# An Algorithm for Fast Multiplication of Kaluza Numbers
## 2 Preliminary Remarks
\begin{array}{cccccccccccccccc}b_{13}&b_{19}&-b_{ 3}&b_{7}&-b_{11}&-b_{17}&b_{2}&-b_{6}&-b_{10}&-b_{16}&-b_{22}&b_{26}&-b_{0}&-b_{ 1}&-b_{4}&b_{8}\\ b_{19}&b_{13}&b_{7}&-b_{3}&-b_{17}&-b_{11}&-b_{6}&b_{2}&-b_{16}&-b_{10}&b_{26}&-b_{ 22}&-b_{1}&-b_{0}&b_{8}&-b_{4}\\ -b_{22}&-b_{26}&-b_{10}&b_{16}&-b_{4}&-b_{8}&-b_{0}&b_{1}&-b_{3}&-b_{7}&b_{13}&- b_{19}&b_{2}&b_{6}&-b_{11}&b_{17}\\ -b_{26}&-b_{22}&b_{16}&-b_{10}&-b_{8}&-b_{4}&b_{1}&-b_{0}&-b_{7}&-b_{3}&-b_{19} &b_{13}&b_{6}&b_{2}&b_{17}&-b_{11}\\ b_{11}&b_{17}&-b_{2}&b_{6}&-b_{13}&-b_{19}&b_{3}&-b_{7}&-b_{0}&-b_{1}&-b_{4}&b_{ 8}&-b_{10}&-b_{16}&-b_{22}&b_{26}\\ b_{17}&b_{11}&b_{6}&-b_{2}&-b_{19}&-b_{13}&-b_{7}&b_{3}&-b_{1}&-b_{0}&b_{8}&-b_{ 4}&-b_{16}&-b_{10}&b_{26}&-b_{22}\\ -b_{4}&-b_{8}&-b_{0}&b_{1}&-b_{22}&-b_{26}&-b_{10}&b_{16}&-b_{2}&-b_{6}&b_{11} &-b_{17}&b_{3}&b_{7}&-b_{13}&b_{19}\\ -b_{8}&-b_{4}&b_{1}&-b_{0}&-b_{26}&-b_{22}&b_{16}&-b_{10}&-b_{6}&-b_{2}&-b_{17} &b_{11}&b_{7}&b_{3}&b_{19}&-b_{13}\\ -b_{10}&-b_{16}&b_{22}&-b_{26}&-b_{0}&-b_{1}&b_{4}&-b_{8}&b_{13}&b_{19}&b_{3}&- b_{7}&-b_{11}&-b_{17}&-b_{2}&b_{6}\\ -b_{16}&-b_{10}&-b_{26}&b_{22}&-b_{1}&-b_{0}&-b_{8}&b_{4}&b_{19}&b_{13}&-b_{7} &b_{3}&-b_{17}&-b_{11}&b_{6}&-b_{2}\\ -b_{3}&-b_{7}&-b_{13}&b_{19}&b_{2}&b_{6}&b_{11}&-b_{17}&-b_{22}&-b_{26}&b_{10}& -b_{16}&-b_{4}&-b_{8}&b_{0}&-b_{1}\\ -b_{7}&-b_{3}&b_{19}&-b_{13}&b_{6}&b_{2}&-b_{17}&b_{11}&-b_{26}&-b_{22}&-b_{16} &b_{10}&-b_{8}&-b_{4}&-b_{1}&b_{0}\\ -b_{0}&-b_{1}&b_{4}&-b_{8}&-b_{10}&-b_{16}&b_{22}&-b_{26}&b_{11}&b_{17}&b_{2}& -b_{6}&-b_{13}&-b_{19}&-b_{3}&b_{7}\\ -b_{1}&-b_{0}&-b_{8}&b_{4}&-b_{16}&-b_{10}&-b_{26}&b_{22}&b_{17}&b_{11}&-b_{6}&b _{2}&-b_{19}&-b_{13}&b_{7}&-b_{3}\\ b_{2}&b_{6}&b_{11}&-b_{17}&-b_{3}&-b_{7}&-b_{13}&b_{19}&b_{4}&b_{8}&-b_{0}&b_{1} &b_{22}&b_{26}&-b_{10}&b_{16}\\ b_{6}&b_{2}&-b_{17}&b_{11}&-b_{7}&-b_{3}&b_{19}&-b_{13}&b_{8}&b_{4}&b_{1}&-b_{0} &b_{26}&b_{22}&b_{16}&-b_{10}\end{array} \right],\]  
\[\vec{\textbf{B}}_{16}^{(1)}=\left[\begin{array}{cccccccccccccccc}-b_{15}&-b_{ 21}&-b_{5}&b_{9}&-b_{30}&-b_{31}&-b_{23}&b_{27}&-b_{12}&-b_{18}&b_{24}&-b_{28}&b_ {14}&b_{20}&-b_{25}&b_{29}\\ -b_{21}&-b_{15}&b_{9}&-b_{5}&-b_{31}&-b_{30}&b_{27}&-b_{23}&-b_{18}&-b_{12}&-b_{ 28}&b_{24}&b_{20}&b_{14}&b_{29}&-b_{25}\\ b_{24}&b_{28}&-b_{12}&b_{18}&-b_{25}&-b_{29}&b_{14}&-b_{20}&-b_{5}&-b_{9}&-b_{15} &b_{21}&-b_{23}&-b_{27}&-b_{30}&b_{31}\\ b_{28}&b_{24}&b_{18}&-b_{12}&-b_{29}&-b_{25}&-b_{20}&b_{14}&-b_{30}&-b_{5}&b_{2 1}&-b_{15}&-b_{27}&-b_{23}&b_{31}&-b_{30}\\ -b_{30}&-b_{31}&-b_{23}&b_{27}&-b_{15}&-b_{21}&-b_{5}&b_{9}&-b_{14}&-b_{20}&b_{ 25}&-b_{29}&b_{12}&b_{18}&-b_{24}&b_{28}\\ -b_{31}&-b_{30}&b_{27}&-b_{23}&-b_{21}&-b_{15}&b_{9}&-b_{5}&-b_{20}&-b_{14}&-b_{ 29}&b_{25}&b_{18}&b_{12}&b_{28}&-b_{24}\\ b_{25}&b_{29}&-b_{14}&b_{20}&-b_{24}&-b_{28}&  
The matrices \(\mathbf{B}_{16}^{(-)}\), \(\mathbf{B}_{16}^{(+)}\) and \(\mathbf{B}_{16}^{(1)}\) have similar structures. If we now change the signs of all of the elements of the sixth and seventh rows, as well as all of the elements of the second, third, sixth and seventh columns, to the opposite, then the matrices \(\mathbf{B}_{16}^{(-)}\), \(\mathbf{B}_{16}^{(+)}\) and \(\mathbf{B}_{16}^{(1)}\) will have structures of type \( \begin{bmatrix}\mathbf{A}_{N/2}&\mathbf{B}_{N/2}\\ \mathbf{B}_{N/2}&\mathbf{A}_{N/2}\end{bmatrix} \), which leads to reducing the number of real multiplications during matrix-vector product calculation. We can write the sign transformation matrices for rows \(\mathbf{S}_{16}^{(r)}\) and columns \(\mathbf{S}_{16}^{(c)}\) as:  
\[\mathbf{S}_{16}^{(r)}=\text{diag}\big{(}1,\ \ 1,\ \ 1,\ \ 1,\ \ 1,\ \ 1,\ \ -1,\ \ -1,\ \ 1,\ \ 1,\ \ 1,\ \ 1,\ \ 1,\ \ 1,\ \ 1,\ \ 1 \big{)},\]  
\[\mathbf{S}_{16}^{(c)}=\text{diag}\big{(}1,\ \ 1,\ \ -1,\ \ -1,\ \ 1,\ \ 1,\ \ -1,\ \ -1,\ \ 1,\ \ 1,\ \ 1,\ \ 1,\ \ 1,\ \ 1,\ \ 1,\ \ 1,\ \ 1 \big{)}.\]  
Then, we obtain new standardized matrices:  
\[\mathbf{\tilde{B}}_{16}^{(-)}=\mathbf{S}_{16}^{(r)}\mathbf{B}_{16}^{(-)} \mathbf{S}_{16}^{(c)}= \begin{bmatrix}\mathbf{B}_{8}^{(0)}&\mathbf{B}_{8}^{(1) }\\ \mathbf{B}_{8}^{(1)}&\mathbf{B}_{8}^{(0)}\end{bmatrix}