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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.

vvg
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  • The mathematical term is “primitive root” which may be a helpful search keyword: https://en.m.wikipedia.org/wiki/Primitive_root_modulo_n – Qiaochu Yuan Sep 26 '20 at 17:02
  • There are quite a few primes of 512 bits. We have $\ln (2^{512})\approx350< 2^9$. Meaning that among the $2^{511}$ integers with exactly $512$ bits we expect at least $2^{511-9}$ primes. That's a long table. – Jyrki Lahtonen Sep 26 '20 at 18:13
  • :) The table was a joke. I am looking for exemplar GF(p) with large p and its primitive root or generator. – vvg Sep 26 '20 at 18:23
  • Ok. You got me there :-) – Jyrki Lahtonen Sep 26 '20 at 18:32

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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.

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.

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 $83$.

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.

Jyrki Lahtonen
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  • This is exactly what I was after. Special primes where it is easy to find the primitive root. I looked at Fermat primes, alas the last known is 65537 and we are still searching :) – vvg Sep 26 '20 at 18:33