On the Computational Hardness of Quantum OneWayness
Abstract
There is a large body of work studying what forms of computational hardness are needed to realize classical cryptography. In particular, oneway functions and pseudorandom generators can be built from each other, and thus require equivalent computational assumptions to be realized. Furthermore, the existence of either of these primitives implies that $\rm{P} \neq \rm{NP}$, which gives a lower bound on the necessary hardness. One can also define versions of each of these primitives with quantum output: respectively oneway state generators and pseudorandom state generators. Unlike in the classical setting, it is not known whether either primitive can be built from the other. Although it has been shown that pseudorandom state generators for certain parameter regimes can be used to build oneway state generators, the implication has not been previously known in full generality. Furthermore, to the best of our knowledge, the existence of oneway state generators has no known implications in complexity theory. We show that pseudorandom states compressing $n$ bits to $\log n + 1$ qubits can be used to build oneway state generators and pseudorandom states compressing $n$ bits to $\omega(\log n)$ qubits are oneway state generators. This is a nearly optimal result since pseudorandom states with fewer than $c \log n$qubit output can be shown to exist unconditionally. We also show that any oneway state generator can be broken by a quantum algorithm with classical access to a $\rm{PP}$ oracle. An interesting implication of our results is that a $t(n)$copy oneway state generator exists unconditionally, for every $t(n) = o(n/\log n)$. This contrasts nicely with the previously known fact that $O(n)$copy oneway state generators require computational hardness. We also outline a new route towards a blackbox separation between oneway state generators and quantum bit commitments.
 Publication:

arXiv eprints
 Pub Date:
 December 2023
 DOI:
 10.48550/arXiv.2312.08363
 arXiv:
 arXiv:2312.08363
 Bibcode:
 2023arXiv231208363C
 Keywords:

 Computer Science  Cryptography and Security;
 Computer Science  Computational Complexity;
 Quantum Physics
 EPrint:
 Abstract modified to fit ArXiv requirements