Cryptography Reference
In-Depth Information
Figure 14.13: Another solution is to order the coefficients in signifi-
cance. This figure shows the popular zig-zag method to order coef-
ficients from a two-dimensional transform. The ones in the upper
lefthand corner correspond to the lowest frequencies and are thus
the most significant for the eye.
most significant parts of the data stream. They are the frequencies
that have the most “energy” or that do the most for carrying the in-
formation about the final image.
Cox and colleagues suggest that hiding the information in the
largest coefficients may sound counterintuitive, but it is the only
choice. At first glance, the most logical place to hide the data is in
the noise— that is, the smallest coefficients. But this noise is also the
most likely to be modified by compression, printing, or using a less
than perfect conversion process. The most significant parts of the
signal, on the other hand, are unlikely to be damaged without dam-
aging the entire signal.
This philosophy has many advantages. The data is spread out
over numerous data elements. Even if several are changed or deleted,
the information can be recovered. Cox and colleagues demonstrate
that the images carrying this watermark can survive even after being
printed and scanned in again. Of course, the bandwidth is also signif-
icantly smaller than other solutions like tweaking the least signficant
bit.
Their algorithm uses these steps:
1. Use a DCT or FFT to analyze the data.
2. Choose the
{y
0
,y
1
,y
2
,...y
k−1
}
for simplicity. The smaller coefficients are ignored. The first co-
k
largest coefficients and label them
