# Section 2.7 The Friedman and Kasiski Tests by GttTJCU

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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
used.
– 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
Polyalphabetic
• 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
polyalphabetic
• 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
1863.
– 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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