Let's assume $N$ is Paillier encryption public key, where $N=pq$ and $p,q$ are strong prime numbers.
I have a set of plaintext $p_i$ whose domain is $N$. I want to deterministically generate a series of pseudorandom values $r_i$ that have multiplicative inverse. So $r_i$ should be distributed uniformly random over $N$ or its subfield.
I want to blind each plaintext as $b_i=p_i\cdot r_i \bmod N$. So given $b_i$ a semi-honest adversary cannot learn anything about $p_i$.
Question: Is there any way to construct a pseudorandom function that outputs uniformly random values over $N$ (or its subfield) and has multiplicative inverse with a high probability?