A.3.2 Gaussian elimination for O 2ð Þ l þ 1; k

wi ¼ x0,ið Þ �1 xi,0ð Þ1 x0,ið Þ¼ �1 I � 2e0,<sup>0</sup> � ei,i � e�i,�<sup>i</sup> � ei,�<sup>i</sup> � e�i,i:

g ¼

0

B@

α X Y EAB FCD 1

(odd) ith↦ith þ 2t0th � t

(even) ith↦ith þ t

wi (even)

Table A6.

108

(even) ð Þ �i th↦ð Þ �i th þ t

The row-column operations for Oð Þ 2l þ 1; k .

(odd) with a sign change in ith, � ith and 0th rows

Interchange ith and ð Þ �i th row wi

(odd) ð Þ �i th↦ð Þ �i th � 2t0th � t

ces, E and F are l � 1 matrices, α∈ k and β ¼

Modern Cryptography – Current Challenges and Solutions

tion Tgβ<sup>g</sup> <sup>¼</sup> <sup>β</sup>, we get the following relations:

The Gaussian elimination algorithm for O 2ð Þ l þ 1; k follows the earlier algorithm for symplectic and even-orthogonal group closely, except that we need to take care of the zero row and the zero column. We write an element g ∈ Oð Þ 2l þ 1; k as

CA, where A, B, C, and <sup>D</sup> are <sup>l</sup> � <sup>l</sup> matrices, <sup>X</sup> and <sup>Y</sup> are 1 � <sup>l</sup> matri-

0

B@

Let us note the effect of multiplying g by elementary matrices (Table A6).

ER1 ith↦ith þ tjth row EC1 jth↦jth þ tith column (both) �jth↦ � jth � tð Þ �i th row (both) �ith↦ � ith � tð Þ �j th column ER2 ith↦ith þ tð Þ �j th row EC2 �ith↦ � ith � tjth column (both) jth↦jth � tð Þ �i th row (both) �jth↦ � jth þ tith column ER3 �ith↦ � ith � tjth row EC3 jth↦jth þ tð Þ �i th column (both) �jth↦ � jth þ tith row (both) ith↦ith � tð Þ �j th column ER4 0th↦0th � tð Þ �i th row EC4 0th↦0th þ 2tith column

ER5 0th↦0th þ tith row EC5 0th↦0th � 2tð Þ �i th column

ER6 0th↦0th þ tð Þ �i th row EC6 ð Þ �i th↦ð Þ �i th þ t0th þ t

<sup>2</sup>ð Þ �<sup>i</sup> th row (even) ER7 0th↦0th þ tith row EC7 ith↦ith þ t0th þ t

<sup>2</sup>ith row (even) wi Interchange ith and ð Þ �i th rows wi Interchange ith and ð Þ �i th column

(even)

Row operations Column operations

<sup>2</sup>ð Þ �<sup>i</sup> th row (odd) ð Þ �<sup>i</sup> th↦ð Þ �<sup>i</sup> th � <sup>t</sup>0th � <sup>t</sup>

<sup>2</sup>ith row (odd) <sup>i</sup>th↦ith <sup>þ</sup> <sup>t</sup>0th � <sup>t</sup>

1

<sup>2</sup>TXXþTACþTCA <sup>¼</sup> <sup>0</sup> (A.2)

<sup>2</sup>αTXþTAFþTCE <sup>¼</sup> <sup>0</sup> (A.3)

<sup>2</sup>αYþTEDþTFB <sup>¼</sup> <sup>0</sup> (A.4)

<sup>2</sup>TXYþTADþTCB <sup>¼</sup> Il (A.5)

CA. Then from the condi-

<sup>2</sup>ith column

<sup>2</sup>ith column

<sup>2</sup>ð Þ �<sup>i</sup> th column

<sup>2</sup>ð Þ �<sup>i</sup> th column

(odd) with a sign change in ith, � ith and 0th columns

Interchange ith and ð Þ �i th column

	- 1. If r ¼ l then C becomes zero matrix.
	- 2. If r , l then interchange all zero rows of A with corresponding rows of C using wi so that the new C becomes a zero matrix.

Thus the matrix g reduces to diagð Þ �1; 1; …; λ; 1; …; λ , where λ∈k�.

Remark A.4 Let k be a perfect filed of characteristics 2. Note that we can write the diagonal matrix diag 1; …; 1; λ; 1; …; 1; λ�<sup>1</sup> � � as a product of elementary matrices as follows:

diag 1; …; <sup>1</sup>; <sup>λ</sup>; <sup>1</sup>; …; <sup>1</sup>; <sup>λ</sup>�<sup>1</sup> � � <sup>¼</sup> xl,�<sup>l</sup>ð Þ<sup>t</sup> <sup>x</sup>�l,l �<sup>t</sup> �<sup>1</sup> ð Þxl,�<sup>l</sup>ð Þ<sup>t</sup> , where <sup>t</sup> <sup>2</sup> <sup>¼</sup> <sup>λ</sup>, and hence we can reduce the matrix g to identity.

#### A.4 Gaussian elimination in twisted orthogonal groups

In this section we present a Gaussian elimination algorithm for twisted orthogonal groups. The size of the matrix is even; the bilinear form used is c<sup>0</sup> from Section 3.
