MRL’s paper is confusing, it’s better to read the source code.
There is a new output we want to make a “range proof”.
$10$ is the amount, $a$ is the secret key. G and H are different base point.
We split it in four, we get:
$C_0=a_0G+0 \times 1H$
$C_1=a_1G+1 \times 2H$
$C_2=a_2G+0 \times 4H$
$C_3=a_3G+1 \times 8H$
because $2+8=10$. $a_i$ is random.
For the first line, we get $(C_0,C_0 - 1 \times 1H)$ these two points.
We sign a ring signature on these two points, a ring contains only two points.
$L_0= \alpha G$
$\alpha$ is random.
$H()$ is a hash function to covert a point to scalar.
$s_1$ is random, $P_1$ is the second point.
$s_0= \alpha -q_0a_0$ since $L_0= \alpha G=s_0G+q_0P_0=( \alpha G -q_0a_0)G+q_0P_0$
It’s easy to verify this signature because:
The second line is similar but we should change the order of $(P_0,P_1)$ because we only know the second point’s secret key.
At last we make four range proof.
In practice, the code is a little different for space-saving.