Cryptography Reference
In-Depth Information
The first element of (
a, b
) is called the
real part
of the complex number,
whereas the second element of (
a, b
) is called the
imaginary part
.Thispartis
usually written as a multiple of
i
=
√
−
1, meaning that the imaginary part of
a
+
bi
is written as
b
(instead of
bi
).
./
Figure 3.1
A function
f
:
X → Y
.
In this topic, we assume some familiarity with functions and function families.
As illustrated in Figure 3.1, a
function
f
:
X
→
Y
is a mapping from a
domain
X
to a
codomain
Y
assigning to every
x
Y
.The
range
of
f
is
the subset of values of
Y
that are actually reached by the function (i.e.,
f
(
X
)
∈
X
a unique
f
(
x
)
∈
⊆
Y
).
A function
f
:
X
→
Y
may be injective, surjective, or bijective.
•
The function
f
is
injective
(or
one to one
)ifforall
x
1
,x
2
∈
X
it holds that
x
1
=
f
(
x
2
) (i.e., if two preimages are different, then the
corresponding images are also different).
=
x
2
⇒
f
(
x
1
)
•
The function
f
is
surjective
(or
onto
)ifforall
y
X
with
y
=
f
(
x
), meaning that
f
(
X
)=
Y
(i.e., the codomain and the range are the
same).
∈
Y
there is an
x
∈
•
The function
f
is
bijective
if it is both injective and surjective.
Y
that takes a key as an additional
input parameter, then we are talking about function families. Formally, a
function
family
is a mapping
If we consider a set of functions
f
:
X
→
F
:
K
×
X
→
Y
where
X
and
Y
are the domain and codomain of the functions, and
K
is a set of
possible keys. For every
k
∈
K
,themap
f
k
:
X
→
Y
is defined as
f
k
(
x
)=
f
(
k, x
)
