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Randomness Extraction and Privacy Amplification with quantum eavesdroppers Thomas Vidick UC Berkeley Based on joint work with Christopher Portmann, Anindya De, and Renato Renner

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Outline 1.Privacy amplification and randomness extraction 2.A one-bit extractor 3.Trevisan’s construction

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Quantum Key Distribution quantum channel classical channel Eve

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Privacy amplification [BBR’88] Eve F

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Examples F

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Aside: randomness extraction (1) Fundamental concept from TCS [NZ’96] Weak randomness is “readily” available Many applications require “perfect” randomness Can we convert one to the other? x P X (x) x Randomized algorithms Crypto Modeling x P U (x) x P X (x) Public source X: Ideal uniform source: Ext?

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Aside: randomness extraction (2) x P U (x) x P X (x) Ext? + x P Y (x)

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Extractors for privacy amplification F

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Example: the perfect matching extractor x1x1 x3x3 x n-1 x2x2 x4x4 xnxn Classical adversary: cannot do better than birthday paradox → need ≈ √n bits of information about x Quantum adversary: on seeing x, store when matching revealed, measure in → only need ≈ log n qubits! X: n-bit string Y: perfect matching chosen among n 2 Ext Ext: {0,1} n x {0,1} 2log n → {0,1} n/2 Output is uniformly random [GKKRW’07]

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Summary of known constructions SeedOutputRef. Inner-productn1[Ben-Or ’02] 2-universal hashingn[KMR’05] One-bit extractorslog n1[KT’06] n[FS’07] Almost 2-universal hashing m[TSSR’10] Trevisan’s extractor[T-S’09],[DV’10], [DPRV’11]

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Outline 1.Privacy amplification and randomness extraction 2.A one-bit extractor 3.Trevisan’s construction

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A one-bit extractor

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Quantum eavesdroppers

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Outline 1.Privacy amplification and randomness extraction 2.A one-bit extractor 3.Trevisan’s construction

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Trevisan’s construction (1) 0 000111 y

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0 000111 y x + Trevisan’s construction (2)

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Some parameters

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Overview of security proof 0 000111 y: t bits + x: n bits

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Summary Privacy amplification is an important step in QKD Well-understood classically, but quantum eavesdropper is a challenge Some constructions proved to carry over – 2-universal hashing most often used: efficient (matrix multiplication), extracts most key. – All previous const. require as many “fresh” random bits as length of key Trevisan’s construction has many advantages – Efficient (local XOR computation) – Extracts longest possible key, only polylog random bits required Proof of security based on reconstruction argument + [KT’06]

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Open problems

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Thank you!

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