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Multiplication

The multiplication group $\mathbb{F}_{2^n}^*=\mathbb{F}_{2^n}-\{0\}$ of the Galois field $\mathbb{F}_{2^n}$ is cyclic, hence for any primitive element $x_0\in\mathbb{F}_{2^n}^*$ we have that for all $x\in\mathbb{F}_{2^n}^*$ there is a $y\in [\![0,2^n-2]\!]$ such that $x=x_0^y$. Obviously, it is written $y=\log_{x_0}(x)$. Through this function, the multiplication can be expressed additively: $\log_{x_0}(x_1 x_2) = \log_{x_0}(x_1) + \log_{x_0}(x_2)$. Let $J: \mathbb{F}_{2^n}\to[\![0,2^n-1]\!]$ be the map such that $J(0)=0$ and $J(x) = \log_{x_0}(x) +1$. Hence $J$ is a bijection and the multiplication in $\mathbb{F}_{2^n}$ determines an operation $*:[\![0,2^n-1]\!]\times[\![0,2^n-1]\!]\to[\![0,2^n-1]\!]$, which is the multiplication using the discrete logarithm representation. These operations are displayed in tables 17-21.

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

(a)	(b)

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

(a)	(b)

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

(a)

(b)

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

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

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