Cryptography Reference
In-Depth Information
1
1
1
1
0.8
0.5
0.5
0.5
0.6
0.4
50
100
150
200
250
50
100
150
200
250
50
100
150
200
250
0.2
-0.5
-0.5
-0.5
50
100
150
200
250
-1
-1
-1
Figure 14.5: The first four basic cosine functions used in Fourier se-
ries expansions:
cos(
x
)
,
cos(2
x
)
,
cos(3
x
)
and
cos(4
x
)
.
1
0.5
1
2
3
4
5
6
-0.5
-1
-1.5
Figure 14.6: The author's last initial recreated as a Fourier series
adding together the four functions shown in Figure 14.5:
1
.
0cos(
x
)+
.
5cos(2
x
)
− .
8cos(3
x
)+
.
3cos(4
x
)
.
be represented as the sum of the others. That is, there are no values
of
{α
1
,α
2
,...α
i−1
,α
i+1
,...}
that exist so that
f
i
=
i=j
α
j
f
j
.
The set of cosine functions also forms a
basis
for the set of suffi-
ciently continuous functions. That is, for all sufficiently continuous
functions,
f
, there exists some set of coefficients,
{α
1
,α
2
,...}
,such
that
=
f
α
j
f
j
.
The fact that the set of cosine functions is both orthogonal and a ba-
sis means that there is only one unique choice of coefficients for each
function. In this example, the basis must be infinite to represent all
sufficiently continuous functions, but most discrete problems never
require such precision. For that reason, a solid discussion of what it
means to be “sufficiently continuous” is left out of this topic.
Both of these features are important to steganography. The fact
that each function can be represented only by a unique set of values
{α
1
,α
2
,...}
means that both the encoder and the decoder will be
