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I have the following problem.

Here's a rephrased version of your problem, keeping the LaTeX commands unchanged:

We are given $n \in \mathbb{N}$, $p, q \in \mathbb{N}$, and $y \in \mathbb{N}^{n+2}$. Assume the existence of a constant $K$ and a vector $x \in \mathbb{Z}^{n}$, fulfilling the relation:

$$y_{j+2} = \left((K \times y_{j+1}) \oplus (K + y_j \times x_j)\right) \quad \text{mod} \quad 2^{2p},$$

where each element $x_j$ (unknown) falls within the range $[0, 2^{q}[$ and similarly, $K \in [0, 2^{p}[$.

Can we find a solution of this system? If it can help $q < p$

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