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This question already has an answer here:

When choosing the public key for ElGamal, $\alpha$ must be chosen as a primitive mod p. What if $\alpha$ is not a primitive root ? How will it influence the encryption and decryption ?

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marked as duplicate by fgrieu, e-sushi Oct 26 '18 at 7:29

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  • $\begingroup$ nice link $\endgroup$ – kelalaka Oct 25 '18 at 22:05
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Everything works, except the key and signature space, reduced greatly;

The result is that the signature space (when the key is used for signing) or the public key space (when the key is used for encryption) may be greatly reduced from its expected size of log(p) bits, possibly down to 1 bit (the worst case if the order of g is 2)1.

It not hard to detect a $g$ is the generator or not.

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