Mono-alphabetic substition

Associate letters of the english alphabet with numbers 0,...,25. Let $p_{i}$ , for $0 \leq i \leq 25$ denote probability of ith letter in normal English text
Simple calculation gives $\sum_{i = 0}^{25} p_{i}^{2} \approx 0.065$
Say we are given some ciphertext and let $q_{i}$ denote the probability of the ith letter. ( $q_{i}$ is simple the number of occurences of the ith letter divided by the length of the ciphertext). If the key is k then we can expect that $q_{i + k}$ should be roughly equal to $p_{i}$ for every $i$ [They are using i+k instead of the more cumbersome [i+k mod 26]]. Equivalent if we compute

📝 Notes