accordance with a numerical key. If, for instance, the initial position has been established and the first letter enciphered in that position, and if the numerical key is 3-5-2-1-6, the disk will now be revolved 3 positions for encipherment of the second letter, 5 positions for encipherment of the third letter, 2 positions for encipherment of the fourth letter, and so on, so that the disk must move 17 positions during encipherment of five letters. This can produce a very long period indeed, especially when the collective shifts result in an odd number.
Figure 155
Progressing Key: C U L P E P E R D V M Q F Q F S E W N R ... Plaintext: T H E R E I S O T H E R C A U S E F O R ... Partial Encipherment: V B P G . . . . W C Q H . . . . . . . I ...
Substantially the same encipherment as the foregoing can be had with a slide and a key-word, as indicated in Fig. 155. The progression index, in this figure, is 1. The preliminary key-word, CULPEPER, enciphers the first eight letters, then moves forward in the alphabet and becomes _D V M Q F Q F S_ for the encipherment of the next eight, _E W N R G R G T_ for the encipherment of the third eight, and so on. In its practical application, one column could be taken at a time. Notice, however, in Fig. 156, that when a key-letter progresses in the alphabet, the possible substitutes for any one letter will also progress, and to exactly the same extent. If the encipherment is Vigenère or Beaufort proper, this progression is in the same alphabetical direction as that of key-letters, while the variant encipherment causes the substitutes to progress in the contrary direction. Probably, then, the most convenient method of application, and the one least likely to result in errors, would be that of Fig. 157. The cryptogram is first enciphered as an ordinary periodic, and the progression is added later, using group-by-group encipherment. Thus, as we receive the cryptogram, our repeated _ther_ has been enciphered once as _V B P G_, again as _W C Q H_, and possibly, later on, as _A G U L_, and the only period we shall be able to find, using the regular methods, will be 26 x 8, or, if the progression index is an even number, 13 x 8. But notice, in the same figure, comparisons (a) and (b).
Figure 156
Vigenère and Beaufort Progression Variant Progression
Progressing Key: A B C D E..... Progressing Key: A B C D E..... Plaintext letter: H H H H H..... Plaintext Letter: H H H H H.....
Cipher letter (V) H I J K L..... Cipher letter: H G F E D..... (B) T U V W X.....
Vigenère, it will be remembered, has been compared to the mathematical process of addition. If the key-digram _CU_ be added to the plaintext digram _TH_, their sum is the cipher-digram _VB_. The alphabetical distance from _C_ to _U_ is 18, the alphabetical distance _TH_ is 14, and the alphabetical distance _VB_ is 6 or could be 26 plus 6, 52 plus 6, and so on. It is a fact that when we “add” the two digrams _CU_ and _TH_, we actually do add their separating intervals, 18 and 14, since we obtain a sum 32 in that of the cipher digram _VB_. It is also an easily verified fact that the same reasoning applies to the subtractions of the two Beauforts. The student who cares to investigate may make use of the tableau shown as Fig. 158; to find quickly the distance from one letter to another, find the first of these at the left, the second at the top, and the alphabetical interval between the two is shown in the cell of intersection. If it is desired to know the reverse interval, find the first letter at the top and the second at the side. Now notice, carefully, that when any digram progresses in the alphabet, as _CU_ would become _DV_, _EW_, _FX_, and so on, in a series of periods, _it does not change its alphabetical interval_; in all of these digrams, the distance apart of the two component letters is still 18. Thus, while our period vanishes, the alphabetical intervals which represent it are still present in the cryptogram; we have only to find these intervals, subject them to a Kasiski examination, and convert the cryptogram to an ordinary Vigenère.
Figure 157
Initial Key-word: C U L P E P E R C U L P E P E R C U L P E P E R Plaintext: T H E R E I S O T H E R C A U S E F O R T H I N... PRIMARY Cryptogram: V B P G I X W F V B P G G P Y J G Z Z G X W M E...
Progression Key: A B C FINAL Cryptogram: V B P G I X W F W C Q H H Q Z K I B B I Z Y N G...
(a) C U (key) plus T H (plaintext) equals V B (cipher). (b) Interval 18 plus interval 14 equals (32 less 26) = 6
Figure 158
Tableau for Finding ALPHABETICAL INTERVALS
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 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 A B 25 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 B C 24 25 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 C D 23 24 25 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 D E 22 23 24 25 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 E F 21 22 23 24 25 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 F G 20 21 22 23 24 25 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 G H 19 20 21 22 23 24 25 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 H I 18 19 20 21 22 23 24 25 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 I J 17 18 19 20 21 22 23 24 25 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 J K 16 17 18 19 20 21 22 23 24 25 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 K L 15 16 17 18 19 20 21 22 23 24 25 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 L M 14 15 16 17 18 19 20 21 22 23 24 25 0 1 2 3 4 5 6 7 8 9 10 11 12 13 M N 13 14 15 16 17 18 19 20 21 22 23 24 25 0 1 2 3 4 5 6 7 8 9 10 11 12 N O 12 13 14 15 16 17 18 19 20 21 22 23 24 25 0 1 2 3 4 5 6 7 8 9 10 11 O P 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 0 1 2 3 4 5 6 7 8 9 10 P Q 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 0 1 2 3 4 5 6 7 8 9 Q R 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 0 1 2 3 4 5 6 7 8 R S 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 0 1 2 3 4 5 6 7 S T 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 0 1 2 3 4 5 6 T U 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 0 1 2 3 4 5 U V 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 0 1 2 3 4 V W 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 0 1 2 3 W X 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 0 1 2 X Y 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 0 1 Y Z 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 0 Z
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
Fig. 159 shows the preparation of the cryptogram: The alphabetical interval from _K_ to _O_ is 4, that from _O_ to _S_ is 4, that from _S_ to _X_ is 5, and so on. If these numbers are placed directly below the first letter, as shown, the computation of their _lineal_ intervals apart is less confusing than when they are placed between the two. As to repetitions, each repeated single number may represent a repeated digram, each repeated sequence of two numbers may represent a repeated trigram, and so on. Only the longer of these possibilities have been underscored.
Figure 159
(5) (10) (15) (20) K O S X M Y M M Q Y T K N G Z W L T Z L 4 4 5 15 12 14 0 4 8 21 17 3 19 19 23 15 8 6 12 22
(25) (30) (35) (40) H C G F A P J Y K W A T Z P Q X U J Z P 21 4 25 21 15 20 15 12 12 4 19 6 16 1 7 23 15 16 16 6
(45) (50) (55) (60) V C Z Q A R F P V Y U Y C R C X M X G I 7 23 17 10 17 14 10 6 3 22 4 4 15 11 21 15 11 9 2 21
(65) (70) (75) (80) D U X Q Y M T E V V S C X J L J D A E Y 17 3 19 8 14 7 11 17 0 23 10 21 12 2 24 20 23 4 20 4
(85) (90) (95) (100) C S F P J W F V J V Q V E G A N G K B B 16 13 10 20 13 9 16 14 12 21 5 9 2 20 13 19 4 17 0 17
(105) (110) (115) (120) S C P Z B H G I D Z J A N Z I Y E Z P T 10 13 10 2 6 25 2 21 22 10 17 13 12 9 16 6 21 16 4 -
Figure 160
Repeated Intervals Lineal Interval Possible Factors
4-4 KOS-UYC 51 - 1 = 50 2 5 10 15-12 XMY-JYK 27 - 4 = 23 21-17-3-19 YTKNG-IDUXQ 60 - 10 = 50 2 5 10 23-15 ZWL-XUJ 36 - 15 = 21 3 7 21-15 FAP-CXM 55 - 24 = 31 7-23 QXU-VCZ 41 - 35 = 6 2 3 6 16-6 ZPV-IYE 115 - 39 = 76 2 4 19 17-10 ZQA-BSC 100 - 43 = 57 3 19 10-17 QAR-ZJA 110 - 44 = 66 2 3 6 11 15-11 CRC-XMX 56 - 53 = 3 3 9-2 XGI-VEG 92 - 58 = 34 2 17 2-21 GID-GID 107 - 59 = 48 2 3 4 6 8 12 17-0 EVV-KBB 98 - 68 = 30 2 3 5 6 10 13-10 SFP-CPZ 102 - 82 = 20 2 4 5 10 20-13 PJW-GAN 94 - 84 = 10 2 5 10 9-16 WFV-ZIY 114 - 86 = 28 2 4 7
Fig. 160 shows the application of a modified Kasiski examination. Notice the prominence of small factors 2 and 3, caused, often, by repeated alphabetical intervals in the key itself. In the given case, the period 10 would probably be the choice, though period 5 is correct; in practice, we should probably consider possible digrams as well as longer sequences. Accepting period 10, we have still to learn the progression index, and for this we must consider letters, all of which are shown in the second column of the same figure. Taking the longest repetition, most likely to be reliable, the two first letters are _Y_ and _I_; their alphabetical distance apart is 10, and their lineal distance apart in the cryptogram is 50. If the accepted period, 10, is correct, it has taken five periods to produce the alphabetical shift of 10, therefore the shift per period (the progression index) is 10 divided by 5; or 2. This, of course, has taken for granted that the encipherment is either Vigenère or Beaufort. Considered as a possible variant Beaufort, where the progression is backward, the alphabetical interval from _Y_ to _I_ is 16, which is not divisible by 5, the number of periods. But this progression might have covered the entire alphabet and then included 16, or it might have covered the alphabet twice, and so on, before including 16. We must make it divisible by 5, adding 26, then another 26, and so on, until we obtain a total progression of 120. This, divided by 5, gives the progression index as “minus 24” — the same as a normal progression of 2. In Fig. 161, the cryptogram has been re-written into the accepted period 10, and the figures in parentheses at the right of each group will indicate the amount of alphabetical shift when the progression index is 2. A constant progression of 2 per group would correspond to the application of a Vigenère key _A C E G_. . . . . , so that the Saint-Cyr slide will serve for quickly converting the cryptogram to its periodic form, and this is shown in Fig. 162. The period, as mentioned, is actually 5, though this makes no difference in the final results.
Figure 161 Figure 162
K4 O4 S5 X15M12Y14M0 M4 Q8 Y21 (A) K O S X M Y M M Q Y
T17K3 N19G19Z23W15L8 T6 Z12L22 (2) (C) R I L E X U J R X J
H21C4 G25F21A15P20J15Y12K12W4 (4) (E) D Y C B W L F U G S
A19T6 Z16P1 Q7 X23U15J16Z16P6 (6) (G) U N T J K R O D T J
V7 C23Z17Q10A17R14F10P6 V3 Y22 (8) (I) N U R I S J X H N Q
U4 Y4 C15R11C21X15M11X9 G2 I21 (10) (K) K O S H S N C N W Y
D17U3 X19Q8 Y14M7 T11E17V0 V23 (12) (M) R I L E M A H S J J
S10C21X12J2 L24J20D23A4 E20Y4 (14) (O) E O J V X V P M Q K
C16S13F10P20J13W9 F16V14J12V21 (16) (Q) M C P Z T G P F T F
Q5 V9 E2 G20A13N19G4 K17B0 B17 (18) (S) Y D M O I V O S J J
S10C13P10Z2 B6 H25G2 I21D22Z10 (20) (U) Y I V F H N M O J F
J17A13N12Z9 I16Y6 E21Z16P4 T- (22) (W) N E R D M C I D T X
Figure 163
MATHEMATICAL FORMULA - C. H. PRICE
X = AD x P LD P = Period AD = Alphabetical Distance X = Progression Index LD = Lineal Distance
As Applied to the Supposed Repeated Trigram KOS-UYC, Positions 1, 51:
X = 10 P = P 50 5
BUT: P and X must be integers • • If P = 5, then X = 1 (and P must be a divisor of 50) • If P = 10, then X = 2 (Periods of 25, 50, are unlikely)
For those who like mathematics, Fig. 163 shows a method used by one of our collaborators for determining both the period and the progression index directly from the cryptogram. Price also preferred to find alphabetical intervals by writing the normal alphabet into a block, five letters to the line, with _Z_ standing alone on the last line; thus, except for watching _Z_ occasionally, the distance from one letter to another could be counted by fives. It is understood, of course, that we do not accept the evidence obtained from only one of the supposed repeated sequences; too many of these will be accidental, and many of those which are actually periodic have not represented repeated digrams, but merely repeated intervals. Naturally, too, the progression index need not be a small number; the disk encipherment, mentioned in the beginning, showed a progression of 17 for each period 5. This disk encipherment, incidentally, has been dealt with in a most interesting manner in Givierge’s _Cours de cryptographie_.
* * *
We have seen, then, all of the essentials of polyalphabetical encipherment. With the cipher alphabets known to the decryptor, practically all of the multiple-alphabet ciphers will be solved by suitable modifications of processes described for Vigenère. When alphabets are not known, his problem, always, is that of collecting as many as possible of the substitutes belonging to each alphabet, so that he may determine both the order of the letters and the relationship of alphabets to one another.
144. By THE SQUIRE.
S O V F O G S G U F V I J R I F M O U I C F T T I K Z Y Z Z Z U I F Q Q L O W U V A F J F I W W L N C R G J F E M V V N N C D H W T A J N W A R D B. yallyyayyayalyyaaallayaaaylalllaylyayyallyyayaylalllyyall.
145. By NEMO.
Y Y I Z C U O F Y V H Q Y H T B E B S X P T S Y C R M R X L X E A G U Y L P U Q B U U Q N Y U S O Q M O O S P U G I J I I F F F A L I R G G F G E H H N T E G Y Z S M C O F U D E M X O G I K K V B N K W K P Q X M G D L A I F N H M X T U M E Z X Y Z G N A P D W C D M N C T T H N J F D.
146. By PICCOLA. (We wouldn't throw monkey-wrenches for anything!)
X E I T B B B B V M X R S J P Y L K E N Y K S Z K F R W L G S A Y E A V I X I X D U V D U R J G E I B A N Z F H D C C Y C O Y R V A B K W B R H F K K F X S E J Y T F N L R N I V K V K Q H I Q H I J L P G O U J V F C F T S H L I D V D D M P.
147. By DAN SURR. (Might try this without bothering about its progression!)
E C G M H T Y T A J B T H N G A W K L I B E M N R H T D G N G P D A O Q A X R P Z P F H D D X I E A U B S Y C I X C W V R H P B O I X Y P Y D V W N R N X O O K K I H F O X D S V L V W W C L I H Z H V W R L H W M M I E E A H G Q Y R S R L K L W Z T J A Y W F N S S U C V Z L P X P S E E E Y R T H D H T Z N U P U R M G K Z N T Y E Q D E Z E N N H W M I N R L P S S W P Y M C R U B J Z Y C R N L M A S M E U C L R M D Y R N E S T O B V J E U D V L O T S Q B J H B N R L B V D X J P X N I G F I C Q J Y Q Z X Q G K B L F Q U B Q K N E L S S L Y G T L F L T D Z Z Y K E E R H K L W L I M R N J S O O J P Q C A U D M E I B B Q X A H C V A J C M G X B I C D K V C L G Q I B S C F V F W Q N A X I D R Z S X R B I W R C Q R.
148. By PICCOLA. (When is a tramp not a tramp?)
E R N I C D M R A S T A A P H T P I L T Q V A A S N E A E E R O O L R.