RSA assumption and relationship given by generating elemts of a Cayley graph
For finding the maximum length cycle, it is taken as finding the maximum solution to the two congruence relations. But since these are congruent relations how can we justify that the maximum will give the answer relevant to the longest cycle?


It looks like it shouldn't. I'm pretty sure I was visualizing cycles in the Cayley Graph as being paths in the lattice $\mathbb{Z}^2$. This corresponds to viewing points in the RSA group as tuples $(g_p^{r_p}, g_q^{r_q})$ (where $(r_p, r_q)\in\mathbb{Z}^2$). One could hope to make some statement about maximum elements in this representation relating to maximum length cycles, but this is different than the congruences mentioned.
One could maybe hope to prove some statement about maximal elements in the above representation $(r_p, r_q)$, and then transfer (via exponentiation) to the &quot;standard representation&quot; and hope that they are &quot;close to maximal&quot;. I would strongly expect this to be false though --- this property would be akin to being something like Lipschitz normally, which is much more regularity than I'd expect from these functions.


Can we write other groups such as $(\mathbb{Z}_p\times\mathbb{Z}_p)\rtimes_\varphi(\mathbb{Z}_q\times\mathbb{Z}_q)$, where $p, q$ are odd distinct primes, in terms of a free group as above? I am very glad if some guidance/steps to do so can be explained.


For $G\rtimes_\varphi H$ I explicitly include the homomorphism $\varphi : H \to \mathsf{Aut}(G)$ that the semi-direct product is defined with respect to.
The concept you're looking for is that of a group presentation.
This is a way of writing a group $G$ as a set of generators $R$ and relations $S$ the generators satisfy (denoted $\langle R | S\rangle$).
Equivalently, this is a way of writing a group $G$ as a free group on the generators $R$, quotiented by the normal subgroup generated by the relations $S$.
In this terminology, your question becomes &quot;What is a group presentation of $(\mathbb{Z}_p\times\mathbb{Z}_p)\rtimes_\varphi(\mathbb{Z}_q\times\mathbb{Z}_q)$?&quot;
It will be useful to know how group presentations behave under direct products and semi-direct products.

Let $G_1 = \langle R_1 | S_1\rangle$ and $G_2 = \langle R_2 | S_2\rangle$. Then:

$G_1 \times G_2 = \langle R_1, R_2 | S_1, S_2, [R_1, R_2]\rangle$

$G_1\rtimes_\varphi G_2 =\langle R_1, R_2 | S_1 , S_2, \forall (r_1, r_2)\in R_1\times R_2 : r_2 r_1^{-1}r_2 = \varphi(r_2)(r_1)\rangle$



Here $[A, B]$ is the commutator subgroup.
One should be able to use the above &quot;transformation rules&quot; (along with the presentation $\mathbb{Z}_p = \langle g | g^p = e\rangle$ for prime $p$) to compute a group presentation for the group you're interested in (or any other group built from direct and semi-direct products of cyclic groups).