What are the solutions for discrete integers b, d to $a^b \equiv c^d \pmod p$ where $p$ is a large prime number? Is there a way to efficiently discover or choose the integers $b$, $d$ for the congruence relationship below where $p$ is a large prime number? Is there a name for this relationship?
$$
a^{b} = c^{d} \pmod p
$$
 A: As was mentioned in the comments, this is essentially the discrete logarithm problem.
Since $a^{p-1}=c^{p-1}=1$, $b,d$ are naturally thought of best as modulo $p-1$.
Now for any solution, we can factor $d= d'\cdot gcd(d,p-1)$. Then $d'$ is invertible modulo $p-1$, so we can find some $e$ for which $d'e = 1$ modulo $p-1$. Setting $c' = c^{gcd(d,p-1)}$, taking the $e$-th power of the original equation gives you
$$
a^{be} = c'
$$
mod $p$. Now finding $be$ here is exactly the discrete logarithm problem.
You can get your original equation back by powering with $d'$ again (since $d'e=1$ mod $p-1$),
so this describes all solutions. More precisely, let $k$ run through divisors $k | p-1$, let $x$ run through all solutions of the discrete logarithm problem $a^x = c^k$, and $d'$ through all things invertible mod $p-1$, then $(c,d) = (d'x, d'k)$ describes all solutions mod $p-1$.
As was also mentioned in the comments, this discrete logarithm problem is believed to be not efficiently solvable, and a lot of modern cryptography depends on that.
