A Galois Field Arithmetic Primer

Introduction A Galois Field Introduction by Example Base Galois Fields A Real World Example of a Base Galois Field GF(2) Extended Galois Fields Extended Galois Field Addition Extended Galois Field Multiplication A Field Defining Irreducible Polynomial A Primitive Polynomial From Abstract Alpha to a Real Value Selecting Primitive Polynomials The Benefit of Primitive Polynomials Linear Feedback Shift Register Multiplication through Addition of Exponents References Footnotes Introduction In my blog post about Reed-Solomon coding, I used regular integers for all calculations. These are impractical for a real-world implementation, but since everybody knows integer math since first grade, it made things easier to learn things one step at a time. Instead of working with pure integers, actual Reed-Solomon implementations use elements from a Galois or finite field as symbols. I’ve been sitting on implementing and writing about a Reed-Solomon decoder for almost 4 years now1, and I’m still not…