Siguiente: Algorithms for field arithmetic Arriba: Field operations with the Anterior: Multiplication

Addition

Also, in terms of a primitive element $x_0\in\mathbb{F}_{2^n}^*$ we may express the elements of $\mathbb{F}_{2^n}$ through the map $J: \mathbb{F}_{2^n}\to[\![0,2^n-1]\!]$ such that $J(0)=0$ and $J(x) = \log_{x_0}(x) +1$. For any $i,j\in[\![0,2^n-2]\!]$, with $i\leq j$, we have $x_0^i+x_0^j = x_0^i(1+x_0^{j-i}) = x_0^{i+k}$, where the addition is taken modulus $2^n-1$ and $x_0^k=(1+x_0^{j-i})$, or $p_{nk}(X)\vert\,1+x_0^{j-i}+x_0^k$. Thus, addition involves the irreducible polynomial $p_{nk}(X)$. The corresponding form of addition for $n=2,3,4,5$ are displayed in tables 22-26.

Table: Logarithmic addition table in $[\![0,2^2-1]\!]$ using map $J$.

(a)	(b)

Table: Logarithmic addition table in $[\![0,2^3-1]\!]$ using map $J$.

(a)	(b)

Table: Logarithmic addition table in $[\![0,2^4-1]\!]$ using map $J$.

(a)

(b)

Table: Logarithmic addition table in $[\![0,2^5-1]\!]$ using map $J$ (numeric values).

Table: Logarithmic addition table in $[\![0,2^5-1]\!]$ using map $J$ (density plot).

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