Index of Coincidence

The index of coincidence is an indicator used in cryptanalysis which makes it possible to evaluate the global distribution of letters in encrypted message for a given alphabet.

A text written in English language has an index of coincidence of 0.0667, but some ciphers modify this value which can allow them to be recognized.

Index of coincidence uses the formula:

$$ IC = \sum_{i=A}^{i=Z} \frac{n_{i}(n_{i}-1)}{N(N-1)} $$

with $ n_i $ the number of occurrences of the letter $ i $ in the text and $ N $ the total number of letters.

For a given ciphered message, the value for the IoC allows to filter the list of ciphering methods to use. It is one of the first cryptanalysis technique.

If the Index of coincidence is high (close to $ 0.070 $), i.e. similar to plain text, then the message has probably been crypted using a transposition cipher (letters were shuffled) or a monoalphabetic substitution (a letter can be replaced by only one other).

If the Index of coincidence is low (close to $ 0.0385 $), i.e. similar to a random text, then the message has probably been crypted using a polyalphabetic cipher (a letter can be replaced by multiple other ones).

The more the coincidence count is low, the more alphabets have been used.

Example: Vigenere cipher with a key of length 4 to 8 letters have an IC of about $ 0.045 \pm 0.05 $

For an non encrypted text, coincidence indexes are

A text where each letter has the same probability of appearance than another, IC is then of $ 1/N $ (where $ N $ is the number of letters in the alphabet)

Example: $ IC = 0.0385 $ for $ N=26 $

Examples of codes in programming languages // Python
def ic(self):
num = 0.0
den = 0.0
for val in self.count.values():
i = val
num += i * (i - 1)
den += i
if (den == 0.0):
return 0.0
else:
return num / ( den * (den - 1))