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Fast double exponentiation in finite fields
Let $p$ be a prime, and let $\mathbb{F}_p$ be the finite field with $p$ elements. Let $a$ be a non-zero element of $\mathbb{F}_p$. Can we quickly evaluate $a^{2^r} \mod{p}$? Using repeated squaring, ...
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Irreducible algebraic sets via irreducible polynomials
There are many results about irreducible polynomials over finite fields:
we know a cardinality of all irreducible polynomials with given degree, we know explicit examples of irreducible polynomials, ...