We know $$$f(x)\equiv \exp(\ln f(x))\equiv \pmod Q$$$ so $$$f^k(x) \equiv \exp(k\ln f(x))\equiv \exp(k\bmod Q\ln f(x))\equiv f^{k\bmod Q} (x) \pmod Q$$$ (Here, it is assumed that $$$\exp$$$ and $$$\ln$$$ are polynomials), but when $$$f(x)=(x+1),Q=2,k=2$$$, $$$f^k(x)\equiv 1\pmod Q$$$, but $$$f^{k\bmod Q}(x)\equiv x^2+1\pmod Q$$$, WHY??