Known table of $GF(p)$ and characteristics

Question

Is there a list of Galois Fields, $GF(p)$ and its known multiplicative generator(s), $g$? I know the general case of finding the generators may take long especially for large $p$, but was wondering if there was OEIS entry or other resource? I am specifically looking for primes of the order of 512 bits or greater. Larger the better.

Answer 1

It is unreasonable to expect a table, that would not fit into the universe. Below I suggest that you look for primes of a special form, when finding a generator is a lot simpler.

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Find a pair of primes $p$ and $q=2p+1$ (the buzzword safe prime probably gives you search hits) Then modulo the bigger prime $q$ every element is of order $1,2,p$ or $2p$. The only residue classes $x$ satisfying $x^2\equiv1\pmod q$ are $x\equiv\pm1$. The residue classes of order $p$ are exactly the quadratic residues. We thus conclude that modulo a safe prime $q$, the residue class of $a\not\equiv-1$ is a primitive root if and only if it is a quadratic non-residue modulo $q$. Those are quick to find. The law of quadratic reciprocity is your friend.

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For a small example consider $p=41$, $q=83$. We have $83\equiv3\pmod5$. Three is a quadratic non-residue modulo five, so quadratic reciprocity tells us that $5$ is a quadratic non-residue modulo $83$. Because $q=2p+1$ with $p$ a prime, the argument above shows that $5$ is a primitive root modulo $83$.

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I'm afraid I don't know how difficult it is to find a $(p,q=2p+1)$ pair of primes (aka Sophie Germain primes). Random poking might be fast enough at this range.