Given: columnar transposition ciphertext y;
Find: the transposition width N and transposition τ.
Our plan is to test N as a possible width by computing a Marko score for the adjacency of columns in the ciphertext, assuming each of the N! transpositions of width N are equally likely to have been used.
Testing a width N is formulated as a hypotheses testing problem; for each pair (i, j) with i ≠ j, decide which of the two hypotheses is the most likely to be true.
|,||jth column is read from X immediately after theith column is read from X.|
|,||jth column is not from X immediately after theith column is read from X.|
When ADJ(i, j) is true, the ith and jth columns must be columns (k, k + 1) in X for some k with 0 ≤ k < n−1. As the N! transpositions τ have been chosen with equal probability, the a priori4 probabilities of the hypotheses ADJ(i, j) and are
The ratio of these probabilities is the a priori odds of ADJ(i, j) over
The term ODDS has the same interpretation ...