Discrete Logarithm problem with inverse

Question

Let $\mathbb G$ be a cyclic group of order $q$. The Discrete Logarithm Problem (DLP) is, given $g, g^x \in \mathbb G$, to compute $x \in \mathbb Z_q $.

I'm interested to know if there is a known variant of DLP as follows:

Given $g, g^x, g^{x^{-1}\bmod q} \in \mathbb G$, compute $x \in \mathbb Z_q$

It is clear that this variant would not be harder than the DLP, since a DLP solver could be used to solve this problem simply by ignoring the third input.

Answer 1

This problem seem to be equivalent to a particular instance of the $k$-Strong Discrete Logarithm Problem.

Let $\mathbb G$ be a cyclic group of order $q$. The $k$-Strong Discrete Logarithm Problem ($k$-DLP) is, given $h, h^x, h^{x^2}, .., h^{x^k} \in \mathbb G$, to compute $x \in \mathbb Z_q $. Taking a simple change of variable ($h^x = g$), one obtains an equivalent version of the problem, which is as follows:

Given $g^{x^{-1}\bmod q}, g, g^x, ..., g^{x^{k-1}} \in \mathbb G$, compute $x \in \mathbb Z_q $

It can be seen then that the problem in the question is the case of $k=2$.

The $k$-Strong Discrete Logarithm Problem is discussed in Correlated-Input Secure Hash Functions by Goyal, O'Neill, and Rao, for instance.

Answer 2

This is the Strong-DDH: strong decision Diffie-Hellman problem. See Final Report on Main Computational Assumptions in Cryptography. Assumption number 19.

If the DDH is hard then this implies that it must also be hard to calculate the discrete log.