Quantum Position Verification in the random oracle model Dominique - - PowerPoint PPT Presentation

Dominique Unruh

Quantum Position Verification in the random oracle model

Dominique Unruh

University of Tartu

Dominique Unruh

Position Verification

Speed of light  Position verified

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A generic protocol

time space

x y f (x,y) g(x,y)

prover verifier 1 verifier 2

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A generic attack

time space

x y f (x,y) g(x,y)

adv 1 verifier 1 verifier 2 adv 2

x y

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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

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Way out: quantum crypto

• In attack: adversary copies x,y
• If x or y quantum: No cloning!
• 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

x y

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Quantum crypto: A secure protocol

time

|Ψ〉 Basis B

prover verifier 1 verifier 2 [TFKW13] Tomamichel, Fehr, Kaniewski, Wehner: One-Sided Device- Independent QKD and Position-Based Cryptography from Monogamy Games, Eurocrypt 2013 (and [BCF+11])

Assumption: No entangled photons Only 1D proof

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Our protocol

time prover verifier 1 verifier 2

B := H(x1⊕x2)

• Avoids attack
• Provably secure (in random oracle model)

|Ψ〉

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Right region Left region

Security proof (overview, 1D)

time verifier 1 verifier 2

|Ψ〉

x1⊕x2 not known Program H(x1⊕x2):=B

Light barrier

Measuring |Ψ〉 in 2 separated space regions for random B  Impossible [TFKW13]

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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

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Proof technique: Space-time circuits

• How to reason about

events happening along curved space-time surfaces? Tricky!

• Tool: Space-time circuits

– No wire leaves light cone

• Then forget about

geometry, only connectivity

AND OR OR AND OR AND OR

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Open problems

• Improve error tolerance (3.7%)
• Improve precision in 3D case
• Security in standard model (no random oracle)?

Or even without hardness assumptions?

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I thank for your attention

This research was supported by European Social Fund’s Doctoral Studies and Internationalisation Programme DoRa

Logo soup

Dominique Unruh

Attack on [TFKW13]

entan- gled

• No entanglement = strong assumption
• Does not work in 3D (bug in [BCF+09] proof)

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Monogamy game

Alic Alice Bo Bob Re Refer eree

ρ

Basis Basis Basis

x1 x2 x3

Pr[x1=x2=x3] small

[TFKW13]

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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

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Programming later?

Assume that adv is not in δ radius of P. Then achieve non-

• verlapping

regions  apply monogamy Quality: δ = 0.38 * |V-P|

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Multiparty Monogamy Game

A B Re Refer eree

Basis

x1

ρ

C D

commuting simultaneous x2 x3 x4 x5

Pr[all xi equal] small???

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Security proof

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Result:

• Our protocol is secure if:
• Geometric condition, e.g. honest prover in the

middle of verifier-tetrahedron Only the honest prover is at a point in spacetime such that:

– Can be reached from all verifiers – Can reach V1, V2

Because monogamy- games for two recipients only

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Proof technique: Space-time circuits

• How to reason about

events happening along curved space-time surfaces? Tricky!

• Tool: Space-time circuits

– No wire leaves light cone

• Then forget about

geometry, only connectivity

AND OR OR AND OR AND OR

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