Fixed Points of Columnar Transpositions

Abstract

A columnar transposition cipher encrypts messages by inserting the plaintext into a rectangular grid one row at a time and then removing the contents of the grid one column at a time to produce the ciphertext. The grid is created by choosing a natural number C and then making a grid with C columns and [L/C] rows, where L is the length of the message. We denote the columnar transposition with these parameters by πC,L. It is evident that πC,L is a permutation on the positions of characters in the message. In this paper, we prove several results about the fixed points of this permutation for general C and L. We first locate positions in the message that are always fixed and prove that no column in the enciphering grid can contain more than one fixed point. We then give several results that provide a method of iteratively checking columns for nontrivial fixed points. Our results culminate in an algorithm for finding the fixed points of πC,L whose time complexity depends only on the number of columns chosen.