Quantum Position Verification in the random oracle model Dominique Unruh University of Tartu Dominique Unruh
Position Verification Speed of light Position verified Quantum Position Verification 2 Dominique Unruh
A generic protocol time f ( x,y ) g ( x,y ) x y space verifier 1 verifier 2 prover Quantum Position Verification 3 Dominique Unruh
A generic attack time f ( x,y ) g ( x,y ) x y y x space verifier 1 verifier 2 adv 1 adv 2 Quantum Position Verification 4 Dominique Unruh
Impossibility • Applies to 3D-protocols as well • Any number of verifiers • Any computational assumptions (exception: transfer capacity limitations) [CGMO09] Chandran, Goyal, Moriarty, Ostrovsky, Position Based Cryptography, Crypto 2009 Quantum Position Verification 5 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 computational assumptions: Generic attack (exponential entanglement) [BCF + 11] Buhrman, Chandran, Fehr, Gelles, Goyal, Ostrovsky, Schafftner: Position-Based Quantum Crypto , Crypto 2011 Quantum Position Verification 6 Dominique Unruh
Quantum crypto: A secure protocol time Assumption: No entangled photons | Ψ 〉 Only 1D proof 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 7 Dominique Unruh
Our protocol time B := H ( x 1 ⊕ x 2 ) | Ψ 〉 verifier 1 verifier 2 prover • Avoids attack • Provably secure (in random oracle model) Quantum Position Verification 8 Dominique Unruh
Security proof (overview, 1D) Right region Left region Measuring | Ψ 〉 time Light barrier in 2 separated space regions for random B Impossible [TFKW13] Program H ( x 1 ⊕ x 2 ):= B x 1 ⊕ x 2 not | Ψ 〉 known verifier 1 verifier 2 Quantum Position Verification 9 Dominique Unruh
3D case • 3D proof: regions overlap! • Need to program RO at different times in different locations! • Leads to curved “programming surface” • New tool: spacetime circuits Quantum Position Verification 10 Dominique Unruh
Proof technique: Space-time circuits • How to reason about AND events happening along curved space-time OR OR surfaces? Tricky! • Tool: Space-time circuits AND OR – No wire leaves light cone • Then forget about AND OR geometry, only connectivity Quantum Position Verification 11 Dominique Unruh
Open problems • Improve error tolerance (3.7%) • Improve precision in 3D case • Security in standard model (no random oracle) ? Or even without hardness assumptions? Quantum Position Verification 12 Dominique Unruh
I thank for your attention Logo soup This research was supported by European Social Fund’s Doctoral Studies and Internationalisation Programme DoRa Dominique Unruh
Attack on [TFKW13] entan- gled • No entanglement = strong assumption • Does not work in 3D (bug in [BCF + 09] proof) Quantum Position Verification 14 Dominique Unruh
Monogamy game ρ Re Refer eree Alic Alice Bo Bob Basis Basis Basis x 1 x 2 x 3 Pr[ x 1 = x 2 = x 3 ] small [TFKW13] Quantum Position Verification 15 Dominique Unruh
Security in higher dimensions? Programming the random oracle: When all signals reach honest P (no later!) Picture: Which space-point reaches which verifier after programming Quantum Position Verification 16 Dominique Unruh
Programming later? Assume that adv is not in δ radius of P . Then achieve non- overlapping regions apply monogamy Quality: δ = 0.38 * | V - P | Quantum Position Verification 17 Dominique Unruh
Multiparty Monogamy Game ρ Re Refer eree simultaneous Basis commuting A B C D x 1 x 3 x 2 x 4 x 5 Pr[all x i equal] small??? Quantum Position Verification 18 Dominique Unruh
Security proof Quantum Position Verification 19 Dominique Unruh
Result: • Our protocol is secure if: Only the honest prover is at a point in spacetime such that: Because monogamy- – Can be reached from all verifiers games for two – Can reach V 1 , V 2 recipients only • Geometric condition, e.g. honest prover in the middle of verifier-tetrahedron Quantum Position Verification 20 Dominique Unruh
Proof technique: Space-time circuits • How to reason about AND events happening along curved space-time OR OR surfaces? Tricky! • Tool: Space-time circuits AND OR – No wire leaves light cone • Then forget about AND OR geometry, only connectivity Quantum Position Verification 21 Dominique Unruh
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