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When selecting $g$ and $p$ for the Diffie-Hellman modular exponentiation:

$ s= g^x \bmod p $

Which pairs of $g$ and $p$ are considered good nowadays, are recommended by bibliography?

What I try to do is to write a simple Python script that chain calculates Diffie-Hellman Modular exponentiation:

def dh_public(secret):
    g=5;p=23 //@todo select better values

    return g**secret % p

For now, I have REALLY POORLY selected values but I want standardized and tested values for calculating a Diffie Hellman key exchange. As far as I know, both $g$ and $p$ should be primes.

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    $\begingroup$ There is no specific reason why $g$ must be a prime $\endgroup$
    – poncho
    Nov 14, 2018 at 14:44
  • $\begingroup$ $p$ should be a prime and $g$ a primitive root $\endgroup$ Nov 14, 2018 at 14:53
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    $\begingroup$ IANA has a registery of them $\endgroup$
    – DannyNiu
    Nov 14, 2018 at 15:00
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    $\begingroup$ @YoussefElHousni: $g$ needn't be a primitive root; it must generate a subgroup which has a size with a large prime factor $\endgroup$
    – poncho
    Nov 14, 2018 at 15:03
  • $\begingroup$ @poncho I agree. $g$ only needs to be of order multiple of a large enough prime. $\endgroup$ Nov 14, 2018 at 15:36

1 Answer 1

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The most commonly used values are in this RFC; it includes groups that range from 1536 to 8192 bits; the size 2048 is actually the most common.

The properties of these groups include:

  • The modulus is chosen using a 'nothing up my sleeve' method (and hence is unlikely to be specifically chosen to be especially vulnerable)

  • The modulus has the high 32 bits and the low 32 bits all set; this can be useful for some modmul algorithms

  • It has $g=2$; this can make the initial computation of $g^x \bmod p$ rather more efficient

  • The order of the subgroup generated by $g$ is a prime of size $(p-1)/2$; a size which is quite sufficient for security

DannyNiu's reference to the IANA TLS groups (actually, the groups 256-260; the others are various elliptic curves which aren't what you are asking about) are another set; they're actually generated using the same procedure (and share the above properties), but are deliberately slightly different (to avoid potential factor-based style attacks against the original group)

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    $\begingroup$ Addition: the fragment g**secret % p must be changed to pow(g, secret, p) for the code to work with sizable arguments. Making it secure against side-channel attacks is uh, much harder. $\endgroup$
    – fgrieu
    Nov 14, 2018 at 16:58
  • $\begingroup$ Well I do not want to mmake it ultra secure just to do some tests/research on chaining Diffie Hellman calculations on a Rachet tree Implementation. $\endgroup$ Nov 15, 2018 at 0:07

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