We consider two existing FFT-based fast-convolution iterative solution techniques for the scalar T-matrix multiple-scattering equation [1]. The use of the FFT operation requires field values be expressed on a regular Cartesian grid and the two techniques differ in how to go about achieving this. The first technique [6, 7] uses the non-diagonal translation operator [1, 9] of the spherical multipole field, while the second method [11] uses the diagonal translation operator of Rokhlin [10]. Because of its use of the non-diagonal translator, the first technique has been thought to require a greater number of spatial convolutions than the second technique. We establish that the first method requires only half as many convolution operations as the second method for a comparable numerical accuracy and demonstrate, based on an actual CPU time comparison, that it can therefore perform iterations faster than the second method. We then consider the respective symmetry relations of the non-diagonal and diagonal translators and discuss a memory-reduction procedure for both FFT-based methods. In this procedure, we need to store only the minimum sets of near-field and far-field translation operators and generate missing elements on the fly using the symmetry relations. We show that the relative cost of generating the missing elements becomes smaller as the number of scatterers increases.