Let $G$ be a group, $g ∈_R G, x ∈_R Z_q$, and $e: G \times G \rightarrow G_T$ be a bilinear paring.
Then, given $g, g^x$, is it still hard to compute $g^{x^2}$?
1. In other words is Square Diffie-Hellman hard in Bilinear Groups?
I am trying to look for references that have used this problem or a reduction which proves this is hard.
[I am aware of a variant of this called the Flexible Square Diffie-Hellman [Laguillaumie et al], which asks to compute $(h,h^x,h^{x^2})$ given $g^x$.]
2. What about the decisional version of the same problem?