Additive Differential of XOR

Question

I'm studying about Additive differential of XOR

I saw two papers that are "H.Lipmaa et al., On the Additive Differential Probability of Exclusive-or" and "V.Velichkov et al., The Additive Differential Probability of ARX"

In this two paper, Limaa et al. and Velichkov et al. compute the probability of additive differential of xor in similar way.

But there is one difference between them.

That is matrix $A_i$ is different.

For example $A_{000}$ in first paper is $A_{000}=\begin{pmatrix} 4& 0& 0& 1& 0& 1& 1& 0 \\0& 0& 0& 1& 0& 1& 0& 0 \\ \vdots & & & \cdots & \cdots & & & \vdots \\ 0& 0& 0& 0& 0& 0& 0& 0 \end{pmatrix}$

and $A_{000}$ in second paper is $A_{000}=\begin{pmatrix} 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\0& 1& 0& 0& 0& 0& 0 & 0 \\ \vdots & & & \cdots & \cdots & & & \vdots \\ 0& 0& 0& 0& 0& 0& 0& 1 \end{pmatrix}$

I hope someone can tell me why there exists a difference.

Answer 1

The OP has provided the answer in a comment.

"In the second paper(V.Velichkov et al., The Additive Differential Probability of ARX), the table 2's order is not same as the original order.

For example, $S[i] =0$ when $(s_1[i],s_2[i],s_3[i])=(0,0,-1)$. So, there exists a difference.