Block substitutions using orthomorphic mappings
Let \(\mathbb{Z}^n_2\) be the group of binary \(n\)-tuples under coordinatewise addition modulo 2. An orthomorphism or complete mapping of \(\mathbb{Z}^n_2\) is a bijective mapping \(R:\mathbb{Z}^n_2\to\mathbb{Z}^n_2\) such that \(\{x+R(x)\}_{x\in\mathbb{Z}^n_2}=\mathbb{Z}^n_2\). The main result of the paper is a characterization of orthomorphisms by the property that each maximal subgroup of \(\mathbb{Z}^n_2\) is mapped half into itself and half into its complement. After clearing up further details (number of fixed points, non-linearity etc.) orthomorphisms could be used as cipher functions for messages of binary \(n\)-tuples.
scientific article; zbMATH DE number 778908