Cryptography Reference
In-Depth Information
3.1.1
Preliminary Remarks
be a binary operation on the elements of this set.
3
For
example,
S
may be one of the following sets of numbers (that are frequently used in
mathematics):
Let
S
be a nonempty set and
∗
•
The set
of
natural numbers
(also known as
nonnegative
or
positive integers
). In some literature, the term
N
:=
{
0
,
1
,
2
,...
}
+
is used to refer to
N
N
without
+
:=
zero (i.e.,
N
N
\{
0
}
).
•
The set
of
integer numbers
,or
integers
in
short. In addition to the natural numbers, this set also comprises the negative
numbers.
Z
:=
{
...,
−
2
,
−
1
,
0
,
1
,
2
,...
}
•
Q
The set
of
rational numbers
. Roughly speaking, a rational number is a
number that can be written as a ratio of two integers. More specifically, a
number is rational if it can be written as a fraction where the numerator and
denominator are integers and the denominator is not equal to zero. This can
be expressed as follows:
a
b
|
Q
:=
{
a, b
∈
Z
and
b
=0
}
•
R
The set
of
real numbers
. Each real number can be represented by a converg-
ing infinite sequence of rational numbers (i.e., the limit of the sequence refers
to the real number). There are two subsets within the set of real numbers:
algebraic numbers and transcendental numbers. Roughly speaking, an
alge-
braic number
is a real number that is the root of a polynomial equation with
integer coefficients, whereas a
transcendental number
is a real number that is
not the root of a polynomial equation with integer coefficients. Examples of
transcendental numbers are
π
and
e
. Real numbers are the most general and
most frequently used mathematical objects to model real-world phenomena.
A real number that is not rational is called
irrational
, and hence the set of
irrational numbers is
+
is also used to
R
\
Q
. In some literature, the term
R
refer to the real numbers that are nonnegative.
•
The set
of
complex numbers
. Each complex number can be specified by a
pair (
a, b
) of real numbers, and hence
C
C
can be expressed as follows:
and
i
=
√
−
C
:=
{
a
+
bi
|
a, b
∈
R
1
}
The choice of the symbol
∗
is arbitrary. The operations most frequently used in algebra are addition
(denoted as +) and multiplication (denoted as
3
·
).
