Intermediate Computational Number Theory

Robert Campbell


Contents

  1. Introduction
  2. Extension Fields
  3. Integer Algorithms
  4. Preliminaries

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:

(x2 + x + ) + (x2 + x + ) + = (x2 + x + )

2.2 Number Fields

3. Integer Algorithms

3.1 Factoring: Pollard Rho

4. Preliminaries

4.1 Random Maps

4.2 Polynomials

Robert Campbell
Last modified: Dec 9, 2000