Cryptography Reference
In-Depth Information
For every natural number
n
∈
N
there exists a function
e
n
from
N
N
to
such
that
(i)
e
n
(0)=1
,
(ii)
e
n
x
+
=
e
n
(
x
)
· n
for every natural number
x ∈
N
.
The value of the function
e
n
(
x
)
is called the
x
th
power
n
x
of
n
. With complete
induction we can prove the
power law
n
x
n
y
=
n
x
+
y
,n
x
· m
x
=(
n · m
)
x
,
(
n
x
)
y
=
n
xy
,
to which we shall return in Chapter 6.
In addition to the calculational operations, the set
N
of natural numbers
has defined on it an order relation “
<
” that makes it possible to compare two
elements
n, m
∈
N
. Although this fact is worthy of our great attention from a
set-theoretic point of view, here we shall content ourselves with noting that the
order relation has precisely those properties that we know about and use in our
everyday lives.
Now that we have begun with establishing the empty set as the sole
fundamental building block of the natural numbers, we now proceed to consider
the materials with which we shall be concerned in what follows. Although number
theory generally considers the natural numbers and the integers as given and
goes on to consider their properties without excessive beating about the bush, it
is nonetheless of interest to us to have at least once taken a glance at a process
of “mathematical cell division,” a process that produces not only the natural
numbers, but also the arithmetic operations and rules with which we shall be
deeply involved from here on.
About the Software
The software described in this topic constitutes in its entirety a package, a
so-called function library, to which frequent reference will be made. This library
has been given the name FLINT/C, which is an acronym for “functions for large
integers in number theory and cryptography.”
The FLINT/C library contains, among other items, the modules shown in
