Operations in $\mathbb{F}_{32}$

Question

I've some difficulties about sums in the field $\mathbb{F}_{32}$. In particular I'm studying an example of a cryptographic attack, where there are a lot of sums in this field, which I don't understand. One of these is:

$1+7=6$ in $\mathbb{F}_{32}$.

Anyone coud clarify me the reason of this result?

Answer 1

You are actually adding polynomials of degree less than $5$ with coefficients in $\mathbb F_2$. Thus $$1+7 = 1 + (1+x+x^2) = (1+1) + x + x^2 = x+x^2 = 6$$ Note that this operation is equivalent to binary XOR of the numbers. $$1 \oplus 7 = 00001_2 \oplus 00111_2 = 00110_2 = 6$$