I am engineer/applied mathematician who is quite new to most things number theory and also groups so please bear with me if this is obvious.
Consider the $b\times b$ matrix constructed by:
$$M_{ij} = \delta(j-(\text{mod }(a^i,b)))$$
- For which $a,b$ can we be sure $M$ is a permutation matrix.
- For which can we be sure $M$ will be a permutation matrix permutation-similar to cyclic matrix: $$P_{ij} = \delta(i,j+1) : P=QMQ^{-1}$$, where $Q$ permutation matrix.
I think I understand the question.
I assume that when you write $\mod(x,m)$ you mean what a mathematician would write as $x \pmod{m}$, the remainder when you divide $x$ by $m$.
Then you are asking when the powers of some number $a$ run through the possible remainders modulo $b$ once each with no repetition.
If that is the case then what you are looking for is a primitive root. That wikipedia page can get you started.
Every prime has a primitive root. For example, $a = 3$ is one for $b=7$, but $a=2$ is not. The other numbers that have primitive roots are the powers of primes and twice the powers of primes.
It's easy to count how many primitive roots there are, but very hard to find one.