TITLE: Addition and multiplication in a Galois Field
QUESTION [19 upvotes]: I am attempting to generate QR codes on an extremely limited embedded platform. Everything in the specification seems fairly straightforward except for generating the error correction codewords. I have looked at a bunch of existing implementations, and they all try to implement  a bunch of polynomial math that goes straight over my head, particularly with regards to the Galois fields. The most straightforward way I can see, both in mathematical complexity and in memory requirements is a circuit concept that is laid out in the spec itself:
 
With their description, I am fairly confident I could implement this with the exception of the parts labeled GF(256) addition and GF(256) Multiplication.
They offer this help: 

The polynomial arithmetic for QR Code shall be calculated using bit-wise modulo 2 arithmetic and byte-wise
  modulo 100011101 arithmetic. This is a Galois field of 2^8
  with 100011101 representing the field's prime modulus
  polynomial x^8+x^4+x^3+x^2+1.

which is all pretty much greek to me.
So my question is this: What is the easiest way to perform addition and multiplication in this kind of Galois field arithmetic? Assume both input numbers are 8 bits wide, and my output needs to be 8 bits wide also. Several implementations precalculate, or hardcode in two lookup tables to help with this, but I am not sure how those are calculated, or how I would use them in this situation. I would rather not take the 512 byte memory hit for the two tables, but it really depends on what the alternative is. I really just need help understanding how to do a single multiplication and addition operation in this circuit.

REPLY [6 votes]: I might be a little late, but in case you are still looking at methods for modular multiplication (ie. Multiplication of finite field elements followed by modular reduction), you might want to look at the Wikipedia page for Finite Field Arithmetic.  
You want to read the modified version of 'Peasant's algorithm', described right below the long-division figure.