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In Shoup's proof of the hardness of discrete log in the generic group in this paper, he mentions that:

At any step in the game, the algorithm has computed a list $F_1,\dots,F_k$ of linear polynomials in $Z/p^t[X]$ along with a list of values $z_1,\dots,z_k$ in $Z/s$, and a list $\sigma_1,\dots,\sigma_k$ of distinct values in $S$.

The algorithm is initially given the encodings of $1,x$ and access to the group operation + inverses so it is clear that anything the algorithm computes can be expressed as a linear polynomial in $Z/n[X]$, where $n=p^t s$. However, I don't see how this breaks down into a linear polynomial in $Z/p^t[X]$ and a constant in $Z/s$.

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$n$ and $p^t$ are coprime numbers. Then, we can use the Chinese remainder theorem to deduce a ring isomorphism $\phi$ from $\mathbb{Z}_n$ to $\mathbb{Z}_s\times \mathbb{Z}_{p^t}.$

And $\phi$ is just $y \mapsto (y \mod s, y \mod p^t).$

In the proof, the value of $(x\mod s)$ is chosen uniformly at random at the beginning of the game : it's $z_2$ in the paper. But $x \mod p^t$ is written as the indeterminate $X$ by the challenger (it's $F_2$). Because $1 \mod s = 1$ and $1 \mod p^t =1$, $(z_1,F_1)=(1,1).$

Then for all the new elements, they are written as a linear combination of $x$ and $1$ in $\mathbb{Z}_n$ and also (by using the chinese remainder theorem) as a linear combination of $(z_2, X)$ and $(1,1)$ in $\mathbb{Z}_n$.

To compute the new coefficients for the $k^{th}$ element which is decomposed as $\lambda_1 \cdot 1 + \lambda_x \cdot x$ in $\mathbb{Z}_n$, you have to use the ring isomorphism $\phi.$

$$ \phi(\lambda_1 \cdot 1 + \lambda_x \cdot x) = \phi(\lambda_1 \cdot 1) + \phi(\lambda_x \cdot x) = \phi(\lambda_1) +\phi(\lambda_x) \cdot(z_2, X) $$

$$=(\lambda_1 \mod s, \lambda_1 \mod p^t) + (\lambda_x \cdot z_2 \mod s, (\lambda_x \mod p^t)\cdot X)$$ $$= ((\lambda_1 +\lambda_x \cdot z_2) \mod s, \lambda_1 \mod p^t +(\lambda_x \mod p^t)\cdot X) $$

We deduce that $z_k=(\lambda_1 +\lambda_x \cdot z_2) \mod s$ and $F_k =\lambda_1 \mod p^t +(\lambda_x \mod p^t)\cdot X).$

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