Intermediate Computational Number Theory

Robert Campbell

Contents

1. Introduction
2. Extension Fields
   • Finite Fields
   • Number Fields
3. Integer Algorithms
   • Factoring: Pollard Rho
4. Preliminaries
   • Random Maps
   • Polynomials

1. Introduction

2. Extension Fields

2.1 Finite Fields

A simple example is the 8 element Galois Field, GF8. We choose the polynomial f(x)=x³+x²+1, checking that it is irreducible mod 2. Now we construct GF8 by considering all quadratic polynomials with coefficients mod 2. Thus the field has the elements GF8 = {0, 1, x, x+1, x², x²+1, x²+x, x²+x+1}. Addition is done the usual way. Multiplication is done by multiplication, then reducing the result mod f(x). Try adding and then multiplying some elements:

2.2 Number Fields

3. Integer Algorithms

3.1 Factoring: Pollard Rho

4. Preliminaries

4.1 Random Maps

4.2 Polynomials