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If the order of group ($p$)selected by attacker then discrete logarithm is still hard?

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  • $\begingroup$ Welcome to Cryptography. What is the source of this question? What is the order of the group? What about pre-computation with a small group like the small group attack on the elliptic curve cryptography. $\endgroup$
    – kelalaka
    Nov 8 '20 at 15:37
  • $\begingroup$ Thanks for your answer but my means that the group order p is a large prime number. In this situation what's the answer? $\endgroup$ Nov 9 '20 at 13:00
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If the attacker can pick, say, an order $p=3$, then it's easy.

However, if the attacker is constrained to pick a large order, that may not be enough. If the attacker is able to select a smooth group order, that is, an order $p$ which is the product of a multiset of small primes, then the discrete log problem is easy (using the Pohlig-Hellman algorithm).

Hence, unless the flexibility that the attacker is granted is more constrained than the question indicates, the answer is "no, the discrete log problem may be easy"

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  • $\begingroup$ Thanks for your answer but my means that the group order $p$ is a large prime number. In this situation what's the answer? $\endgroup$ Nov 9 '20 at 12:53
  • $\begingroup$ @mehdimahdavioliaiy Did you read the last paragraph on the Poncho's answer? And next time, please be more specific about your problem. Writing $p$ doesn't mean it is a prime, we say like "for a prime $p$". Still, what is the source of this question? why do you need this question? $\endgroup$
    – kelalaka
    Nov 9 '20 at 13:04

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