How does $g^{x^2} \bmod p$ help you find $x$? I was thinking about the Diffie-Hellman key exchange. One fact that we seem to know is that given a group generator $g$, a prime $p$ and the expression $g^x \bmod p$ its believed to be hard to find $x$; or more accurately $x \bmod(p-1)$.
Now the question is, what if we gave away some extra information. Such as $g^{x^2} \bmod p$.
My guess is that surely this should help us find $x$ in some fashion, but it’s not clear how.
Finding an exponent that sends $g^x$ to $g^{x^2}$ is as hard as finding $x$ given just our usual initial data. So that’s probably not the right way to exploit this information.
It is not known that knowing $g^{x^2}$ would help in any meaningful way to solve the discrete logarithm problem.
What you are describing can be generalized to asking whether in a group $\mathbb{G}$ giving an attacker $$(g, g^x,g^{x^2},\dots,g^{x^q})$$ for some $q$ and a uniformly chosen $x \in |\mathbb{G}|$ makes it feasible to recover $x$.
It is generally assumed that this is not the case. In fact the assumption that recovering $x$ remains hard is called the $q$-strong discrete logarithm ($q$-SDL) assumption and has been used before. [GOR11] Similarly, the bilinear variant is also assumed to be hard. [FS20]
While the $q$-SDL assumption itself is not used very often, it is actually implied by a variety of much more commonly used variants of $q$-strong Diffie Hellman assumptions. [TS10]

[GOR11] Vipul Goyal, Adam O'Neill, and Vanishree Rao. &quot;Correlated-Input Secure Hash Functions&quot;. TCC 2011
[FS20] Nils Fleischhacker and Mark Simkin. &quot;Robust Property-Preserving Hash Functions for Hamming Distance and More&quot;. Cryptology ePrint Archive, Report 2020/1301, 2020
[TS10] Naoki Tanaka and Taiichi Saito, &quot;On the $q$-Strong Diffie-Hellman Problem&quot;. Cryptology ePrint Archive, Report 2010/215, 2010