Block substitutions using orthomorphic mappings
type: Item
instance of: scholarly article
title: Block substitutions using orthomorphic mappings
zbMATH Open document ID: 0863.20012
author: Lothrop Mittenthal
published in: Advances in Applied Mathematics
publication date: 1997-04-15T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 94A60
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D25
Mathematics Subject Classification ID: 20B35
zbMATH DE Number: 778908
zbMATH Keywords: complete mappings
zbMATH Keywords: orthomorphisms
zbMATH Keywords: maximal subgroups
zbMATH Keywords: fixed points
MaRDI profile type: Publication
full work available at URL: aama.1995.1003
OpenAlex ID: W2071789297
DOI: 10.1006/AAMA.1995.1003
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description: scientific article; zbMATH DE number 778908