Better Gutenberg
Free, beautifully formatted public-domain ebooks
‹Better GutenbergElementary CryptanalysisChapter XX — Vigenère With Key-Progression · 21 of 26Contents›

Elementary Cryptanalysis

by Helen Fouché Gaines

21 of 26 · Contents · Download EPUB

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,

            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.

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.

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

            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...

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.

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

            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.....(B) T U V W X..... Cipher letter: H G F E D.....

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).

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.

Fig. 159 shows the preparation of the cryptogram: The alphabetical interval

Figure 157
Initial Key-word:C U LPE P E R C U L PE P E RC U L P E P E R
Plaintext:T H ERE I S O T H E RC A U SE F O R T H I N...
PRIMARY Cryptogram:V B PGI X W F V B P GG P Y JG Z Z G X W M E...
Progression Key:ABCFINAL Cryptogram:VBPGIXWFWCQHHQZKIBBIZYNG...
(a)
C U (key)plusT H (plaintext)equalsV B (cipher).
(b)
Interval 18plusinterval 14equals(32 less 26) = 6
Figure 158
Tableaufor Finding ALPHABETICAL INTERVALS
ABCDEFG H IJ K L M N O P Q R S T UVWXYZ
A0123456 7 89 10 11 12 13 14 15 16 17 18 19 202122232425A
B25012345 6 78 9 10 11 12 13 14 15 16 17 18 192021222324B
C242501234 5 67 8 9 10 11 12 13 14 15 16 17 181920212223C
D2324250123 4 56 7 8 9 10 11 12 13 14 15 16 171819202122D
E22232425012 3 45 6 7 8 9 10 11 12 13 14 15 161718192021E
F212223242501 2 34 5 6 7 8 9 10 11 12 13 14 151617181920F
G2021222324250 1 23 4 5 6 7 8 9 10 11 12 13 141516171819G
H19202122232425 0 12 3 4 5 6 7 8 9 10 11 12 131415161718H
I18192021222324 25 01 2 3 4 5 6 7 8 9 10 11 121314151617I
J17181920212223 24 250 1 2 3 4 5 6 7 8 9 10 111213141516J
K16171819202122 23 2425 0 1 2 3 4 5 6 7 8 9 101112131415K
L15161718192021 22 2324 25 0 1 2 3 4 5 6 7 8 91011121314L
M14151617181920 21 2223 24 25 0 1 2 3 4 5 6 7 8910111213M
N13141516171819 20 2122 23 24 25 0 1 2 3 4 5 6 789101112N
O12131415161718 19 2021 22 23 24 25 0 1 2 3 4 5 67891011O
P11121314151617 18 1920 21 22 23 24 25 0 1 2 3 4 5678910P
Q10111213141516 17 1819 20 21 22 23 24 25 0 1 2 3 456789Q
R9101112131415 16 1718 19 20 21 22 23 24 25 0 1 2 345678R
S891011121314 15 1617 18 19 20 21 22 23 24 25 0 1 234567S
T78910111213 14 1516 17 18 19 20 21 22 23 24 25 0 123456T
U6789101112 13 1415 16 17 18 19 20 21 22 23 24 25 012345U
V567891011 12 1314 15 16 17 18 19 20 21 22 23 24 2501234V
W45678910 11 1213 14 15 16 17 18 19 20 21 22 23 24250123W
X3456789 10 1112 13 14 15 16 17 18 19 20 21 22 232425012X
Y2345678 9 1011 12 13 14 15 16 17 18 19 20 21 2223242501Y
Z1234567 8 910 11 12 13 14 15 16 17 18 19 20 21222324250Z
ABCDEFG H IJ K L M N O P Q R S T UVWXYZ

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.

Figure159
(5)(10)(15)(20)
KOSXM YMMQY TKNGZ WLTZL
4451512 1404821 173191923 15861222
(25)(30)(35)(40)
HCGFA PJYKW ATZPQ XUJZP
214252115 201512124 1961617 231516166
(45)(50)(55)(60)
VCZQA RFPVY UYCRC XMXGI
723171017 14106322 44151121 15119221
(65)(70)(75)(80)
DUXQY MTEVV SCXJL JDAEY
17319814 71117023 102112224 20234204
(85)(90)(95)(100)
CSFPJ WFVJV QVEGA NGKBB
1613102013 916141221 5922013 19417017
(105)(110)(115)(120)
SCPZB HGIDZ JANZI YEZPT
10131026 252212210 171312916 621164-
Figure160
Repeated IntervalsLineal IntervalPossible Factors
4-4 KOS-UYC51 - 1 = 5025 10
15-12 XMY-JYK27 - 4 = 23
21-17-3-19 YTKNG-IDUXQ60 - 10 = 5025 10
23-15 ZWL-XUJ36 - 15 = 2137
21-15 FAP-CXM55 - 24 = 31
7-23 QXU-VCZ41 - 35 = 6236
16-6 ZPV-IYE115 - 39 = 762419
17-10 ZQA-BSC100 - 43 = 57319
10-17 QAR-ZJA110 - 44 = 66236 11
15-11 CRC-XMX56 - 53 = 33
9-2 XGI-VEG92 - 58 = 34217
2-21 GID-GID107 - 59 = 482346 8 12
17-0 EVV-KBB98 - 68 = 30235 6 10
13-10 SFP-CPZ102 - 82 = 20245 10
20-13 PJW-GAN94 - 84 = 1025 10
9-16 WFV-ZIY114 - 86 = 28247

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

Figure 161Figure162
K4 O4 S5 X15M12Y14M0 M4 Q8 Y21(A)KOS XM YMMQY
T17K3 N19G19Z23W15L8 T6 Z12L22(2)(C)RIL EX UJRXJ
H21C4 G25F21A15P20J15Y12K12W4(4)(E)DYC BW LFUGS
A19T6 Z16P1 Q7 X23U15J16Z16P6(6)(G)UNT JK RODTJ
V7 C23Z17Q10A17R14F10P6 V3 Y22(8)(I)NUR IS JXHNQ
U4 Y4 C15R11C21X15M11X9 G2 I21(10)(K)KOS HS NCNWY
D17U3 X19Q8 Y14M7 T11E17V0 V23(12)(M)RIL EM AHSJJ
S10C21X12J2 L24J20D23A4 E20Y4(14)(O)EOJ VX VPMQK
C16S13F10P20J13W9 F16V14J12V21(16)(Q)MCP ZT GPFTF
Q5 V9 E2 G20A13N19G4 K17B0 B17(18)(S)YDM OI VOSJJ
S10C13P10Z2 B6 H25G2 I21D22Z10(20)(U)YIV FH NMOJF
J17A13N12Z9 I16Y6 E21Z16P4 T-(22)(W)NER DM CIDTX

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

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)

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.

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.

← Back to Elementary Cryptanalysis

Scroll to Top