A prime $p$ is Sophie Germain if $2p+1$ is also prime. Wiki says if $p\equiv3\bmod4$ then $2p+1|2^p-1$. This seems to put a huge restriction on density of Sophie-Germain prime if they are $3\bmod4$. For this reason do all systems use Sophie-Germain primes with $1\bmod 4$ for discrete logarithm?
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This seems to put a huge restriction on density of Sophie-Germain prime if they are $3 \bmod 4$.
I'm not sure how you come to that conclusion; whether $2p+1$ is a factor of $2^p-1$ would not have any immediately obvious consequence to $2p+1$ being prime.
For this reason do all systems use Sophie-Germain primes with $1 \bmod 4$ for discrete logarithm?
Actually, Sophie-Germain primes of the form $3 \bmod 4$ would appear to be more prevalent; for example, the primes listed here are all of the form $3 \bmod 4$. One reason is that it makes the value $g=2$ be a quadratic residue, that is, its use during the DH operation doesn't leak the lsbit of the private exponent, and that's appreciated by some people.
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$\begingroup$ could you explain on 'its use during the DH operation doesn't leak the lsbit of the private exponent'? I though lsbit is always leaked no matter what. $\endgroup$– TurboCommented May 14, 2017 at 21:45
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2$\begingroup$ @Turbo: sure; in DH, Alice picks a random value $a$ and transmites $g^a \bmod p$; if $g$ is a primitive element, anyone can test if $g^a$ is a quadratic residue; if it is, then the lsbit of $a$ is a 0; otherwise it is a 1. If $g$ is a quadratic residue, then that test tells people nothing (as $g^a \bmod p$ is always a quadratic residue). $\endgroup$– ponchoCommented May 14, 2017 at 21:47
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$\begingroup$ so $1\bmod 4$ and $3\bmod 4$ Sophie-Germain prime densities are believed similar? $\endgroup$– TurboCommented May 14, 2017 at 21:49
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2$\begingroup$ @Turbo: as far as I know; a quick count of the ones under 100,000,000 shows a slight edge to the $3 \bmod 4$ (211,439 vs 211,700)... $\endgroup$– ponchoCommented May 15, 2017 at 2:53
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1$\begingroup$ 3 mod 4 has a very small lead at 1e9 and 1e10 as well, but loses it by 2e10 (20,000,000,000). It's extremely close at 1e11 with 3 mod 4 leading by a very small amount. $\endgroup$– DanaJCommented May 15, 2017 at 19:44