# inverse problem about scalar multiplication on elliptic curve

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?

• You can recover $P$ by computing $(n^{-1} \bmod l)\cdot Q$, where $l$ is the order of $Q$. Commented Jun 29, 2013 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). Commented Jun 29, 2013 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 ? Commented Jul 14, 2013 at 17:56
• @SamuelNeves does it works on secp256k1 which is a koblitz curve? Commented Jan 27 at 22:16
• Yes, it will work on any prime order curve. Commented Jan 28 at 2:26

As Samuel Neves described in the comments it is trivially possible to obtain $$P$$ from $$Q=[n]P$$, given $$Q$$ and the integer $$n$$.
Simply compute $$k=n^{-1} \bmod \ell$$, with $$\ell=|E(\mathbb F_p)|$$ being the order (= the number of points) of the curve. $$\ell$$ is usually known.
Then, to obtain the desired point $$P$$, calculate $$P=[k]Q=[kn]P$$ and you're done.