Cramer-Shoup asymmetric encryption (an extension of ElGamal)

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Is Cramer-Shoup backdoorable?

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 ...
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Can a Cramer-Shoup public key have two different private keys?

Since I am new to the Cramer-Shoup algorithm, I would like to know: Is it possible to have two different private keys for a public key due the key pair generation algorithm that Cramer-Shoup uses? I ...
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Key construction in the Full Cramer-Shoup cryptosystem

The Full Cramer-Shoup Encryption scheme needs to choose six random numbers from $\mathbb{Z}_q$, which we denote them by $ x_1,x_2,y_1,y_2,z_1,z_2 $. Then the scheme hides these random numbers in the ...