Cryptanalysis of the Vigenere Cipher Using Signatures and Scrawls To break a Vigenere cipher you need to know the keyword length. – The Kasiski and Friedman tests give keyword lengths, but not the keyword. – The method described here not only estimates the keyword length but also allows the determination of the keyword. The method was described by Charles Babbage 1854…

Mathematical Description of the Vigenere Cipher and Cosets Fact: When a plaintext letter is enciphered using a keyword letter, the result is equivalent to adding the two characters together mod 26. Example 1: Encipher with Vigenere Example 1 The first step in decrypting a Vigenere cipher is to know the length of the keyword. Definition: A coset are all the letters of the Vigenere ciphertext that are enciphered by the same letter of the keyword. – Note: A coset of the Vigenere ciphertext has the same encipherment as a shift cipher…

Mathematical Description of the Vigenere Cipher and Cosets Example 2: Keyword PET Example 2 Fact: The number of cosets is the length of the keyword. Note: If the length of the keyword is n, then there are 26^n possible ways the Vigenere cipher can be constructed. Note: Because a coset is formed from a shift cipher, we can perform a frequency analysis on that coset, which frequency is the same as the regular alphabet only shifted according to the letter of the keyword.frequency Example 3 – Three letter keyword decipher… Example 3

Mathematical Description of the Vigenere Cipher and Cosets The method described in Example 3 describes how to break the Vigenere Cipher. Let’s now describe the method in more detail: – Definition: English Signature Graph– the plot of the probabilities of printed standard English plotted in increasing sorted order (from smallest to largest).English Signature – Definition: English Sample Signature Graph– The signature plot of the probability frequencies of a sample of letters (small set). Note:English Sample Signature Some letters may not appear in the sample. Some letters will have probability zero. Other letters will have probability greater than in the English Signature The sum of all the probabilities is 1. The sample signature curve should be lower than the English Signature initially, then become higher then the English Signature for the higher occurring frequencies. The sample signature of the plaintext message should have the same sample signature as the ciphertext, when the shift on the cosets is correct…

Mathematical Description of the Vigenere Cipher and Cosets Example 3A: Sample SignatureSample Signature – The Table of frequency of lettersTable of frequency – The Sample and Signature GraphsSample and Signature – Notice that for the graphs the Sample signature probabilities start out below the English Signature but end up above. As mentioned before this is to be expected. (This behavior is typical of small sample signatures)…

Mathematical Description of the Vigenere Cipher and Cosets How can we use the concept of a signature to determine the length of the keyword used in a Vigenere cipher (Each letter of the keyword performs a simple shift cipher and forms a coset) – Each coset should perform like a simple shift cipher. Since there are few letters in each coset we would expect the first few letters to be below the English Signature, and those at the end to be above the English Signature. – The number of cosets is the same as the number of letters in the keyword. – Example 5 Comparing Coset Signatures… Example 5 Comparing Coset Signatures

Determining The Keyword Using Scrawls Scrawl – The graph of the probability distribution of letters in alphabetical order. English Scrawl – The scrawl of the entire English Language. English Scrawl Sample Scrawl – The scrawl of a sample of English (for example a ciphertext message) Sample Scrawl Facts about Scrawls: – For a shift cipher, the scrawl of the ciphertext will be roughly the shape of the English Scrawl except it is shifted to the right. For the Vigenere this means each coset – For each coset produced by a keyword letter in the Vigenere cipher, we will draw successive shifts (from 0 to 25) until we find the sample coset scrawl with the closest match to the English Scrawl. – The alphabet letter in our MOD 26 alphabet assignment corresponding to the shift amount number found in step 2 is the shift amount when ciphering the text. The inverse of the number will be the keyword letter needed in the Vigenere encipherment. – Repat steps 2 and 3 for each coset. Example 6: Continuation of Example 5…! Example 6