# Is Square DH hard in Bilinear Groups?

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?

• I'm assuming DH is diffie helman ? is there a link to a description of the problem ? is this well known ? – Suresh Venkat Mar 11 '14 at 17:34
• Yes, DH is Diffie Hellman. And no, I don't think the problem is well known (at least in the bilinear setting). Otherwise, traditional square DH is quite popular and is known to be as hard as CDH. – Subhayan Mar 12 '14 at 4:21

• So, from the paper, can I infer, given $g,h∈G,\ g^a$, it is hard to compute $e(g,h)^{a^2}$ ? – Subhayan Mar 27 '14 at 16:17