By what modulo calculations with discrete logarithms are performed?

Question

For odd prime $p$, I have been given a group $\mathbb{Z}_p^*$ of all invertible elements from $\mathbb{Z}_p$. Basically, $\mathbb{Z}_p^* = \{1,2,\ldots , p-1 \}$. I also have $a$ and $b$, which are two generators of $\mathbb{Z}_p^*$. My question is, by what modulo calculations with the discrete logarithm are performed for $\mathbb{Z}_p^*$? Or in other words, does it hold: $$ a^k \equiv b \pmod p$$ $$k \equiv \log_a b \pmod{p}$$ Or does it hold: $$ a^k \equiv b \pmod{p-1}$$ $$k \equiv \log_a b \pmod{p-1}$$

And also, a followed up question:
If $\mathbb{Z}_p^*$, $a$ and $b$ are as described above, does it holds, that if $k=\log_a b$ then $k\in \mathbb{Z}_p^*$?

$\mod p$, This was intended to be an educative answer : Discrete Logarithm: Given a p, what does it mean to find the discrete logarithm of x to base y? if not clear, please indicate, Also see Does classifying an integer as a discrete log require it be part of a multiplicative group? — kelalaka, Dec 20 '20 at 14:00
So this answer, where user @fgrieu♦ suggested that operations with discrete logarithm are performed by $\bmod{(p-1)}$ is wrong? — Jan, Dec 20 '20 at 14:39
It is the power, see Euler's Theorem. You should take an introduction to abstract algebra course. — kelalaka, Dec 20 '20 at 14:42
Yes and no. $a^{\varphi(k)} = 1 \bmod n$ where $k \in \mathbb Z$ and $\gcd(a,p)=1$. There is a representative of $k \in \varphi(n)$ that is $k' = k + t\cdot \varphi(n) $. Since $n$ is a prime than $\varphi(n) = n-1$. See a recent RSA question about Euler's theorem. — kelalaka, Dec 20 '20 at 15:01

fgrieu · Accepted Answer · 2020-12-20T21:36:09.737

Neither of the question's alternative hold.

For any $n>1$, and any $a$ with $\gcd(a,n)=1$, $$a^k \equiv b \pmod n\quad\iff\quad k \equiv \log_a b \pmod{\text{ord}_n(a)}$$

where $\text{ord}_n(a)$ is the order of $a$ in the multiplicative group $\mathbb Z_n^*$, that is the smallest $r\ge1$ with $a^r\equiv1\pmod n$.

The order of an element divides the order of the group, which is $\varphi(n)$ for group $\mathbb Z_n^*$, where $\varphi$ is Euler's totient function. Therefore $$k\equiv\log_a b \pmod{\varphi(n)}\quad\implies\quad a^k \equiv b \pmod n$$

When $a$ is a generator of $\mathbb Z_n^*$, which is testable as $a^{\varphi(n)/q}\not\equiv1\bmod p$ for every prime $q$ dividing $\varphi(n)$, we simply have $$a^k \equiv b \pmod n\quad\iff\quad k \equiv \log_a b \pmod{\varphi(n)}$$

Restating this for prime $p$, and any $a$ with $a\not\equiv0\pmod p$, $$k\equiv\log_a b \pmod{(p-1)}\quad\implies\quad a^k \equiv b \pmod p$$ and, for the lowest $r\ge1$ dividing $p-1$ such that $a^r\equiv1\pmod p$, $$a^k \equiv b \pmod p\quad\iff\quad k \equiv \log_a b \pmod{r}$$

When $a$ is a generator of $\mathbb Z_p^*$, which is testable as $a^{(p-1)/q}\not\equiv1\bmod p$ for every prime $q$ dividing $(p-1)$, we simply have $$a^k \equiv b \pmod p\quad\iff\quad k \equiv \log_a b \pmod{(p-1)}$$

Good, another educative answer. – kelalaka Dec 20 '20 at 23:06 — kelalaka, Dec 20 '20 at 23:06

By what modulo calculations with discrete logarithms are performed?

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