The IC of a text with the counts written out, Friedman's estimate, and the average IC of the columns for every period, to find a Vigenère key length.
Procedure
How it works
The index of coincidence is the probability that two letters taken from two different positions of a text are equal. Counting $n_i$ letters of each kind among $N$, it is $\mathrm{IC} = \sum_i n_i (n_i - 1) / (N (N - 1))$. For English it is close to $\kappa_p = \sum_i p_i^2 \approx 0.066$; for letters chosen uniformly at random it is $\kappa_r = 1/26 \approx 0.0385$.
Relabelling the letters, as a substitution does, leaves the IC unchanged. A Vigenère cipher, however, uses $L$ alphabets in turn, and its IC falls toward $\kappa_r$. William Friedman turned this into an estimate of $L$ in the 1920s.
The more reliable test splits the ciphertext into $p$ columns (every $p$-th letter) and averages the IC of the columns. When $p$ equals the key length, each column was enciphered with a single Caesar shift, so its IC jumps back to the English value. Multiples of the key length score high as well, so the smallest high period is the answer.
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