Quantum Position Verification in the Plane Serge Fehr and Dominique Unruh CWI University of Tartu Dominique Unruh
Position Verification Speed of light Position verified Quantum Position Verification in 2D 2 Dominique Unruh
A generic protocol time f ( x,y ) g ( x,y ) x y space verifier 1 verifier 2 prover Quantum Position Verification in 2D 3 Dominique Unruh
A generic attack time f ( x,y ) g ( x,y ) x y x y space verifier 1 verifier 2 adv 1 adv 2 [CGMO09] Chandran, Goyal, Moriarty, Ostrovsky, Position Based Cryptography , Crypto 2009 Quantum Position Verification in 2D 4 Dominique Unruh
Way out: quantum crypto • In attack: adversary copies x,y • If x or y quantum: No cloning! x y • Attack does not work • Other attacks? – Without extra assumptions: Generic attack (exponential entanglement) [BCF + 11] Buhrman, Chandran, Fehr, Gelles, Goyal, Ostrovsky, Schaffner: Position-Based Quantum Crypto , Crypto 2011 Quantum Position Verification in 2D 5 Dominique Unruh
Quantum crypto: A secure protocol time Assumption: No entangled photons in 𝒚 𝑪 Basis B verifier 1 verifier 2 prover [TFKW13] Tomamichel, Fehr, Kaniewski, Wehner: One-Sided Device- Independent QKD and Position-Based Cryptography from Monogamy Games , Eurocrypt 2013 (and [BCF + 11]) Quantum Position Verification in 2D 6 Dominique Unruh
2D/3D case verifier 2 verifier 1 verifier 0 time Measure Ψ in basis 𝜄 1 ⊕ 𝜄 2 space 𝜄 2 verifier 2 𝜄 1 space verifier 0 verifier 1 (There is a secure 3D protocol in the random oracle model, though [Unr14]) Quantum Position Verification in 2D 7 Dominique Unruh
Our result • Security proof in 2D-case • Sufficient for position verification “on earth” • 3D-case: open problem Quantum Position Verification in 2D 8 Dominique Unruh
Why is 2D/3D tricky? • Events (like getting all three messages) along complicated space-time surfaces • In some space-time areas, some but not all messages known • Complicated mix geometry + quantum Quantum Position Verification in 2D 9 Dominique Unruh
Proof technique: Space-time circuits • Tool: Space-time circuits AND – Gates have positions in space-time – No wire leaves light cone OR OR • Derive connectivity from geometry AND OR • Then forget about geometry, only use connectivity AND – Normal game-based proof OR [Unruh, Quantum Pos. Verif. in the RO model , Crypto 14] Quantum Position Verification in 2D 10 Dominique Unruh
Proof – analyzing space-time regions before protocol reachable by reaches reaches reachable by two verifiers two verifiers one verifier one verifier 𝜄 1 𝜄 2 Analyzed in [TFKW13]: 𝜄 1 𝜄 2 Pr[both verifiers guess 𝒚 ] many copies exponentially small. 𝜄 1 𝜄 2 many copies many copies many copies Quantum Position Verification in 2D 11 Dominique Unruh
Conclusion • 2D case solved • Lesson learned: Relativistic protocols complicated in 2D/3D – [BCF + 11] got it wrong. • Use space-time circuits! (Also for relativistic commitments) • 3D case: open problem Quantum Position Verification in 2D 12 Dominique Unruh
Thank you for your attention 13 Dominique Unruh
Postdoc Positions (also phd) Verification of Quantum Crypto Formal verification of quantum crypto protocols (“ QuEasyCrypt ” tool) http://tinyurl.com/postdoc-vqc Dominique Unruh
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