Preface
Number
theory
has long
been a favorite subject
for
students and teachers of
mathematics. It is a classical
subject and has a reputation for being
the
"purest" part
of
mathematics,
yet
recent
developments
in
cryptology and
computer science are based on elementary number
theory.
This
book
is
the
first text to
integrate
these important
applications of elementary
number
theory
with
the traditional topics covered in an introductory number
theory
course.
This book is suitable
as a text in
an undergraduate number theory
course at
any
level.
There are no
formal
prerequisites
needed
for most of
the
material
covered,
so that even a bright high-school
student could
use this book.
Also,
this book is designed
to be a useful
supplementary book for computer
science
courses, and as a number
theory
primer
for computer
scientists interested in
learning
about the new
developments in
cryptography.
Some of the important
topics that will interest
both
mathematics
and computer science students
are
recursion,
algorithms and their
computationai
complexity, computer
arithmetic
with
large integers,
binary
and hexadecimal
representations
of integers,
primality
testing,
pseudoprimality,
pseudo-random
numbers,
hashing functions,
and cryptology, including
the recently-invented
area of
public-key
cryptography.
Throughout
the book various
algorithms
and
their
computational
complexities
are discussed.
A
wide
variety
of
primality
tests
are
developed
in the
text.
Use
of the Book
The
core material
for
a course
in number
theory is
presented
in
Chapters 1,
2,
and 5,
and in Sections
3.1-3.3
and
6.1. Section
3.4 contains
some linear
algebra;
this section
is
necessary
background
for
Section 7.2;
these
two
sections
can be
omitted
if desired.
Sections 4.1, 4.2,
and 4.3
present
traditional
applications
of number
theory
and Section 4.4 presents
an
application
to computer
science;
the instructor
can decide which
of
these
sections
to cover.
Sections
6.2
and 6.3
discuss
arithmetic
functions.
Mersenne
primes,
and
perfect
numbers;
some
of this material
is used
in
Chapter
8.
Chapter 7
covers
the applications
of number
theory to cryptology.
Sections
7.1, 7.3,
and 7.4, which
contain
discussions
of classical
and
public-key
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