# Elliptic Curve - distinguish between two points after multiplication

If $$P$$ and $$Q$$ are two points on an elliptic curve of large prime order, given $$P, Q$$, and a point $$R$$ which is either (a) $$nP$$ or (b) $$nQ$$, is it possible to determine if $$R$$ is of form (a) or form (b)? Here $$n$$ is a secret.

As there exist both $$n_1$$ and $$n_2$$ such that $$R=n_1P$$ and $$R=n_2Q$$, $$R$$ is of form both a) and b). In general all elements of a cyclic group of prime order are generators and so all elements are multiples of all other elements (if the group operation is written additively).
If anybody can find out $$n$$ given $$R$$ and $$[n]P$$ or $$[n]Q$$ then they can break the discrete logarithm on this curve. To solve the DLog, just provide them $$(R,[n]P, [n]P)$$ and you solved DLog. So, this is equivalent to DLog.
One can distinguish weather $$R = [n]P$$ or $$R = [n]Q$$ ( i.e. determines that $$R$$ is from generator point $$P$$ or $$Q$$ ) if they are able solve the Dlog. The reverse reduction is not clear yet!.
• The question is not asking if you can find $n$, it is asking if you can distinguish which generator point was used May 22 at 19:18