A generation method for a kind of linear independent Latin arrays
type: Item
instance of: scholarly article
title: A generation method for a kind of linear independent Latin arrays
zbMATH Open document ID: 0834.05016
published in: Science in China. Series A
publication date: 1996-03-31T00:00:00Z
review text: An \((n, k)\)-Latin array means an \(n\times nk\) matrix on \(N= \{1,\dots, n\}\) such that each element of \(N\) occurs exactly once in each column and \(k\) times in each row. If \(x, y\in N\) and if the components of column labels of columns of \(A\) in which the elements at rows \(x\) and \(y\) satisfy some \(r\)-ary polynomial with degree \(\leq c\) over \(\text{GF}(2)\), then \(A\) is said to be \(c\)-dependent with respect to \((x, y)\), otherwise \(A\) is said to be \(c\)-independent with respect to \((x, y)\). If \(A\) is \(c\)-dependent with respect to \((x, y)\) for any \(x, y\in N\), then \(A\) is said to be \(c\)-dependent. If \(A\) is \(c\)-independent with respect to \((x, y)\) for any \(x, y\in N\), then \(A\) is said to be \(c\)-independent. If \(A\) is \(c\)-dependent and not \((c-1)\)-dependent, then \(c\) is said to be the dependent degree of \(A\), denoted by \(c_A\). If \(A\) is \(c\)-independent and not \((c- 1)\)-independent, then \(c\) is said to be the independent degree of \(A\), denoted by \(I_A\). Clearly, \(I_A\leq c_A- 1\text{ and } c_A\leq \min\{c\mid 1+ (\begin{smallmatrix} r\\ 1\end{smallmatrix})+\cdots+ (\begin{smallmatrix} r\\ c\end{smallmatrix})\geq k+ 1\},\text{ where }r= [\log_2 nk].\) The paper discusses the generation problem of Latin arrays with independent degree \(\geq 1\). The authors give a method to generate such Latin arrays, that is, shifting invertible Boolean vector functions with independence degree \(\geq 1\). Also this paper discusses the generation problem of linearly independent and invertible Boolean vector functions, and gives a generation method, based on a truth table, to generate all linearly independent and invertible Boolean vector functions. The authors also discuss some applications of their results in cipher design.
Mathematics Subject Classification ID: 05B15
Mathematics Subject Classification ID: 94C30
zbMATH DE Number: 818344
zbMATH Keywords: polynomial
zbMATH Keywords: dependent degree
zbMATH Keywords: independent degree
zbMATH Keywords: generation problem
zbMATH Keywords: Latin arrays
zbMATH Keywords: Boolean vector functions
zbMATH Keywords: generation method
author: Tao Renji
author: Shi-Hua Chen
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 818344