Let $T$ denote the smallest exponent such that $b^T \equiv 1 \pmod{N}$, then we call $T$ as fundamental period of the sequence satisfying the equation.
As an example for $b=2, N=2731$ we have $T=26$ since $T$ is the smallest number of the following sequence:
$ {26,52,78,104,...} $
It's obvious that $26$ is made of factors of $N-1$, here $\{2,13\}$. Assuming that we know all factors of $N-1$ even if $N$ is very large, without calculating the modular exponentiation for all $t$ up to $N$:
- How we can know the existence of any $T$ less than $N$?
- Is there any composite number having a $T$ less than N?
- If there exist one, then how we can find it among the divisors of $N-1$?