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I purposefully did not look at the details of the change you are proposing because whatever the change is, the answer is a resounding YES. If you make any change to a cryptographic construction, then you must prove the security of the modified scheme. If you are lucky, you may be able to reduce the security of the modified scheme to the original scheme, or ...


1

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 ...



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