Existence of solutions to DLP and Primitive roots mod $p$

Asked Apr 30 '18 at 18:49

Active May 01 '18 at 19:28

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Discrete log problem - finding $x \ge 0$ for prime $p$ , generator $g>0$ and $h>0$ such that:

$$g^x \cong h \pmod{p}$$

Define $G$ as the group generated by all values of $g^x \pmod{p}$. Eg $G=${2,3,4}

My understanding is that if $h$ is in $G$ then solutions to the DLP exist.

Question:

Where solutions exist, does this mean that one does not need to check if $g$ is a primitive root mod $p$?

Answer (based on comments)

I don't believe one needs to check for primitive roots when solutions exist for the DLP

edited May 01 '18 at 19:28

asked Apr 30 '18 at 18:49

unseen_rider

Read a bit about DLP and primitive roots, e.g., here. If $g$ is a primitive root modulo $p$, then by definition $g^x\equiv h \bmod p$ is always uniquely solvable for $h\in G$. – Dietrich Burde Apr 30 '18 at 19:03
@dietrichburde I've read the linked question and answers. From your point, is the converse true? – unseen_rider Apr 30 '18 at 19:15
The converse need not be true in general, of course. Just take $h=e$. Then $x=0$ is a solution, but $g$ need not be a primitive root. – Dietrich Burde Apr 30 '18 at 19:18
Is $e$ the identity element & in this case has to be 1 since $g^0 = 1$? – unseen_rider May 01 '18 at 18:48

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