Cryptography Reference
In-Depth Information
7. Change the values of the
k
changeable bits so that they're equal
to
ˆ
b
. Therecipientwillbeabletouse
D
to find
m
without
knowing which of the bits were actually changed.
In practice, it oftenmakes sense towork with subsets of the image
when the implementation computes
ˆ
b
O
(
k
3
)
time.
in
14.5 Fast Fourier Solutions
Many of the spread-spectrum solutions use a branch of mathemat-
ics known
Fourier Analysis
or a more modern revision of this known
as
Wavelet Analysis
. The entire branch is based on the work of Jean-
Baptiste Fourier, who came up with a novel way of modeling a func-
tions using a set of sine and cosine functions. This decomposition
turned out to be quite useful for finding solutions to many differen-
tial equations, and it is often used to solve engineering, chemistry
and physics problems.
The mechanism he proposed is also quite useful for stegano-
graphy because it provides a basic way to embed several signals to-
gether in a larger one. The huge body of scholarship devoted to the
topic makes it easier to test theories and develop tools quickly. One
of the greatest contributions, the so-called
Fast Fourier Transform
is
an algorithm optimized for the digital data files that are often used to
hide information today.
The basic idea is to take a mathematical function,
,andrep-
resent it as the weighted sum of another set of functions,
f
α
1
f
1
+
α
2
f
2
α
3
f
3
...
f
i
is something of an art and
different choices work better for solving different problems. Some of
the most common choices are the basic harmonic functions like sine
and cosine. In fact, the very popular
discrete cosine transform
which
is used in music compression functions like the MP3 and the MPEG
video compression functions uses
. The choice of the values of
f
1
=
cos
(
πx
)
,
f
2
=
cos
(2
πx
)
,
f
3
=
)
, etc. (Figure 14.6 shows the author's last initial recoded
as a discrete cosine transform.) Much research today is devoted to
finding better andmore sophisticated functions that are better suited
to particular tasks. The section on wavelets (Section 14.8) goes into
some of the more common choices.
Much of themathematical foundation of Fourier Analysis is aimed
at establishing several features that make them useful for different
problems. The cosine functions, for instance, are just one set that is
orthogonal
, a term that effectivelymeans that the set is as efficient as
possible. A set of functions is orthogonal if no one function,
cos
(3
πx
f
i
,can
