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I've been reading for Pohlig-Hellman Algorithm on Elliptic Curve.

My question is, why the simultaneous congruence gives the answer we want, could somebody please provide a proof of it?

i.e., why

\begin{align*} x \equiv x_i \mod p_i^{e_i}, \forall i\in \{1,2,...,n\} \end{align*}

Reference: https://risencrypto.github.io/PohligHellman/

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This is the case of the chinese remainder theorem (CRT) for the relatively prime moduli $m_i=p_i^{e_i}.$ They have decomposed the group order $p-1$ into $p-1=m=m_1 m_2 \cdots m_k.$

There are many proofs available of this. For example, the Wikipedia page has details of different algebraic structures but here we are just using integer residue rings.

See the direct construction section here.

It is also easy to see that as long as after integer reconstruction, we reduce the obtained expression $\pmod m$ we have a one-to-one map between $(x_1,\ldots,x_k)$ and $x.$

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