Fast computation of the number of solutions to x₁²+cdots+x_k² equiv λ pmod{n}

In this paper we study the multiplicative function $\rho_{k,\lambda}(n)$ that counts the number of incongruent solutions of the equation $x_1^2+\cdots+x_k^2 \equiv \lambda\pmod{n}$. In particular we give closed explicit formulas for $\rho_{k,\lambda}(p^s)$ with a arithmetic complexity of constant order.