Order-finding problem / period finding?

Question

For positive integers x and N, such that x<N, with no common factors, the order of x modulo N is defined to be the least positive integer r such that:

$x^r≡1(modN)$

The order-finding problem is determining the order r of a specified x and N. Link: https://www.youtube.com/watch?v=YNq7x6z8F1g

My questions:

1. Is it not 1modN, always 0 (then we want an r such that $x^r$ is 0, this sounds weird)? Maybe I am misunderstanding the ≡ symbol.
2. Is it this problem exactly the period finding problem with another name?

Thanks!

Answer 1

1. No, it is 1 mod N: $x^0=1$, and then you want to know the smallest $r>0$ where $x^r=1$ again (modulo $N$). Ten, of course, also $x^{2r}=1$, $x^{3r}=1$, etc. (all modulo $N$), so this has a period $r$.

2. No. Period finding is more general. This is finding the period of a specific type of function, namely a function of the form $f_{x,N}(r)=x^r\mod N$.