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Private Proofs of When and Where

Uma Girish, Greg Gluch, Shafi Goldwasser, Tal Malkin, Leo Orshansky, Henry Yuen

TL;DR

This work defines zero-knowledge position verification, enabling privacy-preserving proofs about where and when a prover was located. It constructs position commitments from post-quantum one-way functions and nice singleton position-verification protocols, then upgrades these into zero-knowledge proofs for any finite spacetime region under honest verifiers. The central technique, position commitments, provides hiding during commitment and binding at reveal, which can be leveraged with NP-zero-knowledge proofs to produce ZK location proofs. The paper also discusses practical optimizations to improve efficiency and analyzes the honest-verifier setting, outlining challenges and avenues for extending privacy against malicious verifiers. Overall, the results advance privacy-preserving, location-based proofs with potential applications in alibi verification, treaty compliance, and location-based cryptographic credentials.

Abstract

Position verification schemes are interactive protocols where entities prove their physical location to others; this enables interactive proofs for statements of the form "I am at a location $L$." Although secure position verification cannot be achieved with classical protocols (even with computational assumptions), they are feasible with quantum protocols. In this paper we introduce the notion of zero-knowledge position verification, which generalizes position verification in two ways: 1. enabling entities to prove more sophisticated statements about their locations at different times (for example, "I was NOT near location $L$ at noon yesterday"). 2. maintaining privacy for any other detail about their true location besides the statement they are proving. We construct zero-knowledge position verification from standard position verification and post-quantum one-way functions. The central tool in our construction is a primitive we call position commitments, which allow entities to privately commit to their physical position in a particular moment, which is then revealed at some later time.

Private Proofs of When and Where

TL;DR

This work defines zero-knowledge position verification, enabling privacy-preserving proofs about where and when a prover was located. It constructs position commitments from post-quantum one-way functions and nice singleton position-verification protocols, then upgrades these into zero-knowledge proofs for any finite spacetime region under honest verifiers. The central technique, position commitments, provides hiding during commitment and binding at reveal, which can be leveraged with NP-zero-knowledge proofs to produce ZK location proofs. The paper also discusses practical optimizations to improve efficiency and analyzes the honest-verifier setting, outlining challenges and avenues for extending privacy against malicious verifiers. Overall, the results advance privacy-preserving, location-based proofs with potential applications in alibi verification, treaty compliance, and location-based cryptographic credentials.

Abstract

Position verification schemes are interactive protocols where entities prove their physical location to others; this enables interactive proofs for statements of the form "I am at a location ." Although secure position verification cannot be achieved with classical protocols (even with computational assumptions), they are feasible with quantum protocols. In this paper we introduce the notion of zero-knowledge position verification, which generalizes position verification in two ways: 1. enabling entities to prove more sophisticated statements about their locations at different times (for example, "I was NOT near location at noon yesterday"). 2. maintaining privacy for any other detail about their true location besides the statement they are proving. We construct zero-knowledge position verification from standard position verification and post-quantum one-way functions. The central tool in our construction is a primitive we call position commitments, which allow entities to privately commit to their physical position in a particular moment, which is then revealed at some later time.
Paper Structure (32 sections, 8 theorems, 7 equations, 3 figures, 1 algorithm)

This paper contains 32 sections, 8 theorems, 7 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Position verification for regions of spacetime
  3. A canonical position verification protocol.
  4. Zero-knowledge position verification
  5. Technical overview
  6. Constructing position commitments.
  7. From position commitments to zero-knowledge position verification.
  8. Optimizing the position commitment protocol.
  9. A brief overview of quantum position verification
  10. Summary and future directions
  11. The honest-verifier assumption.
  12. Towards stronger position verification guarantees.
  13. Towards more realistic models.
  14. Preliminaries
  15. Notation
  16. ...and 17 more sections

Key Result

Theorem 1.1

Suppose there exist post-quantum one-way functions and nice singleton position verification protocols which have position security against spoofers sharing at most $E$ entangled qubits. Then for all finite $R \subseteq \mathbb{R}^d \times \mathbb{R}$, there is an (honest-verifier) zero-knowledge pos

Figures (3)

  • Figure 1: A spacetime diagram of the $f$-BB84 protocol bluhm2022single, specialized to 1 dimension position verification. Time goes up, position is horizontal, and signals travel along 45$^{\circ}$ angles.
  • Figure 2: The position commitment described in \ref{['def:optimized-pc']} pictured for 1D (left), and 2D (right). In $d$ spatial dimensions, intersections between $d+1$ different verifier messages represent spacetime points at which an honest prover might receive a set of position verification challenges, which they will respond to in an encrypted way. A few example points have been bolded on the right for clarity.
  • Figure 3: Illustration of the optimized position commitments in \ref{['def:optimized-pc']} for $d=2$, with intersection points highlighted in red. When these arcs are as dense as one per timestep, the set of intersection points will be a fine-grained mesh.

Theorems & Definitions (24)

  • Theorem 1.1: Zero-knowledge position verification, informal
  • Remark 1.2
  • Theorem 1.3: informal version of \ref{['thm:construction-secure']}
  • Definition 2.1: Computational Indistinguishability
  • Definition 2.2: Secret-Key Encryption
  • Definition 2.3: Commitment Scheme
  • Definition 2.4: Zero-Knowledge Proof
  • Remark 2.6
  • Definition 2.7: Position Verification
  • Definition 2.8: Nice Protocols
  • ...and 14 more