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PreparedbyD Almeida
School of Mathematics and Statistics (SoMaS)
Vigen��re ciphers

PreparedbyD Almeida
What is a Vigen��re cipher?
? Consider this sentence:-
? Mathematics is the engine of all science. It enables physical phenomena to
be modelled in a structured way. The model enables scientists to make
predictions about the phenomena.
? In a Vigen��re cipher a key word (generally larger than 3 letters) is used to
scramble a message. We will use the key word surd.
? The word surd is repeated above the given message (with no spaces) thus:
? surdsurdsurdsurdsurdsurdsurdsurdsurdsurdsurdsurdsurdsurdsur
? MathematicsistheengineofallscienceItenablesphysicalphenomenato

PreparedbyD Almeida
What is a Vigen��re cipher?
The key surd defines by how much each letter in the message is translated.
This means a Vigen��re code is variable Caesar code.
0 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
1 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
2 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
3 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
4 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 D
5 F G H I J K L M N O P Q R S T U V W X Y Z A B C D E
6 G H I J K L M N O P Q R S T U V W X Y Z A B C D E F
7 H I J K L M N O P Q R S T U V W X Y Z A B C D E F G
8 I J K L M N O P Q R S T U V W X Y Z A B C D E F G H
9 J K L M N O P Q R S T U V W X Y Z A B C D E F G H I
10 K L M N O P Q R S T U V W X Y Z A B C D E F G H I J
11 L M N O P Q R S T U V W X Y Z A B C D E F G H I J K
12 M N O P Q R S T U V W X Y Z A B C D E F G H I J K L
13 N O P Q R S T U V W X Y Z A B C D E F G H I J K L M
14 O P Q R S T U V W X Y Z A B C D E F G H I J K L M N
15 P Q R S T U V W X Y Z A B C D E F G H I J K L M N O
16 Q R S T U V W X Y Z A B C D E F G H I J K L M N O P
17 R S T U V W X Y Z A B C D E F G H I J K L M N O P Q
18 S T U V W X Y Z A B C D E F G H I J K L M N O P Q R
19 T U V W X Y Z A B C D E F G H I J K L M N O P Q R S
20 U V W X Y Z A B C D E F G H I J K L M N O P Q R S T
21 V W X Y Z A B C D E F G H I J K L M N O P Q R S T U
22 W X Y Z A B C D E F G H I J K L M N O P Q R S T U V
23 X Y Z A B C D E F G H I J K L M N O P Q R S T U V W
24 Y 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
25 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
This table tells us that
A letter corresponding to D is translated by 3:
e.g. A ?D, B ?E, C ?F, ��
A letter corresponding to R is translated by 17:
e.g. A ?R, B ?S, C ?T, ��
A letter corresponding to S is translated by 18:
e.g. A ?S, B ?T, C ?U, ��
A letter corresponding to U is translated by 20:
e.g. A ?U, B ?V, C ?W, ��
According to these rules our message becomes
the last line below:
surd surd surd surd surd surd surd surd surd surd surd surd surd surd surd su
math emat icsi sthe engi neof alls cien ceit enab lesp hysi calp heno mena to
eukk wgrw awjl knyh whxl fyfi sfcv ucvq uyzw whre dyjs zsjl uucs zyer eyed li
? The rules for translation are given by this table
m corresponds
to s, so it is
translated by
18 to e

PreparedbyD Almeida
Objective
We will break this Vigenere cipher presented in blocks of 5 letters:
ANYVG YSTYN RPLWH RDTKX RNYPV QTGHP HZKFE YUMUS AYWVK ZYEZM EZUDL
JKTUL JLKQB JUQVU ECKBN RCTHP KESXM AZOEN SXGOL PGNLE EBMMT GCSSV
MRSEZ MXHLP KJEJH TUPZU EDWKN NNRWA GEEXS LKZUD LJKFI XHTKP IAZMX
FACWC TQIDU WBRRL TTKVN AJWVB REAWT NSEZM OECSS VMRSL JMLEE BMMTG
AYVIY GHPEM YFARW AOAEL UPIUA YYMGE EMJQK SFCGU GYBPJ BPZYP JASNN
FSTUS STYVG YS
The key aim is to find how many letters were used in the key word used to code
the original message.
This does not mean the key word has 5 letters!

PreparedbyD Almeida
1. Find repeating blocks of groups of three or more letters
2. Find the gaps n between repeating blocks of groups of letters
3. Find the non-trivial factors of n (so exclude 1 and n)
4. We must find the common factor (larger than 3) with the highest frequency.
Call this N
5. Group the ciphered text in blocks of N letters.
6. Find the frequencies of the 26 letters a, b, c,��.., x, y, z in the first place in each
block; repeat for the second place, third place, ��.Nth place
7. Compare the frequencies of the 26 letters a, b, c,��.., x, y, z in the first place with
the expected frequencies of the 26 letters: observation and judgement will imply a
Caesar shift of the letters. Replace the ciphered letter in each of the first places
with the unciphered one. Repeat for other places, etc. using common sense and
comparative frequencies for the other places.
8. Partition to reveal message.
The key steps in cracking such ciphers is given below and the steps will
be illustrated by applying it to our ciphered text:
Generally the key word will have more than 3 letters but for practice identifying blocks we include 3

PreparedbyD Almeida
Step 1. Finding repeating blocks of groups of 3 or more
letters.
Use MS Word ��find�� to highlight the letters A, B, ��.
FSTUS STYVG YS
There are no repeating groups of 3 or more letters beginning with A.
So we keep going.

PreparedbyD Almeida
Step 1 (continued) Finding repeating blocks of groups
of 3 or more letters.
Use MS Word ��find�� to highlight the letters A, B, ��.
FSTUS STYVG YS
By (careful) observation we identify the groups GHP and GEE.

PreparedbyD Almeida
Step 1 (continued) Finding repeating blocks of groups
of 3 or more letters.
The groups GHP and GEE:
FSTUS STYVG YS
And here��s the group LEEBMMTG:
FSTUS STYVG YS

PreparedbyD Almeida
Step 2 Finding the gaps n between repeating blocks of
groups of letters
FSTUS STYVG YS
By (careful) counting the gap between the two GHP��s = 198
and the gap between the two GEE��s = 108
Important: the gap between the AMB��s in:
AMBCV GUYPG LPRCT MAMBD is 16 (count between the A��s and add 1)
15
15 + 1

PreparedbyD Almeida
Step 3 Finding the non-trivial factors of n exclude 1 and n)
After examining the repeating groups of letters we will have this table:
Group Gap Factors
GHP 198 2, 3, 6, 9, 11, 18, 22, 33, 66, 99
GEE 108 2, 3, 4, 6, 9, 12, 18, 27, 36, 54
LEEBMMTG 114 2, 3, 6, 19, 38, 57
SEZM 84 2, 3, 4, 6, 7, 12, 14, 21, 28, 42
STY 274 1, 2, 137, 274
YVGYS 280 2, 4, 5, 7, 8, 10, 14, 20, 28, 35,
40, 56, 70, 140
ZMX 48 2, 3, 4, 6, 8, 12, 16, 24
ZUDLJK 96 2, 3, 4, 6, 8, 12, 16, 24, 32, 48

PreparedbyD Almeida
Step 4 Finding the common factor N (larger than 3) with the
highest frequency.
So N = 6
Factor > 3 Frequency
4 5
6 6
7, 8, 9, 14, 16 2
12 4
Others 1
? Highest frequency

PreparedbyD Almeida
Step 5 Group the ciphered text in blocks of N = 6 letters.
ANYVGY STYNRP LWHRDT KXRNYP VQTGHP HZKFEY UMUSAY WVKZYE
ZMEZUD LJKTUL JLKQBJ UQVUEC KBNRCT HPKESX MAZOEN SXGOLP
GNLEEB MMTGCS SVMRSE ZMXHLP KJEJHT UPZUED WKNNNR WAGEEX
SLKZUD LJKFIX HTKPIA ZMXFAC WCTQID UWBRRL TTKVNA JWVBRE
AWTNSE ZMOECS SVMRSL JMLEEB MMTGAY VIYGHP EMYFAR WAOAEL
UPIUAY YMGEEM JQKSFC GUGYBP JBPZYP JASNNF STUSST YVGYS

PreparedbyD Almeida
Step 6 Find the frequencies of the 26 letters a, b, c,��.., x, y, z in
the first place in each block;
The first place letters:
Frequency table:
A B C D E F G H I J K L M
2 0 0 0 1 0 2 3 0 6 3 3 3
N O P Q R S T U V W X Y Z
0 0 0 0 0 6 1 5 2 5 0 2 4

PreparedbyD Almeida
The second place letters:
AWTNSE ZMOECS SVMRSL JMLEEB MMTGA YVIYGHP EMYFAR WAOAEL
Now construct frequency table:
A B C D E F . . . . . .
4 2 1 0 0 . . . . .
the other places in each block;

PreparedbyD Almeida
1st 2nd 3rd 4th 5th 6th
a 2 4 0 1 4 2
b 0 2 1 1 2 2
c 0 1 0 0 3 3
d 0 0 0 0 1 4
e 1 0 2 6 9 4
f 0 0 0 4 1 1
g 2 0 5 4 1 0
h 3 0 1 1 2 0
i 0 1 1 0 3 0
j 6 3 0 1 0 1
k 3 1 10 0 0 0
l 3 2 2 0 2 4
m 3 10 2 0 0 1
n 0 2 2 5 3 1
o 0 0 2 2 0 0
p 0 3 1 1 0 8
q 0 3 0 2 0 0
r 0 0 1 5 3 2
s 6 0 1 2 6 2
t 1 4 5 1 0 4
u 5 1 2 3 3 0
v 2 3 2 2 0 0
w 5 4 0 0 0 0
x 0 2 2 0 0 3
y 2 1 4 2 3 5
z 4 1 2 4 0 0
the 6 places in each block;
This is the completed table of
frequencies of the letters in the
1st, 2nd, 3rd, 4th, 5th and 6th places

PreparedbyD Almeida
Normal relative
frequency of letters
a 2 4 0 1 4 2 8.17%
b 0 2 1 1 2 2 1.49%
c 0 1 0 0 3 3 2.78%
d 0 0 0 0 1 4 4.25%
e 1 0 2 6 9 4 12.70%
f 0 0 0 4 1 1 2.23%
g 2 0 5 4 1 0 2.02%
h 3 0 1 1 2 0 6.09%
i 0 1 1 0 3 0 6.97%
j 6 3 0 1 0 1 0.15%
k 3 1 10 0 0 0 0.77%
l 3 2 2 0 2 4 4.03%
m 3 10 2 0 0 1 2.41%
n 0 2 2 5 3 1 6.75%
o 0 0 2 2 0 0 7.51%
p 0 3 1 1 0 8 1.93%
q 0 3 0 2 0 0 0.10%
r 0 0 1 5 3 2 5.99%
s 6 0 1 2 6 2 6.33%
t 1 4 5 1 0 4 9.06%
u 5 1 2 3 3 0 2.76%
v 2 3 2 2 0 0 0.98%
w 5 4 0 0 0 0 2.36%
x 0 2 2 0 0 3 0.15%
y 2 1 4 2 3 5 1.97%
z 4 1 2 4 0 0 0.07%
Step 7 Compare the frequencies of the 26 letters a, b, c,��.., x, y,
z in the first place with the expected frequencies of the 26
letters: observation will imply a Caesar shift of the letters
Comparison of 1st place frequency with that
of expected frequency implies a Caesar shift
of 8.
That is
8 0 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

PreparedbyD Almeida
Step 7 Replace the ciphered letter in each of the first places
with the unciphered one.
INYVGY ATYNRP TWHRDT SXRNYP DQTGHP PZKFEY CMUSAY EVKZYE
HMEZUD TJKTUL RLKQBJ CQVUEC SBNRCT PPKESX UAZOEN AXGOLP
ONLEEB UMTGCS AVMRSE HMXHLP SJEJHT CPZUED EKNNNR EAGEEX
ALKZUD TJKFIX PTKPIA HMXFAC ECTQID CWBRRL BTKVNA RWVBRE
IWTNSE HMOECS AVMRSL RMLEEB UMTGAY DIYGHP MMYFAR EAOAEL
CPIUAY GMGEEM RQKSFC OUGYBP RBPZYP RASNNF ATUSST GVGYS
This rule for 1st place letters

PreparedbyD Almeida
Normal relative
frequency of letters
a 2 4 0 1 4 2 8.17%
b 0 2 1 1 2 2 1.49%
c 0 1 0 0 3 3 2.78%
d 0 0 0 0 1 4 4.25%
e 1 0 2 6 9 4 12.70%
f 0 0 0 4 1 1 2.23%
g 2 0 5 4 1 0 2.02%
h 3 0 1 1 2 0 6.09%
i 0 1 1 0 3 0 6.97%
j 6 3 0 1 0 1 0.15%
k 3 1 10 0 0 0 0.77%
l 3 2 2 0 2 4 4.03%
m 3 10 2 0 0 1 2.41%
n 0 2 2 5 3 1 6.75%
o 0 0 2 2 0 0 7.51%
p 0 3 1 1 0 8 1.93%
q 0 3 0 2 0 0 0.10%
r 0 0 1 5 3 2 5.99%
s 6 0 1 2 6 2 6.33%
t 1 4 5 1 0 4 9.06%
u 5 1 2 3 3 0 2.76%
v 2 3 2 2 0 0 0.98%
w 5 4 0 0 0 0 2.36%
x 0 2 2 0 0 3 0.15%
y 2 1 4 2 3 5 1.97%
z 4 1 2 4 0 0 0.07%
Step 7 Compare the frequencies of the 26 letters a, b, c,��.., x, y,
z in the second place with the expected frequencies of the 26
letters: observation will imply a Caesar shift of the letters
Comparison of 2nd place frequency with that
of expected frequency implies a Caesar shift
of 18:
i.e.
18

PreparedbyD Almeida
Step 7 Replace the ciphered letter in each of the second places
with the unciphered one.
INYVGY ATYNRP TWHRDT SXRNYP DQTGHP PZKFEY CMUSAY EVKZYE
HMEZUD TJKTUL RLKQBJ CQVUEC SBNRCT PPKESX UAZOEN AXGOLP
ONLEEB UMTGCS AVMRSE HMXHLP SJEJHT CPZUED EKNNNR EAGEEX
ALKZUD TJKFIX PTKPIA HMXFAC ECTQID CWBRRL BTKVNA RWVBRE
IWTNSE HMOECS AVMRSL RMLEEB UMTGAY DIYGHP MMYFAR EAOAEL
CPIUAY GMGEEM RQKSFC OUGYBP RBPZYP RASNNF ATUSST GVGYS
IFYVGY ALYNRP TOHRDT SPRNYP DITGHP PRKFEY CEUSAY ENKZYE
HEEZUD TBKTUL RDKQBJ CIVUEC STNRCT PHKESX USZOEN APGOLP
OFLEEB UETGCS ANMRSE HEXHLP SBEJHT CHZUED ECNNNR ESGEEX
ADKZUD TBKFIX PLKPIA HEXFAC EUTQID COBRRL BLKVNA ROVBRE
IOTNSE HEOECS ANMRSL RELEEB UETGAY DAYGHP MEYFAR ESOAEL
CHIUAY GEGEEM RIKSFC OMGYBP RTPZYP RSSNNF ALUSST GNGYS
This rule for 2nd place letters

PreparedbyD Almeida
Step 7 Use common sense and/or comparative frequencies for
the other place letters.
IFYVGY ALYNRP TOHRDT SPRNYP DITGHP PRKFEY CEUSAY ENKZYE
HEEZUD TBKTUL RDKQBJ CIVUEC STNRCT PHKESX USZOEN APGOLP
OFLEEB UETGCS ANMRSE HEXHLP SBEJHT CHZUED ECNNNR ESGEEX
ADKZUD LBKFIX PLKPIA HEXFAC EUTQID COBRRL BLKVNA ROVBRE
IOTNSE HEOECS ANMRSL RELEEB UETGAY DAYGHP MEYFAR ESOAEL
CHIUAY GEGEEM RIKSFC OMGYBP RTPZYP RSSNNF ALUSST GNGYS
This E before HE suggests E is a T: a Caesar shift of 15 in the 6th place
15 P Q R S T U V W X Y Z A B C D E F G H I J K L M N O
IFYVGN ALYNRE TOHRDI SPRNYE DITGHE PRKFEN CEUSAN ENKZYT
HEEZUS TBKTUA RDKQBY CIVUER STNRCI PHKESM USZOEC APGOLE
OFLEEQ UETGCH ANMRST HEXHLE SBEJHI CHZUES ECNNNG ESGEEM
ADKZUS TBKFIM PLKPIP HEXFAR EUTQIS COBRRA BLKVNP ROVBRT
IOTNST HEOECH ANMRSA RELEEQ UETGAN DAYGHE MEYFAG ESOAEA
CHIUAN GEGEEB RIKSFR OMGYBE RTPZYE RSSNNU ALUSSI GNGYS

PreparedbyD Almeida
5th
Normal
frequency
a 4 8.17%
b 2 1.49%
c 3 2.78%
d 1 4.25%
e 9 12.70%
f 1 2.23%
g 1 2.02%
h 2 6.09%
i 3 6.97%
j 0 0.15%
k 0 0.77%
l 2 4.03%
m 0 2.41%
n 3 6.75%
o 0 7.51%
p 0 1.93%
q 0 0.10%
r 3 5.99%
s 6 6.33%
t 0 9.06%
u 3 2.76%
v 0 0.98%
w 0 2.36%
x 0 0.15%
y 3 1.97%
z 0 0.07%
This block of letters and the comparative frequency of 5th place
letters suggest��..
There is no shift for 5th place letters

PreparedbyD Almeida
This G and the HE suggests that G should be a T: a Caesar shift of 13
for the 4th place letters
13 N O P Q R S T U V W X Y Z A B C D E F G H I J K L M
IFYIGN ALYARE TOHEDI SPRAYE DITTHE PRKSEN CEUFAN ENKMYT
HEEMUS TBKGUA RDKDBY CIVHER STNECI PHKRSM USZBEC APGBLE
OFLREQ UETTCH ANMEST HEXULE SBEWHI CHZHES ECNANG ESGREM
ADKMUS TBKSIM PLKCIP HEXSAR EUTDIS COBERA BLKINP ROVORT
IOTAST HEORCH ANMESA RELREQ UETTAN DAYTHE MEYSAG ESONEA
CHIHAN GEGREB RIKFFR OMGLBE RTPMYE RSSANU ALUFSI GNGLS

PreparedbyD Almeida
IFYIGN ALYARE TOHEDI SPRAYE DITTHE PRKSEN CEUFAN ENKMYT
HEEMUS TBKGUA RDKDBY CIVHER STNECI PHKRSM USZBEC APGBLE
OFLREQ UETTCH ANMEST HEXULE SBEWHI CHZHES ECNANG ESGREM
ADKMUS TBKSIM PLKCIP HEXSAR EUTDIS COBERA BLKINP ROVORT
IOTAST HEORCH ANMESA RELREQ UETTAN DAYTHE MEYSAG ESONEA
CHIHAN GEGREB RIKFFR OMGLBE RTPLYE RSSANU ALUFSI GNGLS
3rd
Normal freq
a 0 8.17%
b 1 1.49%
c 0 2.78%
d 0 4.25%
e 2 12.70%
f 0 2.23%
g 5 2.02%
h 1 6.09%
i 1 6.97%
j 0 0.15%
k 10 0.77%
l 2 4.03%
m 2 2.41%
n 2 6.75%
o 2 7.51%
p 1 1.93%
q 0 0.10%
r 1 5.99%
s 1 6.33%
t 5 9.06%
u 2 2.76%
v 2 0.98%
w 0 2.36%
x 2 0.15%
y 4 1.97%
z 2 0.07%
Comparative high frequency of K��s in 5th place suggest K must be an
E: a Caesar shift 0f 20.
IFSIGN ALSARE TOBEDI SPLAYE DINTHE PRESEN CEOFAN ENEMYT
HEYMUS TBEGUA RDEDBY CIPHER STHECI PHERSM USTBEC APABLE
OFFREQ UENTCH ANGEST HERULE SBYWHI CHTHES ECHANG ESAREM
ADEMUS TBESIM PLECIP HERSAR EUNDIS COVERA BLEINP ROPORT
IONAST HEIRCH ANGESA REFREQ UENTAN DASTHE MESSAG ESINEA
CHCHAN GEAREB RIEFFR OMALBE RTJMYE RSMANU ALOFSI GNALS
20 U V W X Y Z A B C D E F G H I J K L M N O P Q R S T

PreparedbyD Almeida
Step 8 Partition to reveal message.
IF|SIGNALS|ARE|TO|BE|DISPLAYED|IN|THE|PRESENCE|OF|AN|ENEMY|
THEY|MUST|BE|GUARDED|BY|CIPHERS|THE|CIPHERS|MUST|BE|
CAPABLE|OF|FREQUENT|CHANGES|THE|RULES|BY|WHICH|THESE|
CHANGES|ARE|MADE|MUST|BE|SIMPLE|CIPHERS|ARE|UNDISCOVERABLE|
IN|PROPORTION|AS|THEIR|CHANGES|ARE|FREQUENT|AND|AS|THE|
MESSAGES|IN|EACH|CHANGE|ARE|BRIEF|FROM|ALBERTJMYERS|MANUAL|
OF|SIGNALS|
IFSIGN ALSARE TOBEDI SPLAYE DINTHE PRESEN CEOFAN ENEMYT
HEYMUS TBEGUA RDEDBY CIPHER STHECI PHERSM USTBEC APABLE
OFFREQ UENTCH ANGEST HERULE SBYWHI CHTHES ECHANG ESAREM
ADEMUS TBESIM PLECIP HERSAR EUNDIS COVERA BLEINP ROPORT
IONAST HEIRCH ANGESA REFREQ UENTAN DASTHE MESSAG ESINEA
CHCHAN GEAREB RIEFFR OMALBE RTJMYE RSMANU ALOFSI GNALS

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