Introduction to Number Theory

PART 2: ASYMMETRIC CIPHERS

Chapter 8 INTRODUCTION TO NUMBER THEORY

            Prime Numbers

       Fermat’s and Euler’s Theorems

Fermat’s Theorem Euler’s Totient Function Euler’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 Fermat’s theorem and Euler’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.