Siguiente: Field operations with the Arriba: Field operations with the Anterior: Multiplication

Addition

What has been done for the multiplication can also be done for the addition in the Galois field $\mathbb{F}_{2^n}$. Indeed, addition is performed as the polynomial addition reduced modulus 2. Using the above identification the field addition determines the operation

$$\oplus: [\![0,2^n-1]\!]\times[\![0,2^n-1]\!]\to [\![0,2^n-1]\... ...h $\delta$}}) \mapsto I_{\mbox{\scriptsize\boldmath $\gamma$}},$$

where the vector index $\mbox{\boldmath$\gamma$}$ is such that $p_{\mbox{\scriptsize\boldmath$\gamma$}}(X)=p_{\mbox{\scriptsize\boldmath$\epsilon$}}(X) + p_{\mbox{\scriptsize\boldmath$\delta$}}(X)$. In tables 12-16 these operations are displayed for $n=2,3,4,5$ respectively.

Table: Addition table in $[\![0,2^2-1]\!]$ corresponding to addition in the Galois field $\mathbb{F}_{2^2}$.

(a)	(b)

Table: Addition table in $[\![0,2^3-1]\!]$ corresponding to addition in the Galois field $\mathbb{F}_{2^3}$.

(a)	(b)

Table: Addition table in $[\![0,2^4-1]\!]$ corresponding to addition in the Galois field $\mathbb{F}_{2^4}$.

(a)

(b)

Table: Addition table in $[\![0,2^5-1]\!]$ corresponding to addition in the Galois field $\mathbb{F}_{2^5}$ (numeric values).

Table: Addition table in $[\![0,2^5-1]\!]$ corresponding to addition in the Galois field $\mathbb{F}_{2^5}$ (density plot).

Siguiente: Field operations with the Arriba: Field operations with the Anterior: Multiplication