Let A be a k-dimensional matrix of size d1 × ··· × dk. By a contiguous submatrix B of A we understand the matrix B = {ai1···ik}, i1···ik ∈ I1 × ··· × Ik, where Is is an interval, Is ⊂ {1, ..., ds}, s = 1, ..., k. For a contiguous submatrix B we denote by SUM(B) the sum of all elements of B. The following question has been raised in connection with the security of statistical databases. What is the largest family B of continguous submatrices of A so that knowing the value of SUM(B) for all B in B does not enable one to calculate any of the elements of A? In this paper we show that, for all k, the largest set B is uniquely determined and equals the set of all contiguous submatrices with an even number of elements of A.
