Given a cycle $x \mapsto x^a$ with his starting point $x_1$. Can another starting point $x_2$ be transformed to generate the same cycle?

Question

A cyclic sequence can be produced with

$$s_{i+1} = s_i^a \mod N$$ with $N = P \cdot Q$ and $P = 2\cdot p+1$ and $Q = 2\cdot q+1$ with $P,Q,p,q$ primes.
and $a$ a primitive root of $p$ and $q$.
The starting point $s_0$ is a square ($\mod N$)
It will produce a cycle of length $\mathrm{lcm}(p-1.q-1)$
(except $s_0$ is a $p$-th or $q$-th power $\mod N$)

Given now a starting point $s_0 = x_1$ it will generate such a cyclic sequence.
Given another starting point $s_0 = x_2$ it will generate a cyclic sequence of same length but it an have completely different members.

Is there any way to transform $x_2$ so it will produce the same cyclic sequence as $x_1$ does?
(Edit: the posted answer is if any and not how, same as the question, will mark it as answer here)

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(related to the answer of this)

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Update:
It looks like the number of different cycles $N_c$ is: $$ N_c = (S_N - S_{pq}) /L_c$$ $$ S_N = |\{ v^2 \mod N\}| \text{ with } v\in[1,N-1]$$ $$L_c = \mathrm{lcm}(p-1.q-1)$$ and $S_{pq}$ the number of squares which are also a $p$-th, $q$-th power $\mod N$ .
$S_N$ probably always larger than $\frac{1}{4}N$

In some test for $N=3901$ with $P=47$ , $Q=83$, $a = 7$ (or $11, 17, 19,..$) two cycles are possible with $L_c =440$, $S_N = 1006$, $S_{pq}=127$.
One $x1$ can be transformed to a value from the other cycle (which starts with $x_2$) with an exponent $b$ like $x_1^b \mapsto s_i\in \mathrm{cycle}_{x_2}$
This exponent need to be $b \in [3 , 5 , 6 , 10 , 12 , 13, 20 , 21 , 24 , 26 , 27 , 29 , 33 , 35 , 37, 40, 42 , 43 , 45, 47, ...]$
No idea why exactly those values do work out.

For $N=40633, P= 179, Q= 227$ with $S_N= 10259$ squares, including $S_{pq}= 403$ it has $8$ cycles with length $L_c= 1232$. The exponent $a$ for sequence generation can be $a\in[3, 19, 23, 29, 43,..]$
For this exponent $b$ need to be $b \in [7 , 13, 17 , 21, 28 , 39 , 51 , 52 , 62 , 63 , 68 , 71 , 79 , 84 , 110 , 112 , 117,125,..]$
Applying any of those exponents $b$ to a starting value $x_0$ will result in a cycle of the next sequence. This cyclic sequence order is equal for every exponent $b$.

Answer 1

In analyzing cases like this, it is useful to look at the behaviors modulo $P$ and modulo $Q$ separately, and then see how they combine. I will add in the assumption that $P \ne Q$; you didn't explicitly say so; I believe it is reasonable that you assumed it.

When we look at the cycle structure modulo $P$, we first see that $s_0 \bmod P$ is a quadratic residue modulo $P$ (which is a mathy way of saying "it's a square"); since $a$ is a primitive root of $p$, then we see that there are three cycles:

• A cycle of length 1 (the value 0)

• Another cycle of length 1 (the value 1)

• A cycle of length $p-1$; this is because $a^i \bmod p-1$ are distinct values in the range $[0, p-2]$, and $s_0$ has order $p-1$ (in this group, quadratic residues other than 1 are always that order), and so $s_0^{a^i}$ are $p-1$ distinct values.

We get similar results for looking at the behavior modulo $Q$.

Given those basics, how do they combine?

Well, the cycle modulo $PQ$ repeats only when both the cycle modulo $P$ repeats and the cycle modulo $Q$ repeats; if the $P$-cycle is length $\alpha$ and the $Q$-cycle is length $\beta$, this combined cycle has length $\text{lcm}(\alpha, \beta)$.

This implies that any combination of the two large cycles will have length $\text{lcm}(p-1,q-1)$ (which is a result you have already found). And, there are $\gcd(p-1,q-1)$ ways these two large cycles can combine.

We now consider a combination that includes a small cycle; we have two combinations with cycle length $p-1$, two combinations with cycle length $q-1$, and four combinations with cycle length 1 (which includes $s_0 = 0$ and $s_0 = 1$).

Hence, the total number of cycles is $\gcd(p-1, q-1) + 8$.

And, to answer your question:

Is there any way to transform $x_2$ so it will produce the same cyclic sequence as $x_1$ does?

Well, for any integer $\beta$, we have $s_{i+1}^\beta = (s_i^\beta)^a$. That is, if we take every element of a cycle, and raise it to the power $\beta$, we still have a cycle.

So, if we have the value $\beta$ for which $x_2^\beta = x_1$, then that gives us a way to transform one cycle to another.

It turns out that there is always such a $\beta$ if the two cycles are both large (that is, of length $\text{lcm}(p-1, q-1)$). For the degenerate cycles (all the others), there won't be - however, I don't consider that case that interesting...

Of course, finding such a $\beta$ given $x_1, x_2$ is a nontrivial problem if $P, Q$ are large...