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To compute the following expressions in the finite field $\mathbb{F}_{2^m}$ (say $\mathbb{F}_{2^4}$), using logarithm and antilogarithm tables, we assume that $\gamma$ is a primitive element (generator) of the field. The two expressions to evaluate are: --- (A) $$ \frac{(\gamma^4+\gamma^2)+(\gamma^3+\gamma+1)}{(1+\gamma^2+\gamma^4)(1+\gamma^3)} $$ (B) $$ \frac{\gamma^2(\gamma^2+\gamma+1)}{(\gamma^3+\gamma^2)(1+\gamma^5)} $$ --- ### Step 1: Use Logarithm and Antilogarithm Tables Let’s suppose the field is $\mathbb{F}_{2^4}$, and the irreducible polynomial is $f(x) = x^4 + x + 1$. Then every element in $\mathbb{F}_{2^4}$ can be written as a power of a primitive element $\gamma$, where $\gamma$ is a root of $f(x)$. We will need the exponential/logarithm table of $\mathbb{F}_{2^4}$ with respect to $\gamma$. Here's the standard table for the field defined by $x^4 + x + 1$: | Power $i$ | $\gamma^i$ (polynomial form) | | | --------- | ---------------------------------- | ------------------------------------------------- | | 0 | 1 | | | 1 | $\gamma$ | | | 2 | $\gamma^2$ | | | ____ ________ _________ _____ _________ _____.
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