Computation of the number of representations of the elements of the ring $\mathbb Z/d\mathbb Z$ as a sum of squares

The number of representations of the elements of the ring $\mathbb Z/d\mathbb Z$ as a sum of invertible squares is computed, provided that each square occurs in the sum no more than a fixed number of times. For prime $d$ an exhaustive answer is given in terms of the class number and the fundamental unit of the real quadratic field $\mathbb Q(\sqrt d)$. Bibliography: 5 titles.