Cryptography Reference
In-Depth Information
edges of the objects in images or the transitions in audio files,
making it more difficult for the human perception system to
identify them. [AKSK00]
Quantize the Coefficients to Hide Information
Many of the trans-
form hiding methods hide information by adding in a water-
mark vector. The information is extracted by comparing the co-
efficients with all possible vectors and choosing the best match.
This may be practical for small numbers of watermarks, but it
doesn't work well for arbitrary blocks of information.
A more flexible solution is to tweak the coefficients to hide in-
dividual bits. Let
Q
be some quantization factor. An arbitrary
coefficient,
y
i
, is going to fall between
aQ ≤ y
i
≤
(
a
+1)
Q
for
some integer,
a
.Toencodeabit,roundoff
y
i
to the value where
the least significant bit of
a
is that bit. For example, if the bit
to be encoded is zero and
a
=3
,thenset
y
i
=(
a
+1)
Q
=4
Q
.
[KH98]
Any recipient would extract a bit from
y
i
by finding the closest
value of
. If the transform process is completely accurate,
then there will be some integer where
aQ
y
i
.Ifthetransform
and inverse transform introduce some rounding errors, as they
often do, then
aQ
=
y
i
should still be close enough to some value of
aQ
is large enough.
The value of
—if
Q
should be chosen with some care. If it is too
large, then it will lead to larger changes in the value of
Q
y
i
.Ifit
is too small, then it may be difficult to recover the message in
some cases when error intrudes.
Deepa Kundur and
Dimitrios Hatzinakos
describe a
quantization-based
watermark that also
offers tamper detection.
[KH99]
This mechanism also offers some ability to detect tampering
with the image or sound file.
If the coefficients are close to
some value of
, then this might
be the result of some minor changes in the underlying file. If
the changes are small, then the hidden information can still be
extracted. In some cases, the tamper detection can be useful.
A stereo or televisionmay balk at playing back files with imper-
fect watermarks because they would be evidence that someone
was trying to destroy the watermark. Of course it could also be
the result of some imperfect copying process.
aQ
but not exactly equal to
aQ
Hide the Information in the Phase
The Discrete Fourier Transform
produces coefficients with a real and an imaginary value. These
complex values can also be imagined in polar coordinates as
having a magnitude and an angle.
(If
y
i
=
a
+
bi
,then
a
=
