A generation method for a kind of linear independent Latin arrays
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.
scientific article; zbMATH DE number 818344