Elliptic curves: for $P = aG$ for some $a$, what is $Q = a^{-1}G$?

Question

Given an elliptic curve group with a generator $G$ where $G$ has a prime order, p. Given a point $P=aG$ for some unknown $a$. Is it possible to efficiently calculate $Q=a^{-1}G$ without a discrete log operation?

With a discrete log, the problem is simple: first calculate $a$, then $a^{-1} = a^{p-1} $ mod $p$.

But I can't reduce a diffie-hellman problem to this to break it. Nor do I have the background to prove it directly (I have a background in NP-complete problems).

I see that the possibility of this operations would break a tiny subset of shared secrets but this should be negligible. So unless I'm wrong the existence of this algorithm isn't inconsistent with the original proof.

Answer 1

The name of the problem is `the Inverse Diffie-Hellman problem'. It is as hard as solving the computational Diffie-Hellman problem. A proof can be found in chapter 21, p.448-449 of Mathematics of Public Key Cryptography by Steven Galbraith (2012).

Source: this stackoverflow question.