Codes, Ciphers, and CryptographyCh 2'2

1
Codes, Ciphers, and Cryptography-Ch 2.2

• Michael A. Karls
• Ball State University
• Fall 2004

2
Breaking the Vigenère Cipher

• Suppose we have intercepted the following
  ciphertext (handout)
• DBZMG AOIYS OPVFH OWKBW XZPJL VVRFG NBKIX DVUIM
  OPFQL VVPUD KPRVW OARLW DVLMW AWINZ DAKBW MMRLW
  QIICG PAKYU CVZKM ZARPS DTRVD ZWEYG ABYYE YMGYF
  YAFHL CMWLW LVCHL MMGYL DBZIF JNCYL OMIAJ JCGMA
  IBVRL OPVFW OBVLK OPVUJ ZDVLQ XWDGG IQEYF BTZMZ
  DVRMM ANZWA ZVKFQ GWEAL ZFKNZ ZZVCK VDVLQ BWFXU
  CIEWW OPRMU JZIYK KWEXA IOIYH ZIKYV GMKNW MOIIM
  KADUQ WMWIM ILZHL CMTCH CMINW SBRHV OPVSO
  DTCMGHMKCE ZASYD JKRNW YIKCF OMIPS GAFZK JUVGM
  GBZJD ZWWNZ ZVLGT ZZFZS GXYUT ZBJCF PAVNZ ZAVWS
  IJVZG PVUVQ NKRHF DVXNZ ZKZJZ ZZKYP OIEXX MWDNZ
  ZQIMH VKZHY DVKYD GQXYF OOLYK NMJGS YMRML JBYYF
  PUSYJ JNRFH CISYL N
• We suspect that this message may have been
  enciphered using a Vigenère cipher.
• How can we decipher the message?

3
Breaking the Vigenère Cipher (cont.)

• In order to get an idea of what to do, consider
  the following example.
• In this example, the keyword VENUS is used with
  our Vigenère square handout to encipher a message!

4
Breaking the Vigenère Cipher (cont.)

• Example 3 Use the keyword VENUS and the
  Vigenère square to encipher the cat in the hat
  is near the mat.
• Solution

plaintext
ciphertext
plaintext
ciphertext
5
Breaking the Vigenère Cipher (cont.)

• How is the word the enciphered?
• the ? OLR, NZZ, and OLR

plaintext
ciphertext
plaintext
ciphertext
6
Breaking the Vigenère Cipher (cont.)

• How is the word the enciphered?
• the ? OLR, NZZ, and OLR

plaintext
ciphertext
plaintext
ciphertext
7
Breaking the Vigenère Cipher (cont.)

• The third occurrence of the is the same as the
  first because the same key letters are used to
  encrypt the t, h, and e.
• Different key letters are used for the second
  the.

plaintext
ciphertext
plaintext
ciphertext
8
Breaking the Vigenère Cipher (cont.)

• Notice that the distance between the t in the
  first the and the t in the third the is 20
  letters.
• 20 is a multiple of the length of the keyword
  VENUS, which has 5 letters!

plaintext
ciphertext
plaintext
ciphertext
9
Breaking the Vigenère Cipher (cont.)

• This is true for Vigenère ciphers in general!
• If a plaintext sequence is repeated at a distance
  that is a multiple of the length of the keyword,
  then the corresponding ciphertext sequences are
  also repeated.
• Handout example (Fig. 2.3 from Beutelspacher).

10
Breaking the Vigenère Cipher (cont.)

• In 1854, Charles Babbage used this idea to guess
  the length of a keyword for a Vigenère cipher!
• John Hall Brock Thwaites claimed to have invented
  a new cipher.
• Babbage told Thwaites that his cipher was the
  Vigenère cipher.
• Thwaites challenged Babbage to break his cipher.
• Babbage never published this ideawhy?
• In 1863 Friedrich Wilhelm Kasiski independently
  discovered the idea of looking for repeated
  ciphertext sequences to find a Vigenère ciphers
  key length.

11
The Kasiski Test

• Look for repeated sequences of three or more
  letters in a ciphertext.
• Equal pairs can occur too often to be of use.
• Find the distances between these repeated
  sequences and list the factors of each distance.
• For example, if the distance is d 18 letters,
  then the factors are 1, 2, 3, 6, 9, and 18.

12
The Kasiski Test (cont.)

• Look for the most commonly occurring factors
  greater than 1.
• A keyword of length 1 corresponds to a
  monoalphabetic cipher!
• The most commonly occurring factors are
  candidates for the length L of the keyword.

13
The Kasiski Test (cont.)

• When using the Kasiski Test, keep the following
  in mind
• Ciphertext sequences may repeat accidentally at a
  distance that is not a multiple of the keyword
  length.
• The most commonly occurring factors found in in
  Step 3 of the Kasiski Test may be a factor of the
  keyword length!
• Therefore, one may need to consider multiples of
  possible keyword lengths as potential keyword
  lengths!

14
The Kasiski Test (cont.)

• Example 4 Apply the Kasiski Test to the
  following ciphertext, which is suspected of being
  a Vigenere cipher!
• ZRZNSPCWQGFWLWQGFRGRZNSTZBGTRMNEOXCXRPUBSEYWQGHLSI
  JKGWDSXSSLSIXKKUJPHKBWHW
• For Step 1 of the test, see Mathematica Handout
  Kasiski Test Example 4 .

15
The Kasiski Test (cont.)

• Other ways to find distances between repeated
  sequences of ciphertext
• Use a program on the web, such as this one
• http//www.uni-mainz.de/pommeren/Kryptologie/Clas
  sic.html
• Use Microsoft Words Find and Replace feature.
• The only drawback is you have to find the
  repeated sequences by hand.

16
The Kasiski Test (cont.)

• Step 2 Make a table of the distances and their
  factors greater than or equal to two.

Factors of Distances
17
The Kasiski Test (cont.)

• Step 2 Make a table of the distances and their
  factors greater than or equal to two.

Factors of Distances
18
The Kasiski Test (cont.)

• Step 2 Make a table of the distances and their
  factors greater than or equal to two.

Factors of Distances
19
The Kasiski Test (cont.)

• Step 2 Make a table of the distances and their
  factors greater than or equal to two.

Factors of Distances
20
The Kasiski Test (cont.)

• Step 2 Make a table of the distances and their
  factors greater than or equal to two.

Factors of Distances
21
The Kasiski Test (cont.)

• Step 3 The most commonly occurring factors in
  the table are 2, 3, and 6.

Factors of Distances
22
The Kasiski Test (cont.)

• The greatest common divisor of all distances
  between repeated sequences of ciphertext is 6.
• Guess that the keyword length is L 6.
• Note that L 12, 18, or 30 could be possible
  (these are multiples of 6).

Factors of Distances
23
Using Frequency Analysis to Attack the Vigenère
Cipher

• As Example 4 shows, we can use the Kasiski Test
  to get an idea of the keyword length for a
  Vigenère cipher.
• Now that we know the keyword length, how do we
  use this information to break the cipher?
• Note that each time a certain keyword letter is
  used to encrypt, the same row of the Vigenere
  square is used, i.e. the same cipher alphabet is
  used!
• Therefore, we can apply frequency analysis to the
  subsets of the ciphertext encrypted with each
  keyword letter!

24
Using Frequency Analysis to Attack the Vigenère
Cipher (cont.)

• For example, suppose the keyword in the cipher of
  Example 4 is six letters long.
• Write the keyword as S1S2S3S4S5S6, with unknown
  letters Si to be determined.
• Split up the ciphertext in the following way

25
Using Frequency Analysis to Attack the Vigenère
Cipher (cont.)
Letter Si of keyword
i
i6
Position of letter in keyword
i12
i18
i24
26
Using Frequency Analysis to Attack the Vigenère
Cipher (cont.)
Letter Si of keyword
i
i6
Position of letter in keyword
i12
i18
Column 1 is encrypted by S1
i24
27
Using Frequency Analysis to Attack the Vigenère
Cipher (cont.)
Letter Si of keyword
i
i6
Position of letter in keyword
i12
i18
Column 2 is encrypted by S2
i24
28
Using Frequency Analysis to Attack the Vigenère
Cipher (cont.)
Letter Si of keyword
i
i6
Position of letter in keyword
i12
i18
Column 3 is encrypted by S3
i24
29
Using Frequency Analysis to Attack the Vigenère
Cipher (cont.)
Letter Si of keyword
i
i6
Position of letter in keyword
i12
i18
Column 4 is encrypted by S4
i24
30
Using Frequency Analysis to Attack the Vigenère
Cipher (cont.)
Letter Si of keyword
i
i6
Position of letter in keyword
i12
i18
Column 5 is encrypted by S5
i24
31
Using Frequency Analysis to Attack the Vigenère
Cipher (cont.)
Letter Si of keyword
i
i6
Position of letter in keyword
i12
i18
Column 6 is encrypted by S6
i24
32
Using Frequency Analysis to Attack the Vigenère
Cipher (cont.)

• Since the letters in column 1 are encrypted by
  the same cipher alphabet, we can use frequency
  analysis on this set of cipher letters to figure
  out keyword letter S1.
• Repeat with column 2 letters to find S2, etc.
• In this way, frequency analysis is used to crack
  a Vigenere cipher!

33
Using Frequency Analysis to Attack the Vigenère
Cipher (cont.)

• Example 5 The Kasiski Test has been applied to
  the Vigenère ciphertext below. Assuming that
  the keyword has length 5, find the subsets of
  ciphertext encrypted with each letter. Use
  frequency analysis to determine each keyword
  letter!
• DBZMG AOIYS OPVFH OWKBW XZPJL VVRFG NBKIX DVUIM
  OPFQL VVPUD KPRVW OARLW DVLMW AWINZ DAKBW MMRLW
  QIICG PAKYU CVZKM ZARPS DTRVD ZWEYG ABYYE YMGYF
  YAFHL CMWLW LVCHL MMGYL DBZIF JNCYL OMIAJ JCGMA
  IBVRL OPVFW OBVLK OPVUJ ZDVLQ XWDGG IQEYF BTZMZ
  DVRMM ANZWA ZVKFQ GWEAL ZFKNZ ZZVCK VDVLQ BWFXU
  CIEWW OPRMU JZIYK KWEXA IOIYH ZIKYV GMKNW MOIIM
  KADUQ WMWIM ILZHL CMTCH CMINW SBRHV OPVSO
  DTCMGHMKCE ZASYD JKRNW YIKCF OMIPS GAFZK JUVGM
  GBZJD ZWWNZ ZVLGT ZZFZS GXYUT ZBJCF PAVNZ ZAVWS
  IJVZG PVUVQ NKRHF DVXNZ ZKZJZ ZZKYP OIEXX MWDNZ
  ZQIMH VKZHY DVKYD GQXYF OOLYK NMJGS YMRML JBYYF
  PUSYJ JNRFH CISYL N
• Solution Mathematica Handout for Example 5.