## 4.6 COINCIDENCE

A *coincidence* occurs at the ith position in two samples of plaintext

if . If the length *n* of the samples are the same, the *kappa*-value *κ*[*x*^{(1)}, *x*^{(2)}] is the total number of coincidences

The *normalized kappa*-value *κ**[*x*^{(1)}, *x*^{(2)}] is the average number of coincidences per letter

How many coincidences can one expect in typical plaintext? If the plaintext is generated by the language model consisting of independent and identically distributed random variables with distribution as specified in Equation (4.2), then a coincidence occurs at the *i*th position of two samples *X*^{(1)} and *X*^{(2)} plaintext with probability

The expected number of coincidences is

where *s*_{2} ≈ 0.06875 using the English 1-gram probabilities in Table 4.3. The values of *s*_{2} in some languages are given in Table 4.6. We can use the coincidence rate to detect if two samples of ciphertext result from the same or different ...