Comparing classical and quantum conditional disclosure of secrets

TL;DR

This paper compares classical and quantum conditional disclosure of secrets, establishing bounds and separations that highlight the enhanced capabilities of quantum resources in cryptographic tasks.

Contribution

It provides new lower bounds and separations between classical and quantum CDS, using innovative techniques from quantum computation and communication complexity.

Findings

Quantum CDS has a logarithmic upper bound for forrelation, classical is linear.

Quantum CDS for the not-equals function has an exponential separation from classical.

Quantum CDS can outperform classical in private simultaneous message passing for certain functions.

Abstract

The conditional disclosure of secrets (CDS) setting is among the most basic primitives studied in information-theoretic cryptography. Motivated by a connection to non-local quantum computation and position-based cryptography, CDS with quantum resources has recently been considered. Here, we study the differences between quantum and classical CDS, with the aims of clarifying the power of quantum resources in information-theoretic cryptography. We establish the following results: 1) We prove a $\Omega(\log \mathsf{R}_{0,A\rightarrow B}(f)+\log \mathsf{R}_{0,B\rightarrow A}(f))$ lower bound on quantum CDS where $\mathsf{R}_{0,A\rightarrow B}(f)$ is the classical one-way communication complexity with perfect correctness. 2) We prove a lower bound on quantum CDS in terms of two round, public coin, two-prover interactive proofs. 3) For perfectly correct CDS, we give a separation for a promise version of the not-equals function, showing a quantum upper bound of $O(\log n)$ and classical lower bound of $\Omega(n)$. 4) We give a logarithmic upper bound for quantum CDS on forrelation, while the best known classical algorithm is linear. We interpret this as preliminary evidence that classical and quantum CDS are separated even with correctness and security error allowed. We also give a separation for classical and quantum private simultaneous message passing for a partial function, improving on an earlier relational separation. Our results use novel combinations of techniques from non-local quantum computation and communication complexity.