A combinatorial problem in database security

Let A be a K-dimensional matrix of size d₁ × … × d_k. By a contiguous submatrix B of A we understand the matrix B = {a_i1…i_k}, i_l … i_k ϵ I_l × … × l_k, where I_s is an interval, I_s ⊂ {l, …, d_s, s = l, …, 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 contiguous 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.