Section 2.7 The Friedman and Kasiski Tests by GttTJCU


									Section 2.7 The Friedman and Kasiski Tests

• We will use the probability methods of the last
  section to help get a good estimate of the length
  of the keyword used in the Vigenere Cipher.
• Probability of Selecting Multiple Letters in
  Standard English
  –   P(Single Letter) = (relative frequency of letter) / 100.
  –   Standard Frequency Table
  –   Example 1: Probability of selecting a letter
  –   Example 2: Two Letters
  –   Example 3: Two letters the same…
               Friedman Test
• The Friedman Test is a probabilistic test that can
  be used to determine the likelihood that the
  ciphertext message comes from a
  monoalphabetic or polyalphabetic cipher.
  – If the ciphertext is Vigenere, then Friedman’s test is
    also useful in approximating the length of the keyword
  – Definition: Index of Coincidence: Is denoted by I, and
    represents the probability that two randomly selected
    letters are identical…
       Index of Coincidence for
       Monoalphabetic Ciphers
• In monoalphabetic ciphers, the
  frequencies of letters in standard English
  are preserved when converting from
  plaintext to ciphertext.
  – Why? Using monoalphabetic text in reality
    means that each of the letters is uniquely
    renamed a new name. The names have
    changed, but not the frequency.
• Example 4: Caesar Shift Cipher…
         Index of Coincidence
• In Example 4 we saw that the probability of two
  letters being the same is 0.065. If we then look
  at the ciphertext message and we get 0.065 (or
  nearly this) as the probability of randomly
  selecting two letters and they being the same,
  then our ciphertext was generated from a
  monoalphabetic cipher.
  – I = P(Two randomly selected letters being the same)
    = 0.065, then Monoalphabetic Cipher. (We can try this
    with Mr. Fisher’s Index of Coincidence program)…
         Index of Coincidence for
• The goal for a polyalphabetic cipher is to
  distribute the letter frequencies so that each
  letter has the same likelihood of occurring.
   – Example 5: Polyalphabetic Cipher
   – Conclusion: Index of Coincidence for Polyalphabetic
     Ciphers I = P(Two randomly selected letters are the
     same) = 0.0385
   – Index of Coincidence Bound: 0.0385 ≤ I ≤ 0.065.
      • We derived these numbers under the assumption that the
        ciphertext message was very large.
      • In practice, most messages are short.
      • For this reason we have the bound…
        Index of Coincidence
• Definition: Summation
• Example 6: Compute Sum
• Derivation of Formula for the Index of
  Coincidence for a Given Ciphertext Message
• Example 7: Determine if monoalphabetic or
• Example 8: Determine keyword length…
                 The Kasiski Test
• The Kasiski test is another method that can be used to
  approximate the keyword length to the Vigenere cipher.
   – Kasiski was a Prussian Army officer. Discovered his test in
   – Babbage (from England) made the same discovery in 1854.
• The test:
   – Look for occasional coincidental alignment of letter groups.
   – The distance between these groups (first letter to first letter) of
     letters may be a multiple of the word length.
• Example: Keyword length estimation
• Example 9: Kasiski Analysis Example 7…!

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