# How do i calculate a multiplication table for GF(8)?

Could you please provide the steps involved in calculating a multiplication table for GF(8)?

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 At the risk of blowing my own trumpet: a method (particularly suited for computer implementation) is described in the latter half of my answer to this question. – Jyrki Lahtonen Nov 12 '12 at 12:48

Take an irriducibile polynomial of degree $3$ over the field $\Bbb F_2$ with two elements, e.g. $P(X)=X^3+X+1$. Then you know that $$\Bbb F_8=\Bbb F_2[X]/(P(X))$$ and that its 8 elements are represented by the 8 polynomials of degree $\leq2$. Thus you can construct the multiplication of $\Bbb F_8$ simply by multiplying these polynomials and taking the result modulo $P(X)$.
@phoenix: Looks like "010" represents $0\cdot\tau^2 + 1\cdot\tau + 0\cdot1$, and "100" represents $1\cdot\tau^2 + 0\cdot\tau + 0\cdot1$. – Niel de Beaudrap Oct 22 '12 at 0:46
As Andrea says, the elements of the field are (represented by) the polynomials of degree at most 2. Your source chooses a different representation; the polynomial $ax^2+bx+c$ is represented by the bit-string $abc$ (or maybe $cba$, I didn't read closely enough to tell). – Gerry Myerson Oct 22 '12 at 0:47