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Let $E$ be an elliptic curve over a finite field $F_p$. Given $n$ be a positive integer and $Q$ be a point on $E$, assume that $Q=nP$, how can we find this $P$? We can assume that $n|p-1$. If $n$ is "small", I would imagine that it is possible using division polynomials. Is it a difficult problem if $n$ is large enough? How difficult is it?

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You can recover $P$ by computing $(n^{-1} \bmod l)\cdot Q$, where $l$ is the order of $Q$. –  Samuel Neves Jun 29 '13 at 2:57
Solving $Q=np$ for $n$ is the discrete logarithm problem and expensive. Solving for $P$ is cheap (assuming the order of the curve is known). –  CodesInChaos Jun 29 '13 at 11:20
Why do you assume that $n$ divides $p-1$ ? Is there any specific reason for this condition, if so could you explain what it is ? –  minar Jul 14 '13 at 17:56

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As Samuel Neves described in the comments it is trivially possible to obtain $P$ from $Q=nP$, given $Q$ and $P$.

Simply compute $k=n^{-1} \bmod l$, with $l=|E(\mathbb F_p)|$ being the order (=the number of points) of the curve. $l$ is usually known.

Then, to obtain the desired point $P$, calculate $P=kQ=knP$ and you're done.

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