  There has been a recent interest in proposing quantum protocols whose
security relies on weaker computational assumptions than their classical
counterparts. Importantly to our work, it has been recently shown that
public-key encryption (PKE) from one-way functions (OWF) is possible if we
consider quantum public keys. Notice that we do not expect classical PKE from
OWF given the impossibility results of Impagliazzo and Rudich (STOC'89).
However, the distribution of quantum public keys is a challenging task.
Therefore, the main question that motivates our work is if quantum PKE from OWF
is possible if we have classical public keys. Such protocols are impossible if
ciphertexts are also classical, given the impossibility result of Austrin et
al. (CRYPTO'22) of quantum enhanced key-agreement (KA) with classical
communication. In this paper, we focus on black-box separation for PKE with
classical public key and quantum ciphertext from OWF under the polynomial
compatibility conjecture, first introduced in Austrin et al.. More precisely,
we show the separation when the decryption algorithm of the PKE does not query
the OWF. We prove our result by extending the techniques of Austrin et al. and
we show an attack for KA in an extended classical communication model where the
last message in the protocol can be a quantum state.
