GM is a cryptosystem with a modulus $N = pq$ ($p$, $q$ prime) and quadratic nonresidue $r \in (\Bbb Z / N\Bbb Z)^*$ as the public key. You can see the details on Wikipedia.
Anyway, let's say that $N < 2^{1024}$. Then the ciphertext can be represented as a $1024$-bit string regardless of $N$. In which case, you can take a ciphertext $c$ (which is the encryption of a single bit), and then multiply $c$ by a random $z^2 \pmod{2^{1024}}$, and get an equivalent ciphertext. An adversary should, as far as I can tell, be unable to decide whether this decrypts to the same plainbit as $c$. And you don't need to know the public key $(N,r)$ to do this operation.
Is there any literature on this property?