Number theory related to crypto I have a question related to a piece of coursework that is for cryptography and more for encryption that relies on Number theory, now I have no knowledge of number theory and the tutor did not cover it well enough, and I am starting to learn it slowly, but I have an exam coming up and one question will be like the one below, so any help from you guys would be appreciated.enter image description here
Consider the group $G=\mathbb{Z}^*_{61}$ with respect to multiplication. Compute the following
o Find the inverse element of 53 in $G$.
o Compute the order of the element 15 in $G$.
o Let $H := \langle 8 \rangle$ be a group containt in $G$, i.e. H is generated by the of powers of 8 modulo 61. Find two non-trivial subgroups of H. The trivial subgroups of $H$ are $\langle 1 \rangle$ (which only contains the identity element and $H$ itself.

For the first question, you want to use Euclid's algorithm to find an expression of the form $53a + 61b = 1$, i.e. $53a \equiv 1 \pmod {61}$ Then $a$ is the inverse to 53.
For the second question, you need to find the smallest $n$ such that $15^n \equiv 1 \pmod {61}$, so you would just keep multiplying by 15 and reducing mod 61 until you get to 1. (Note the reducing is important! Otherwise you might have to multiply really big numbers by 15 after a couple of steps.)
For the third question, have you calculated which elements are in $H$? If you do, then you should be able to work out the subgroup generated by each element, and hopefully find two which are different!
