# Use of randomness in an Elgamal like encryption

Suppose I have the following encryption scheme: for a message $$m\in\mathbb{F}_p^*$$, I generate the ciphertext = $$(g^r,f^mh^r)$$ where $$g$$ is the generator of a cyclic group $$G$$ of unknown order $$n$$ and $$F$$ is a cyclic subgroup of $$G$$ of order $$p$$ and $$h=g^x$$ where $$x$$ is the secret key.

Since $$G$$'s order is unknown we randomly generate $$x$$ from a sufficiently large set of numbers $$\{0, \cdots, B\}$$ where $$B$$ is an upper bound on $$n$$ that can be calculated for a $$b$$ bit security level and $$B$$ is chosen in way such that $$g^x$$ is indistinguishable from the uniform distribution in $$G$$ i.e. to ensure $$g^x$$ is a random element from $$G$$.

Now suppose we want to protect this scheme against generic discrete log attacks for 80 bits of security and in this case our $$\log_2B\gg 160$$. Also, for 80 bits security, our secret key $$x$$ has to be at least 160 bits as our generic discrete log attacks have a $$\sqrt n$$ complexity.

Question -- Because I only need $$x$$ to be of 160 bits, can I explicitly choose a random $$x$$ of 160 bits and disregard my theoretical bound $$B$$ which is huge in size. My reasoning is yes because I want to prevent an attack at 80 bit security level, all I need is $$x$$ to be 160 bits and I can ignore $$B$$. Is my reasoning valid? Also, some explanation would be great if I am wrong.