Boyang Chen

We give a meta-complexity characterization of EFI pairs, which are considered the “minimal” primitive in quantum cryptography (due to their equivalence to quantum commitments and for being implied by almost all other known quantum cryptographic primitives). More precisely, we show that the existence of EFI pairs is equivalent to the following: there exists a non-uniformly samplable distribution over pure states such that the problem of estimating a certain Kolmogorov-like complexity measure is hard given a single copy. The complexity measure that we consider is a smoothed version of the algorithmic entropy notion introduced by Gács [Gác01]. A key technical step in our proof, which may be of independent interest, is to show that the existence of EFI pairs is equivalent to the existence of non-uniform single-copy secure pseudorandom state generators (nu 1-PRS). As a corollary, we get an alternative, arguably simpler, construction of a universal EFI pair.

In this work, we focus on the following question: what are the cryptographic implications of having access to an oracle that provides a single Haar random quantum state? We show, perhaps surprisingly, that such an oracle is sufficient to construct quantum pseudorandomness. Pseudorandom states (PRS) are a family of states for which it is hard to distinguish between polynomially many copies of either a state sampled uniformly from the family or a Haar random state. A weaker notion, called single-copy pseudorandom states (1PRS), satisfies this property with respect to a single copy. Our main result is that 1PRS (as well as bit-commitments) exist relative to an oracle that provides a single Haar random state. We build on this result to show the existence of an oracle relative to which 1PRS exist, but PRS do not. This provides one of the first black-box separations between different forms of quantum pseudorandomness.

We show that there exists a quantum oracle relative to which quantum commitments exist but no (efficiently verifiable) one-way state generators exist. Both have been widely considered candidates for replacing one-way functions as the minimal assumption for cryptography—the weakest cryptographic assumption implied by all of computational cryptography. Recent work has shown that commitments can be constructed from one-way state generators, but the other direction has remained open. Our results rule out any black-box construction, and thus settle this crucial open problem, suggesting that quantum commitments (as well as its equivalency class of EFI pairs, quantum oblivious transfer, and secure quantum multiparty computation) appear to be strictly weakest among all known cryptographic primitives.