Cryptography/Breaking Vigenère cipher

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Plain text is encrypted using the Vigenère cipher by first choosing a keyword consisting of letters from the alphabet of symbols used in the plain text. The keyword is then used to encrypt the text by way of the following example.

Using: Plain text: I Like A Book and choosing: Keyword: cta

1. Map all the plain text to numbers 0-25 or however long your alphabet is

  ilikewikibooks converts to 8 11 8 10 4 22 8 10 8 1 14 14 10 18

2. Map your keyword to numbers the same way

  cta maps to 2 19 0

3. add your key to your plain text in the following manner

  8  11  8  10   4  22  8  10  8  1  14  14  10  18
  2  19  0   2  19   0  2  19  0  2  19   0   2  19
  resulting in
  10 30  8  12  23  22 10 29   8  3  33   14  12  37

4. take each resulting number mod 26 ( or for the general case mod the number of characters in your alphabet)

  resulting in
  10 4   8  12  23  22 10 3    8  3  7    14  12   11 

5. map each number back to a letter to get the resulting cypher text


The message can easily be decrypted with the keyword by reversing the above process. The keyword can be any length equal to or less than that of the plain text.

Without the keyword the primary method of breaking the Vigenère cipher is known as the Kasiski test, after the Prussian major who first published it. The first stage is determining the length of the keyword.

Determining the key length[edit]

Given an enciphered message such as:


Upon inspection of the ciphertext, we see that there are a few digraphs repeated, namely DS, SZ, and ZO. It is statistically unlikely that all of these would arise by random chance; the odds are that repeated digraphs in the ciphertext correspond to repetitions in the plaintext. If that is the case, the digraphs must be encoded by the same section of the key both times. Therefore, the length of the key is a factor of the distance in the text between the repetitions.

Digraph First Position Second Position Distance Factors
DS 1 10 9 3
SZ 1 10 9 3
ZO 1 10 3 3

The common factors (indeed, the only factors in this simple example) are 3 and 9. This narrows down the possibilities significantly, and the effect is even more pronounced with longer texts and keys.

Frequency analysis[edit]

Once the length of the key is known, a slightly modified frequency analysis technique can be applied. Suppose the length of the key is known to be three. Then every third letter will be encrypted with the same letter of the key. The ciphertext can be split into three segments - one for each key letter—and the procedure described for the Caesar cipher can be used.

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