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0

Let $r_0$ be the root of $f(x)$, so $r_0$ should be in $\{0,...,p-1\}$ and $f(r_0)=0 \pmod p$. If we cant find such $r_0$, our polynomial haven't root in $F_p$. If $p$ be small we can easily find such $r_i$(root of $f^i(x)$) and compute $f^n(x)=\frac{f^{n-1}(x)}{x-r_{n-1}}$ where $f^0(x)=f(x)$, then repeat this recursively for $n$. Founded $r_i$'s are ...

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Could you please describe what are the difficulties you are encountering? Anyway, the polynomial mentioned is the one used for the AES cipher and I suggest you to read this specification (especially the section 4 which explains briefly the mathematical background).

3

As yyyyyyy mentioned for counting number of points on elliptic curve over $\mathbb F_p$ we can use Elkies method. But for extension of fields use of this theorem make it so easy: Theorem : Let $E$ be an elliptic curve defined over $F_q$, and let $\#E(F_q ) = q +1−t$. Then $\#E(F_{q^n} ) = q^n + 1 − V_n$ for all $n ≥ 2$, where $\{V_n\}$ is the sequence ...

4

When you have a finite field $\mathbb{K}$ of cardinal $q$ (necessarily, $q = p^m$ for some prime $p$ and integer $m$), then a finite field $\mathbb{K}'$ of cardinal $q^k$ can be defined for any integer $k$, in the following way. You consider $\mathbb{K}[X]$, which is the ring of polynomials whose coefficients are in $\mathbb{K}$. Polynomials can be added and ...

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