Recall that the bilinear forms β are the following:


Note that any isometry <sup>g</sup> satisfies Tgβ<sup>g</sup> <sup>¼</sup> <sup>β</sup>. The main reason our algorithm works is the following: Recall that a matrix <sup>g</sup> <sup>¼</sup> A B C D , where <sup>A</sup>, <sup>B</sup>, <sup>C</sup>, and <sup>D</sup> are matrices of size <sup>l</sup>, is orthogonal or symplectic if Tgβ<sup>g</sup> <sup>¼</sup> <sup>β</sup> for the respective <sup>β</sup>. After some usual calculations, for orthogonal group it becomes

$$
\begin{pmatrix}
\begin{array}{cc}
\,^T \mathbf{C} \mathbf{A} + \,^T \mathbf{A} \mathbf{C} & \,^T \mathbf{C} \mathbf{B} + \,^T \mathbf{A} \mathbf{D} \\
\,^T \mathbf{D} \mathbf{A} + \,^T \mathbf{B} \mathbf{C} & \,^T \mathbf{D} \mathbf{B} + \,^T \mathbf{B} \mathbf{D}
\end{pmatrix} = \begin{pmatrix}
\mathbf{0} & I\_l \\
I\_l & \mathbf{0}
\end{pmatrix} \\
\end{pmatrix} \tag{A.1}
$$

<sup>g</sup> <sup>∈</sup> <sup>O</sup>þð Þ <sup>2</sup>l; <sup>k</sup> as <sup>g</sup> <sup>¼</sup> A B

Elementary matrices for Spð Þ 2l; k .

The row-column operations for Oþð Þ 2l; k .

Table A1.

Table A2.

Table A3.

Elementary matrices for Oþð Þ 2l; k .

<sup>g</sup> <sup>∈</sup>Sp 2ð Þ <sup>l</sup>; <sup>k</sup> as <sup>g</sup> <sup>¼</sup> A B

D as well.

105

C D , where A, B, C, and <sup>D</sup> are <sup>l</sup> � <sup>l</sup> matrices.

Step 1: Use ER1 and EC1 to make A into a diagonal matrix. This makes A into a diagonal matrix and changes other matrices A, B, C, and D. For the sake of notational convenience, we keep calling these changed matrices as A, B, C, and

C D , where <sup>A</sup>, <sup>B</sup>, <sup>C</sup>, and <sup>D</sup> are <sup>l</sup> � <sup>l</sup> matrices (Table A4).

xi,�<sup>i</sup>ð Þt I þ tei,�<sup>i</sup> 1≤i ≤l x�i,ið Þt I þ te�i,i 1≤i ≤l

A.2.2 Elementary matrices (Chevalley generators) for symplectic group

Char(k) Elementary matrices

DOI: http://dx.doi.org/10.5772/intechopen.84663

Both xi,�<sup>j</sup>ð Þt I þ t ei,�<sup>j</sup> � ej,�<sup>i</sup>

xi,jð Þt I þ t ei,j � e�j,�<sup>i</sup>

The MOR Cryptosystem in Classical Groups with a Gaussian Elimination Algorithm…

x�i,jð Þt I þ t e�i,j � e�j,i

ER1 ith↦ith þ tjth row EC1 jth↦jth þ tith column

ER2 ith↦ith þ tð Þ �j th row EC2 �ith↦ � ith � tjth column

ER3 �ith↦ � ith � tjth row EC3 jth↦jth þ tð Þ �i th column

Char(k) Elementary matrices

Both xi,�<sup>j</sup>ð Þt I þ t ei,�<sup>j</sup> þ ej,�<sup>i</sup>

wi Interchange ith and ð Þ �i th row Interchange ith and ð Þ �i th column

xi,jð Þt I þ t ei,j � e�j,�<sup>i</sup>

x�i,jð Þt I þ t e�i,j þ e�j,i

<sup>i</sup> 6¼ <sup>j</sup>

i , j

i , j

<sup>i</sup> 6¼ <sup>j</sup>

i , j

i , j

wi I � ei,i � e�i,�<sup>i</sup> þ ei,�<sup>i</sup> þ e�i,i 1≤i ≤l

Row operations Column operations

�jth↦ � jth � tð Þ �i th row �ith↦ � ith � tð Þ �j th column

jth↦jth � tð Þ �i th row �jth↦ � jth þ tith column

�jth↦ � jth þ tith row ith↦ith � tð Þ �j th column

For t ∈k, the elementary matrices are defined as follows (Table A3): Let us note the effect of multiplying g by elementary matrices. We write

A.2.3 Gaussian elimination for Sp 2ð Þ l; k and Oþð Þ 2l; k

The above equation implies among other things, TCAþTAC <sup>¼</sup> 0. This implies that <sup>T</sup> AC is skew-symmetric. In an almost identical way, one can show, if g is symplectic, <sup>T</sup> AC is symmetric. The working principle of our algorithm is simple use the symmetry of <sup>T</sup> AC. The problem is, for arbitrary A and C, it is not easy to use this symmetry. In our case we were able to reduce A to a diagonal matrix, and then it is relatively straightforward to use this symmetry. We will explain the algorithm in details later. First of all, let us describe the elementary matrices and the rowcolumn operations for orthogonal and symplectic groups. The genesis of these elementary matrices lies in the Chevalley basis of simple Lie algebras. We will not go into details of Chevalley's theory in this appendix. Furthermore, we do not need to, the algorithm that we produce will show that these elementary matrices are generators for the respective groups.

Next we present the elementary matrices for the respective groups and then the row-column operations in a tabular form.
