I have been reading a paper of a cryptographic algorithm. At some point, algorithm takes a polynomial f(x) that belongs to $Z_3[x]/\langle x^p-x-1\rangle$ and takes the lifting of this polynomial to $Z[x]/\langle x^p-x-1\rangle$, where p is a prime. However It does not explain what the exact lifting operation is.
Can someone please explain what does "lifting" exactly mean, using an example polynomial say:
$f(x)=2x^8+2x^3+2x\in Z_3[x]/\langle x^3-x-1\rangle$, then what is the lifting of this polynomial to $Z[x]/\langle x^3-x-1\rangle$?
Note that you have a surjective ring homomorphism $\varrho:\Bbb Z[x]/\langle x^3-x-1\rangle\to\Bbb Z_3[x]/\langle x^3-x-1\rangle$ induced by the canonical projection $\Bbb Z\to\Bbb Z_3$. Consequently, for every polynomial $f\in\Bbb Z_3[x]/\langle x^3-x-1\rangle$ there exists a polynomial $g\in\Bbb Z[x]/\langle x^3-x-1\rangle$ such that $f=\varrho(g)$; we say that $f$ lifts to $g$. For polynomials \begin{align} &f=\sum_{i=0}^da_ix^i& &g=\sum_{i=0}^db_ix^i \end{align} the condition $f=\varrho(g)$ is equivalent to $a_i\equiv b_i\pmod 3$ for every $0\leq i\leq d$. A polynomial $f$ can have infinitely many such liftings $g$.