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I've been reading about safe primes and their use in: Cryptography Engineering by Niels Ferguson, Bruce Schneier, and Tadayoshi Kohno.

Having a safe prime $q$ with $q=2p+1$ where $p$ is a Sophie Germain prime

They claim that the elements of the subgroup $p$ are the elements that are preferred to be used. Why?

Why not the elements in $2p$?

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  • $\begingroup$ Yes, I consider to use them in Diffie-Hellman. What is so bad about large group orders? $\endgroup$ – mark23 Aug 24 '20 at 12:37
  • $\begingroup$ No, not really. But the answer given by Occams_Trimmer does $\endgroup$ – mark23 Aug 26 '20 at 14:03
  • $\begingroup$ Yes, That is another reason that I've forgotten. I'll update my answer later... $\endgroup$ – kelalaka Aug 26 '20 at 14:13
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The Decisional Diffie-Hellman assumption, on which the key-exchange would be based on does not hold in $\mathbb{Z}_q^*$. The reason is that the Jacobi symbol "leaks" information about the shared key. Therefore one, instead, works with the subgroup of $\mathbb{Z}_q^*$ of order $p$, which is intuitively obtained by "quotienting out" this information. (This group is sometimes called the Schnorr group.) You can read the details on why DDH doesn't hold here (Exercise 2).

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  • $\begingroup$ Thank you! This was exactly what I was looking for $\endgroup$ – mark23 Aug 26 '20 at 14:03

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