A prime number is an integer that can only be divided without remainder by positive and negative values of itself and 1. Prime numbers play a critical role both in number theory and in cryptography.

Chapter 8 **INTRODUCTION TO NUMBER ****THEORY**

**
****Prime Numbers**

**
****Fermatâ€™s ****and
Eulerâ€™s Theorems**

Fermatâ€™s Theorem Eulerâ€™s Totient Function Eulerâ€™s Theorem

**
****Testing ****for Primality**

Miller-Rabin Algorithm

A Deterministic Primality Algorithm Distribution of Primes

**
****The**** ****Chinese Remainder Theorem**

**
****Discrete Logarithms**

The Powers of an Integer,
Modulo *n *Logarithms for Modular Arithmetic
Calculation of Discrete Logarithms

KEY
POINTS

â—† A prime number is an integer that can only
be divided without remainder by positive and negative values of itself and 1.
Prime numbers play a critical role both in number theory and in cryptography.

â—† Two theorems that play important roles in
public-key cryptography are Fermatâ€™s theorem and Eulerâ€™s theorem.

â—† An important requirement in a number of
cryptographic algorithms is the ability to choose a large prime number. An area
of ongoing research is the development of efficient algorithms for determining
if a randomly chosen large integer is a prime number.

â—† Discrete logarithms are fundamental to a
number of public-key algorithms. Discrete logarithms are analogous to ordinary
logarithms but are defined using modular arithmetic.

A number of concepts from number theory are essential in the design of
public-key cryptographic algorithms. This chapter provides an overview
of the concepts referred
to in other chapters. The reader familiar with these topics can safely
skip this chapter.
The reader should
also review Sections
4.1 through 4.3 before proceeding with this chapter.

As with Chapter 4, this chapter
includes a number of examples, each of which is highlighted in a shaded box.

Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail

Cryptography and Network Security Principles and Practice : Asymmetric Ciphers : Introduction to Number Theory : Introduction to Number Theory |

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