On the Round Complexity of Secure Quantum Computation

Abstract

Can a sender non-interactively transmit one of two strings to a receiver without knowing which string was received? Does there exist minimally-interactive secure multiparty computation that only makes (black-box) use of symmetric-key primitives? We provide affirmative answers to these questions in a model where parties have access to shared EPR pairs, thus demonstrating the cryptographic power of this resource.• First, we construct a one-shot (i.e., single message) string oblivious transfer (OT) protocol with random receiver bit in the shared EPR pairs model, assuming the (sub-exponential) hardness of LWE.Building on this, we show that secure teleportation through quantum channels is possible. Specifically, given the description of any quantum operation Q, a sender with (quantum) input ρ can send a single classical message that securely transmits Q(ρ) to a receiver. That is, we realize an ideal quantum channel that takes input ρ from the sender and provably delivers Q(ρ) to the receiver without revealing any other information. This immediately gives a number of applications in the shared EPR pairs model: (1) noninteractive secure computation of unidirectional classical randomized functionalities, (2) NIZK for QMA from standard (sub-exponential) hardness assumptions, and (3) a noninteractive zero-knowledge state synthesis protocol.

One-Way Functions Imply Secure Computation in a Quantum World

“…We note that while prior work [BCKM21a] has implicitly constructed a two-message zeroknowledge state synthesis protocol in the CRS model, our work is the first to obtain this primitive with one message in the shared EPR pair model. Corollary 4.25.…”

“…We now use a result from [BCKM21a] showing that any unidirectional quantum functionality can be securely computed in one message given access to a secure one-message protocol for arbitrary unidirectional classical functionalities. Since this theorem is not explicitly stated in [BCKM21a], we add some explanation for how it follows from their work.…”

“…It makes use of a type of quantum garbled circuit first described in [BY22] and formalized by [BCKM21a]. This quantum garbled circuit is based on the Clifford + measurement representation of quantum circuits, and supports a quantum input encoding procedure and classical garbling procedure.…”

“…Now, given a circuit Q describing a unidirectional quantum functionality, the sender samples k and sends |ψ ← Encode k (|ψ ) to the receiver. In addition, it inputs k to a unidirectional randomized classical functionality that will sample Q and along with a description of some measurement M that the receiver can apply to |ψ to check that it is well-formed (these are the zero and T checks described in [BCKM21a]). Thus, the receiver obtains |ψ from the sender, and M and Q from the classical functionality, checks that |ψ is well-formed using M , and finally evaluates Q( |ψ ) to obtain their output.…”

Quantum private function evaluation

“…Since MPQC is closely related to QPFE, we give a brief review of MPQC here. A more comprehensive review of the field of MPQC can be found in [9]. In the first work of MPQC [8], Crepeau et al present an information-theoretically secure MPQC protocol when the proportion of malicious parties is less than 1/6.…”