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Conditional disclosure of secrets with quantum resources

Vahid R. Asadi, Kohdai Kuroiwa, Debbie Leung, Alex May, Sabrina Pasterski, Chris Waddell

TL;DR

This work initiates a systematic study of CDQS, the quantum analogue of CDS, by establishing closure under negation, amplification, and fundamental lower bounds drawn from quantum communication complexity and interactive proofs, while connecting CDQS to $f$-routing and quantum position verification. It shows that CDQS can be amplified from one-qubit secrets to $k$-qubit secrets with exponentially small error and linear scaling of resources, using quantum error-correcting codes. The paper also derives tight lower bounds from one-way quantum CC and QIP-type frameworks, and it examines the cost implications under Clifford restrictions, highlighting both parallels and key differences with the classical CDS landscape. The results provide a rigorous resource-based foundation for quantum privacy in CDS/CDQS and point to open questions about entanglement sparsification, potential quantum advantages, and tighter general-function CDQS characterizations.

Abstract

The conditional disclosure of secrets (CDS) primitive is among the simplest cryptographic settings in which to study the relationship between communication, randomness, and security. CDS involves two parties, Alice and Bob, who do not communicate but who wish to reveal a secret $z$ to a referee if and only if a Boolean function $f$ has $f(x,y)=1$. Alice knows $x,z$, Bob knows $y$, and the referee knows $x,y$. Recently, a quantum analogue of this primitive called CDQS was defined and related to $f$-routing, a task studied in the context of quantum position-verification. CDQS has the same inputs, outputs, and communication pattern as CDS but allows the use of shared entanglement and quantum messages. We initiate the systematic study of CDQS, with the aim of better understanding the relationship between privacy and quantum resources in the information theoretic setting. We begin by looking for quantum analogues of results already established in the classical CDS literature. Doing so we establish a number of basic properties of CDQS, including lower bounds on entanglement and communication stated in terms of measures of communication complexity. Because of the close relationship to the $f$-routing position-verification scheme, our results have relevance to the security of these schemes.

Conditional disclosure of secrets with quantum resources

TL;DR

This work initiates a systematic study of CDQS, the quantum analogue of CDS, by establishing closure under negation, amplification, and fundamental lower bounds drawn from quantum communication complexity and interactive proofs, while connecting CDQS to -routing and quantum position verification. It shows that CDQS can be amplified from one-qubit secrets to -qubit secrets with exponentially small error and linear scaling of resources, using quantum error-correcting codes. The paper also derives tight lower bounds from one-way quantum CC and QIP-type frameworks, and it examines the cost implications under Clifford restrictions, highlighting both parallels and key differences with the classical CDS landscape. The results provide a rigorous resource-based foundation for quantum privacy in CDS/CDQS and point to open questions about entanglement sparsification, potential quantum advantages, and tighter general-function CDQS characterizations.

Abstract

The conditional disclosure of secrets (CDS) primitive is among the simplest cryptographic settings in which to study the relationship between communication, randomness, and security. CDS involves two parties, Alice and Bob, who do not communicate but who wish to reveal a secret to a referee if and only if a Boolean function has . Alice knows , Bob knows , and the referee knows . Recently, a quantum analogue of this primitive called CDQS was defined and related to -routing, a task studied in the context of quantum position-verification. CDQS has the same inputs, outputs, and communication pattern as CDS but allows the use of shared entanglement and quantum messages. We initiate the systematic study of CDQS, with the aim of better understanding the relationship between privacy and quantum resources in the information theoretic setting. We begin by looking for quantum analogues of results already established in the classical CDS literature. Doing so we establish a number of basic properties of CDQS, including lower bounds on entanglement and communication stated in terms of measures of communication complexity. Because of the close relationship to the -routing position-verification scheme, our results have relevance to the security of these schemes.
Paper Structure (11 sections, 21 theorems, 71 equations, 4 figures)

This paper contains 11 sections, 21 theorems, 71 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Some quantum information tools
  3. Basic properties
  4. Closure under negation, AND, and OR
  5. Amplification
  6. Lower bounds
  7. Lower bounds from one-way quantum communication complexity
  8. Lower bounds on perfectly private CDQS from TEXT
  9. Lower bounds from quantum interactive proofs
  10. Discussion
  11. Clifford CDQS

Key Result

Theorem 4

A $\epsilon$-correct $f$-routing protocol that routes $n$ qubits implies the existence of a $\epsilon$-correct and $\delta=2\sqrt{\epsilon}$-secure $\textnormal{CDQS}$ protocol that hides $n$ qubits using the same entangled resource state and the same message size. A $\epsilon$-correct and $\delta$-

Figures (4)

  • Figure 1: (a) A CDS protocol. Alice, on the lower left holds input $x\in \{0,1\}^n$ and a secret $z$ from alphabet $Z$. Bob, on the lower right, holds input $y\in \{0,1\}^n$. Alice and Bob can share a random string $r$. The referee, top right, holds $x$ and $y$. Alice sends a message $m_0(x,z,r)$ to the referee; Bob sends a message $m_1(y,r)$. The referee should learn $z$ iff $f(x,y)=1$ for some agreed on choice of Boolean function $f$. (b) A CDQS protocol. The communication pattern is as in CDS. The secret is now a quantum system $Q$, Alice and Bob can share a (possibly entangled) quantum state, and send quantum messages to the referee. The referee should be able to recover $Q$ iff $f(x,y)=1$.
  • Figure 2: Known upper and lower bounds on communication cost in CDQS. Blue entries are new to this work.
  • Figure 3: Known upper and lower bounds on entanglement plus communication cost in CDQS. Blue entries are new to this work.
  • Figure 4: A CDQS protocol, with all system labels and location of each quantum operation.

Theorems & Definitions (36)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 4
  • Theorem 5
  • Remark 6
  • Definition 7
  • Theorem 8
  • Theorem 9
  • Theorem 10
  • ...and 26 more