A recursive method to calculate the number of solutions of quadratic equations over finite fields

Abstract

The number S m ( α ) {S_m}(\alpha ) of solutions of the quadratic equation \[ x 1 2 + x 2 2 + ⋯ + x m 2 = α ( x i 2 ≠ ± x j 2 for i ≠ j ) x_1^2 + x_2^2 + \cdots + x_m^2 = \alpha \quad (x_i^2 \ne \pm x_j^2\quad {\text {for}}\;i \ne j) \] for given m, with α \alpha and x i {x_i} belonging to a finite field, is studied and a recursive method to compute S m ( α ) {S_m}(\alpha ) is established.