Notes - The Vigenere Cipher

The following are notes from Introduction to Cryptography with Coding Theory.

• A variation of the shift cipher
• key for the encryption is a vector of a chosen length
• Example
• key length 6
• (21,4,2,19,14,17)
• corresponds to word vector
• To encrypt we shift by the word vector
• A known plain text attack will succeed if enough characters are known
• Cryptanalysis uses the fact that english letter frequency is not equal
• a - 0.082
• b - 0.015
• c - 0.028
• d - 0.043
• e - 0.127
• f - 0.022
• g - 0.020
• h - 0.061
• i - 0.070
• j - 0.002
• k - 0.008
• l - 0.040
• m - 0.024
• n - 0.067
• o - 0.075
• p - 0.019
• q - 0.001
• r - 0.060
• s - 0.063
• t - 0.091
• u - 0.028
• v - 0.010
• w - 0.023
• x - 0.001
• y - 0.020
• z - 0.001
• Variations can occur but usually takes effort to produce, example the book Gadsby does not contain the letter e
• However, when using a Vigenere Cipher the frequencies are distributed so that the above is not nearly as useful in decryption
• Finding Key Length
• We find the key length by matching the number of coincidences
• write the ciphertext, then rewrite it with a displacement
• Example
•     VVHQ
• VVHQW
• Then we mark each time there is a match between the two rows, and after trying multiple displacements the largest number of coincidences will be closest to the key length
• Finding the Key: First Method
• take a look at the 1+5n letters and find the frequency of letters then repeat for 2+5n, 3+5n, 4+5n, 5+5n we then guess at the code by finding the difference between the most frequent letter and e to find the overall shift that is used as the key
• After finding the key we then test it by decrypting to confirm if the key was correct
• Soft Proof
• place the frequencies of the english letters into a vector A₀ and let A_i represent the frequencies shifted by i
• When taking the dot product of these two vectors we arrive at 0.066 with a full match but significantly less than that for any other shift
• We can also use this knowledge to approximate the number of coincidences we should find for a given match, which would be 0.066*the number of comparisons

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