One of the central questions in cryptology is how efficient generic constructions of cryptographic primitives can be. Gennaro, Gertner, Katz, and Trevisan [SIAM J. of Compt., 2005] studied the lower bounds of the number of invocations of a (trapdoor) one-way permutation in order to construct cryptographic schemes, e.g., pseudorandom number generators, digital signatures, and public-key and symmetric-key encryption.

Recently, quantum machines have been explored to _construct_ cryptographic primitives other than quantum key distribution. This paper studies the efficiency of _quantum_ black-box constructions of cryptographic primitives when the communications are _classical_. Following Gennaro et al., we give the lower bounds of the number of invocations of an underlying quantumly-computable quantum-one-way permutation when the _quantum_ construction of pseudorandom number generator and symmetric-key encryption is weakly black-box. Our results show that the quantum black-box constructions of pseudorandom number generator and symmetric-key encryption do not improve the number of invocations of an underlying quantumly-computable quantum-one-way permutation.

Proving whether it is possible to build IND-CCA public-key encryption (PKE) from IND-CPA PKE in a black-box manner is a major open problem in theoretical cryptography. In a significant breakthrough, Gertner, Malkin and Myers showed in 2007 that shielding black-box reductions from IND-CCA to IND-CPA do not exist in the standard model. Shielding means that the decryption algorithm of the IND-CCA scheme does not call the encryption algorithm of the underlying IND-CPA scheme. In other words, it implies that every tentative construction of IND-CCA from IND-CPA must have a re-encryption step when decrypting.

This result was only proven with respect to classical algorithms. In this work we show that it stands in a post-quantum setting. That is, we prove that there is no post-quantum shielding black-box construction of IND-CCA PKE from IND-CPA PKE. In the type of reductions we consider, i.e. post-quantum ones, the constructions are still classical in the sense that the schemes must be computable on classical computers, but the adversaries and the reduction algorithm can be quantum. This suggests that considering quantum notions, which are stronger than their classical counterparts, and allowing for quantum reductions does not make building IND-CCA public-key encryption easier.