Understanding the marking algorithm

The most natural step to perform with these data is to analyze the difference $D$ between $AM$ and $A$, that is, the quantity which is actually added to the unmarked version.

We performed an autocorrelation on $D$, which is shown on figure 2. The regularly spaced peaks indicate that the signal is periodic: we measured this period $P$, and obtained:

$$P = 1470 = \frac{1}{30}$$ (1)

Then, we compared two successive periods by making their ratios. The graph of the ratio was a stair function, with $10$ different stairs. Figure 3 shows this graph for a specific couple of periods.

The stair structure led us to understand that the same pattern is repeated every $1470$ samples but is multiplied by a different factor every $147$ samples. Let us denote by $w$ this original pattern.

What we know so far, is that in order to compute the $i-th$ chunk (of $1470$ samples) of the final mark that is going to be added to the original song, one has to compute:

$$finalmark_i = \left[ \begin{array}{c} \alpha(s,w,i,1)\\ \alpha(s,w,i,2)\\ \vdots\\ \alpha(s,w,i,10) \end{array} \right] w$$ (2)

where $\alpha$ is a (possibly probabilistic) function depending on the original song $s$, the original pattern $w$, the index of the computed chunk $i$, and the subdivision in this chunk.

We understood that $\alpha$ was essentially the norm of the corresponding $147$ sample long chunk:

$$\alpha(s,w,i,j) = \beta(s,w,i,j) \vert\vert s_i[j]\vert\vert$$

It is quite natural for $\alpha$ to be proportional to this norm. Doing this allows to hide more information when the signal is stronger. Unfortunately, we were not able to exactly figure out the $\beta$ function. $\beta$ probably takes into account the fact that the final result must be between $-1$ and $1$, and also perhaps a psychoacoustic model. We also observed that $\beta$ seems to be the product of a slowly varying function, and of a constant which changes every second. However, we were not able to use these observations to improve our attack.