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?

  • 3
    $\begingroup$ You can recover $P$ by computing $(n^{-1} \bmod l)\cdot Q$, where $l$ is the order of $Q$. $\endgroup$ Jun 29, 2013 at 2:57
  • $\begingroup$ 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). $\endgroup$ Jun 29, 2013 at 11:20
  • $\begingroup$ 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 ? $\endgroup$
    – minar
    Jul 14, 2013 at 17:56

1 Answer 1


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.


Not the answer you're looking for? Browse other questions tagged or ask your own question.