A.2.1 Elementary matrices (Chevalley generators) for orthogonal group Oþð Þ d; k of even size

Following the theory of root system in a simple Lie algebra, we index rows by 1, 2, …, l, � 1, � 2, …, � l. For t∈ k, the elementary matrices are defined as follows (Tables A1 and A2):

Let us note the effect of multiplying g by elementary matrices. We write


The MOR Cryptosystem in Classical Groups with a Gaussian Elimination Algorithm… DOI: http://dx.doi.org/10.5772/intechopen.84663

#### Table A1.

algorithm for all orthogonal groups over a perfect field of even

A.2 Gaussian elimination for matrices of even size—orthogonal group

Note that any isometry <sup>g</sup> satisfies Tgβ<sup>g</sup> <sup>¼</sup> <sup>β</sup>. The main reason our algorithm

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

The above equation implies among other things, TCAþTAC <sup>¼</sup> 0. This implies

AC is symmetric. The working principle of our algorithm is simple—

AC is skew-symmetric. In an almost identical way, one can show, if g is

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

Next we present the elementary matrices for the respective groups and then the

Following the theory of root system in a simple Lie algebra, we index rows by 1, 2, …, l, � 1, � 2, …, � l. For t∈ k, the elementary matrices are defined as follows

Let us note the effect of multiplying g by elementary matrices. We write

A.2.1 Elementary matrices (Chevalley generators) for orthogonal group

TCAþTAC TCBþTAD TDAþTBC TDBþTBD 

�Il 0 

Il 0 

C D 

> <sup>¼</sup> <sup>0</sup> Il Il 0

AC. The problem is, for arbitrary A and C, it is not easy to use

.

.

, where A, B, C, and D are

(A.1)

• Furthermore, we have Gaussian elimination algorithm for orthogonal groups that are given by the above bilinear forms or quadratic forms over arbitrary fields. This algorithm also works for bilinear or quadratic forms

characteristics.

that are equivalent to the above forms.

Modern Cryptography – Current Challenges and Solutions

Recall that the bilinear forms β are the following:

• For symplectic group, Spð Þ <sup>d</sup>; <sup>k</sup> , <sup>d</sup> <sup>¼</sup> <sup>2</sup>l, and <sup>β</sup> <sup>¼</sup> <sup>0</sup> Il

• For orthogonal group, Oþð Þ <sup>d</sup>; <sup>k</sup> , <sup>d</sup> <sup>¼</sup> <sup>2</sup>l, and <sup>β</sup> <sup>¼</sup> <sup>0</sup> Il

works is the following: Recall that a matrix <sup>g</sup> <sup>¼</sup> A B

some usual calculations, for orthogonal group it becomes

that <sup>T</sup>

symplectic, <sup>T</sup>

use the symmetry of <sup>T</sup>

generators for the respective groups.

Oþð Þ d; k of even size

(Tables A1 and A2):

104

row-column operations in a tabular form.

Oþð Þ d; k and symplectic group

Elementary matrices for Oþð Þ 2l; k .


#### Table A2.

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


#### Table A3.

Elementary matrices for Spð Þ 2l; k .

$$\mathbf{g} \in \mathbf{O}^+(2l, k) \text{ as } \mathbf{g} = \begin{pmatrix} A & B \\ C & D \end{pmatrix} \text{, where } A, B, C, \text{ and } D \text{ are } l \times l \text{ matrices.} $$

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

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 <sup>g</sup> <sup>∈</sup>Sp 2ð Þ <sup>l</sup>; <sup>k</sup> as <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 <sup>l</sup> � <sup>l</sup> matrices (Table A4).

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

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 D as well.


However there is one difference between orthogonal and symplectic groups. In

wi ¼ xi,�ið Þ1 x�i,ið Þ �1 xi,�ið Þ1 . In the case of orthogonal groups, that is not the case. This is clear that the elementary matrices come from the Chevalley generators and those generates Ω, the commutator of the orthogonal group. All matrices in Ω have determinant 1. However wi has determinant �1. So we must add wi as an elemen-

Remark A.2 This algorithm proves every element in the symplectic group is of determinant 1. Note the elementary matrices for the symplectic group are of determinant 1, and we have an algorithm to write any element as product of elementary matrices. So

Remark A.3 This algorithm proves if X is an element of a symplectic group then so is

X. The argument is similar to the above; here we note that the transpose of an elementary

A.3 Gaussian elimination for matrices of odd size—the odd-orthogonal

In this case, matrices are of odd size and there is only one family of group to consider; it is the odd-orthogonal group O 2ð Þ l þ 1; k . This group will be referred to

Following the theory of Lie algebra, we index rows by 0, 1, …, l, � 1, …, � l.

Elementary matrices for the odd-orthogonal group in even characteristics differ from that of odd characteristics. In above table we made that distinction and listed them separately in different rows according to the characteristics of k. If char(k) is even, we can construct the elements wi, which interchanges the ith row with �i

wi ¼ ð Þ I þ e0,i þ e�i,i ð Þ I þ e0,�<sup>i</sup> þ ei,�<sup>i</sup> ðI þ e0,i þ e�i,iÞ ¼ I þ ei,i þ e�i,�<sup>i</sup> þ ei,�<sup>i</sup> þ e�i,i:

th and 0th row in odd-orthogonal group as follows:

<sup>i</sup> 6¼ <sup>j</sup>

i , j

i , j

<sup>2</sup>ei,�<sup>i</sup> 1≤i ≤l

<sup>2</sup>e�i,i <sup>1</sup>≤<sup>i</sup> <sup>≤</sup><sup>l</sup>

<sup>2</sup>ei,�<sup>i</sup> <sup>1</sup>≤<sup>i</sup> <sup>≤</sup><sup>l</sup>

<sup>2</sup>e�i,i 1≤i ≤l

Otherwise, we can construct wi, which interchanges the ith row with �i

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

x�i,jð Þt I þ t e�i,j � e�j,i

x0,ið Þt I þ tð�2e�i,<sup>0</sup> þ e0,iÞ � t

x0,ið Þt I þ te0,i þ t

th

th row

A.3.1 Elementary matrices (Chevalley generators) for Oð Þ 2l þ 1; k

symplectic group, wi can be generated by elementary matrices because

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

tary matrix for Oþð Þ 2l; k .

T

group

row as follows:

Table A5.

107

Elementary matrices for Oð Þ 2l þ 1; k .

with a sign change in i

as the odd-orthogonal group.

this proves that the determinant is 1.

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

matrix in symplectic groups is an elementary matrix.

These elementary matrices are listed in Table A5.

th, � <sup>i</sup>

Char(k) Elementary matrices Both xi,jð Þt I þ t ei,j � e�j,�<sup>i</sup>

Odd xi,0ð Þt I þ tð Þ� 2ei,<sup>0</sup> � e0,�<sup>i</sup> t

Even xi,0ð Þt I þ te0,�<sup>i</sup> þ t

#### Table A4.

The row-column operations for symplectic groups.


diag 1; …; 1; λ; 1; …; 1; λ�<sup>1</sup> , where λ∈ k�.

Step 5: (Only for symplectic groups) Reduce the λ to 1 using Lemma A.1.

Lemma A.1 For Sp 2ð Þ <sup>l</sup>; <sup>k</sup> , the element diag 1; …; <sup>1</sup>; <sup>λ</sup>; <sup>1</sup>; …; <sup>1</sup>; <sup>λ</sup>�<sup>1</sup> is a product of elementary matrices.

Proof. Observe that

ð Þ <sup>I</sup> <sup>þ</sup> <sup>λ</sup>el,�<sup>l</sup> <sup>I</sup> � <sup>λ</sup>�<sup>1</sup> e�l,l ð Þ¼ <sup>I</sup> <sup>þ</sup> <sup>λ</sup>el,�<sup>l</sup> <sup>I</sup> � el,l � <sup>e</sup>�l,�<sup>l</sup> <sup>þ</sup> <sup>λ</sup>el,�<sup>l</sup> � <sup>λ</sup>�<sup>1</sup> e�l,l and denote it by wlð Þλ , and then the diagonal element is wlð Þλ wlð Þ �1 .

Remark A.1 As we saw in the above algorithm, we will have to interchange ith and �i th rows for i <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, …, l. This can be done by pre-multiplying with a suitable matrix.

Let I be the 2l � 2l identity matrix over k. To swap ith and �ith row in Oþð Þ 2l; k , swap ith and �ith rows in the matrix I. We will call this matrix wi. It is easy to see that this matrix wi is in Oþð Þ 2l; k and is of determinant �1. Pre-multiplying with wi does the row interchange we are looking for.

In the case of symplectic group Sp 2ð Þ l; k , we again swap two rows ith and �ith in I. However we do a sign change in the ith row and call it wi. Simple computation with our chosen β shows that the above matrices are in Oþð Þ 2l; k and Sp 2ð Þ l; k , respectively.

The MOR Cryptosystem in Classical Groups with a Gaussian Elimination Algorithm… DOI: http://dx.doi.org/10.5772/intechopen.84663

However there is one difference between orthogonal and symplectic groups. In symplectic group, wi can be generated by elementary matrices because wi ¼ xi,�ið Þ1 x�i,ið Þ �1 xi,�ið Þ1 . In the case of orthogonal groups, that is not the case. This is clear that the elementary matrices come from the Chevalley generators and those generates Ω, the commutator of the orthogonal group. All matrices in Ω have determinant 1. However wi has determinant �1. So we must add wi as an elementary matrix for Oþð Þ 2l; k .

Remark A.2 This algorithm proves every element in the symplectic group is of determinant 1. Note the elementary matrices for the symplectic group are of determinant 1, and we have an algorithm to write any element as product of elementary matrices. So this proves that the determinant is 1.

Remark A.3 This algorithm proves if X is an element of a symplectic group then so is T X. The argument is similar to the above; here we note that the transpose of an elementary matrix in symplectic groups is an elementary matrix.

### A.3 Gaussian elimination for matrices of odd size—the odd-orthogonal group

In this case, matrices are of odd size and there is only one family of group to consider; it is the odd-orthogonal group O 2ð Þ l þ 1; k . This group will be referred to as the odd-orthogonal group.
