Where $g$ is a group element in bilinear group $G$, $e(g,g)∈GT$ and $s, y, r, t, d$ are randomly chosen.
I understand it is very similar to the conventional DBDH problem, but $g^t, g^{st-rs}, g^{(yr+d)/t}$ are also known, possibly making it easier???
Does anyone know the answer or suggest some material for reference?
Many Thanks
No, it's no easier than the standard DBDH problem.
Here's the reduction that shows that: suppose that we have an Oracle that solves your problem (given $g^s, g^y, g^r, g^t, g^{st-rs}, g^{(yr+d)/t}, e(g,g)^x$ is $e(g,g)^x = e(g,g)^{syr}$?)
Now, suppose we're given $g^s, g^y, g^r, e(g,g)^x$, and are asked whether $e(g,g)^x = e(g,g)^{syr}$.
What we do is select a random values $t'$ and $z$, and compute $g^{t'} / g^r = g^t$ and $(g^s)^{t'} = g^{st-rs}$; we don't know the value of $t = t' - r$, but we know that it exists). We further compute $g^z = g^{(yr+d)/t}$ (again, we don't know the value $d = zt - yr$; but we don't have to know).
Then, with these values, along with the values we got with the original query, we ask the Oracle (and get the answer).
