In the ElGamal cipher, the key generation algorithm needs to randomly sample a generator g and test whether g is a safe generator or not each time. I wonder whether we can use a fixed safe generator for all users, which may reduce the time of key generation. Will this be safe?


1 Answer 1


The generator $g$ is always assumed to be public and does not need to be generated as part of the KeyGen. Specifically, the whole Diffie-Hellman group description (which includes $g$) can fixed. For example, see https://en.wikipedia.org/wiki/ElGamal_encryption#Key_generation

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    $\begingroup$ What's more, in elliptic-curve ElGamal, the base-point is almost always fixed. $\endgroup$
    – DannyNiu
    Nov 18 at 12:11
  • $\begingroup$ Thanks so much! $\endgroup$
    – Saint Gold
    Nov 18 at 14:18
  • $\begingroup$ I think I understand now ... The reason that ElGamal is described in a random generator way is to prove its security under a weaker assumption (HDH). For a fixed generator, we need a stronger assumption ... $\endgroup$
    – Saint Gold
    Nov 18 at 15:47

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