The Cycle Structure and Order of the Rail Fence Cipher

Abstract

Any transposition cipher used on a message having length L may be thought of as a permutation on the set {0, 1,…, L − 1}. We derive an explicit formula for the permutation implementing a columnar transposition cipher using C columns on a text of length L. We then specialize to the rail fence cipher and derive a formula for an iterated rail fence cipher. Using these formulae, we examine the order of the rail fence cipher in the group S L . Its order is determined to be the order of 2 in the multiplicative group of units ℤ L *. Finally, we use the mathematical results to describe an attack on an iterated rail fence cipher in which the number of iterations is unknown.