We recently had the question whether it's possible to have multiple private keys with one public key for the cramer-shoup cryptosystem.
There it was stated that finding such "secondary" private keys is possible if $log_{g_1}(g_2) \bmod q$ is known. However computing this discrete logarithm is infeasible as both $g$s need to be random.
But the description also mentions that shared group parameters are allowed, meaning a party A could generate the $g$s so other partys could use them.
Now let's assume this party A to be some standardization body, like NIST, getting the $g$s by somebody (for example the NSA). Now let's further assume that there exists some $a$ for which $g^a_1\equiv g_2 \pmod q$ holds and that the supplying party knows $a$.
Can the supplying party (knowing $a$) anyhow break the security of Cramer-Shoup (=IND-CCA2) if honest parties use the backdoored parameters?
Note: "honest parties" use the standardized group and choose their private keys following the specification and assume security of the parameters.