Index of coincidence
The Index of Coincidence measures the probability that two letters selected from a text are identical.
Definition and formula
The Index of Coincidence (IC) is the probability that two letters drawn without replacement from a text are the same. For counts nᵢ and total letter count N, IC = Σ nᵢ(nᵢ−1) / N(N−1). Spaces and punctuation are normally excluded consistently.
How to interpret it
Natural languages have uneven letter frequencies, so their IC is generally higher than that of a uniform random alphabet. Monoalphabetic substitution preserves the frequency counts and therefore the IC. Polyalphabetic encryption mixes several distributions and often lowers the overall value.
Estimating a key period
To test a candidate Vigenère period, divide the ciphertext into columns at that interval and average their IC values. Columns that each resemble one shifted language distribution support the candidate. Expected values depend on language, alphabet, normalization, and text length, so IC is evidence rather than proof.
It indicates an uneven distribution with repeated letters, often compatible with natural language or monoalphabetic substitution, but it does not identify the cipher by itself.
It suggests likely periods. Multiples, divisors, and statistical noise can produce competing peaks, so other tests should confirm the result.
It becomes noisy on short samples because a few letter counts dominate the estimate. Longer ciphertext provides more stable evidence.