# trying to understand multiplication + reduction in binary finite fields [duplicate]

The wikipedia.org article on Finite field arithmetic provides an example of multiplying $83$ and $206$ in $\mathbb{F}_{2^8}$ with $x^8+x^4+x^3+x+1$ as the reducing polynomial (in fact it is the reducing polynomial for AES). Quoting it:

Multiplying $83$ and $206$, whilst ignoring the carries, yields $16,254$ or (in binary) 11111101111110.

The reducing polynomial is then converted to binary (100011011), shifted forward to match the length of the number being reduced (eg. multiplied by a power of two) and then XOR'd (subtracted) against the number to be reduced. It seems like it's supposed to do this until the number being reduced is less than the reducing polynomial (in binary).

Here's the example wikipedia gives:

      11111101111110 (mod) 100011011
^100011011
1110000011110
^100011011
110110101110
^100011011
10101110110
^100011011
0100011010
^100011011
00000001


My question is... why is the last XOR done? 0100011010 is less than 100011011..

The last XOR is done, to get rid of the $x^8$ term. Polynomials in $GF(2^n)$ do have a maximum degree of $n-1$.