Is there a signature scheme for the Cramer-Shoup cryptosystem? I have a project that I have to use cryptosystem and signature scheme for it.

I've read about Cramer-Shoup cryptosystem and I want to use it since it is more secure the ElGamal cryptosystem, but, I need a signature scheme based on it which I did not find.  Is there such a signature scheme?



If it does not exist I thought of two solutions: 


I can use DSA for it but since there is one generator $g$ in DSA and two generators $g_1$,$g_2$ in Cramer-Shoup cryptosystem. So I think I may not be able to use it.
I can generate a separate DH key and use it as a signature public key for the user. Then I could use this key for DSA. But I am not sure if this is considered secure.


Are these options valid?
It is very bad practice to use the same private key for two different schemes. In some cases this is secure but you need to explicitly prove it. One example of this can be seen here: http://www.pinkas.net/PAPERS/combined.ps. 

My suggestion is to take the Cramer-Shoup group and to define a separate key pair for DSA or Schnorr signatures. You can use the generator g_1 and this will be fine (the group can be the same; it's just the key that has to be different).