Cryptography Reference
In-Depth Information
lic key algorithms, but it might still be of use for creating watermarks
that prevent copying. A pirate would find little use in adding a signal
that indicated that copying should be forbidden.
More sophisticated attacks may be possible. The authors credit
Teddy Furon for identifying a trial-and-error approach to removing
watermarks with this tool by changing the scale for the part of the
signal in the space defined by the eigenvectors,
{w
1
,...,w
n
}
,with
length
.
The algorithm can be tuned by choosing different values of
λ
M
.
n × n
The authors particularly like permutation matrices because an
matrix can be stored with
1
values. “Multiplication” by a per-
mutation matrix is also relatively easy. Using the matrix designed to
compute the discrete cosine transform is also a good choice because
the computation is done frequently in image and sound manipula-
tion.
This solution is far from perfect and its security is not great. Still,
it is a good example of how a function might be designed to mini-
mize the camouflaging data while amplifying the hidden data. The
process is also keyed so that the value of
n −
M
must be present to ex-
tract the hidden message.
12.4.4 Removing Parts
Many of the algorithms in this topic hide information by making a
number of changes to a number of different locations in the file and
then averaging these changes to find the signal. The proceding sec-
tion (12.4.3), for instance, may add hundreds of thousands of small
values from the watermark eigenvector into the file. The spread-
spectrum-like techniques fromChapter 14 will spread the signal over
many different pixels or units from a sound file. The signal is ex-
tracted by computing a weighted average over all of them.
One insight due to Frank Hartung and Bernd Girod is that the
information extractor does not need to average all of the locations
to extract a signal. [HG97] The algorithms already include a certain
amount of redundancy to defend against either malicious or acci-
dental modifications to the file. If the algorithms are designed to
carry accurate data even in the face of changes to an arbitrary num-
ber of elements, why not arrange for the receiver to skip that arbitrary
number of elements all together?
Consider this algorithm:
1. Create
n
“keys”,
{k
1
,...,k
n
}
.
2. Use cryptographically secure random number generators to
