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.

Hints:


*

*For the inverse of $53$ mod $61$, you have to find a Bézout's relation between $53$ and $61$: $53 u+61 v=1$. The inverse is $u$, and you should find $38$. The coefficients are found through the extended Euclidean algorithm.

*The order of $15$ is a divisor of $60$, i.e. is is one of $\{1,2,3,4,5,6,10,15,20,30,60\}$. You only have to test these exponents, using for the greatest exponents. the fast exponentiation algorithm.

*You should find with the same method $8$ has order $20$. Hence the (cyclic) subgroup $\langle8\rangle$ is isomorphic to $\mathbf Z/20\mathbf Z$, and an element $8^k$ generates a subgroup of order $\dfrac{20}{\gcd(20,k)}$, so all you have to do is choosing an exponent $k\in(1,20)$, not coprime to $20$.

