Cryptography Reference
In-Depth Information
working with the same information. The fact that the functions form
a basis mean that the algorithm will handle anything it encounters.
Of course, both of these requirements for the set of functions can
be relaxed, as they are later in the topic, if other steps are taken to
providesomeassurance.
Incidentally, the fact that there are many different basis functions
means that there can be many different unique representations of
the data. There is no reason why the basis functions can't be changed
frequently or shared between sender and receiver. A key could be
used to choose a particular basis function and this could hamper the
work of potential eavesdroppers. [FBS96]
14.5.1 Some Brief Calculus
The foundation of Fourier analysis lies in calculus, so a brief intro-
duction is provided in the original form. If we limit
f
(
x
)
to the range
0
≤ x ≤
2
v
, then the function
f
can be represented as the infinite
series of sines and cosines:
∞
)=
c
2
c
j
sin(
jπx
v
d
j
cos(
jπx
v
f
(
x
+
)+
)
.
j=−∞
Fourier developed a relatively straight forward solution for com-
puting the values of
c
j
and
d
j
, again represented as integrals:
2v
2v
1
v
)cos(
jπx
v
1
v
)sin(
jπx
v
c
j
=
f
(
x
)
dx
d
j
=
f
(
x
)
dx
0
0
The fact that these functions are orthogonal is expressed by this
fact:
cos(
iπx
)cos(
jπx
)
dx
=0
, ∀i
=
j.
.
In the past, many of these integrals were not easy to compute for
many functions,
The integral is
1
if
i
=
j
, and entire branches of mathematics developed
around finding results. Today, numerical integration can solve the
problem easily. In fact, with numerical methods it is much easier to
see the relationship between the functional analysis done here and
the vector algebra that is its cousin. If the function
f
f
is only known
at a discrete number of points
{x
1
,x
2
,x
3
,...,x
n
}
, then the equations
for
c
j
and
d
j
look like dot products:
