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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