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Randomness Extraction and Privacy Amplification with quantum eavesdroppers

Randomness Extraction and Privacy Amplification with quantum eavesdroppers. Thomas Vidick UC Berkeley Based on joint work with Christopher Portmann , Anindya De, and Renato Renner. Outline. Privacy amplification and randomness extraction A one-bit extractor Trevisan’s construction.

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Randomness Extraction and Privacy Amplification with quantum eavesdroppers

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

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

  3. Quantum Key Distribution quantum channel classical channel Two phases: • Quantum communication • Classical communication • Parameter estimation: bound Eve’s knowledge • Error correction: A, B compute identical n-bit strings • Privacy amplification: A, B share identical private m-bit strings Final shared string to be used in subsequent protocol:require universallycomposable security: Goals: Security (bound Eve’s knowledge)+Efficiency (bitrate) Eve

  4. Privacy amplification [BBR’88] bits • Goal: given Eve’s (bounded) knowledge about , appears close to uniform: • minimize communication + complexity of applying • Additional rand. necessary: no deterministic process will work • Alice chooses random function from family, tells Bob F Classical communication Eve bits

  5. Examples bits F Classical communication • Output single position: • Output random XOR: (Repeat the above for different positions/XORs.) • Random function, • Apply random 2-universal hash function All are “strong randomness extractors!” bits

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

  7. Aside: randomness extraction (2) Ext? PX(x) • Obvious restriction: • Still, even extracting one bit is impossible in this setting! • No single function will work for every distribution • Need extra randomness to get started: seed • extractor: such thatfor every X with is -close to • Strong extractor: is -close to for • Goals: short seed, large output, efficient construction. x PU(x) x PY(x) + x

  8. Extractors for privacy amplification bits F Classical communication • A,B share X. Classical eavesdropper holds E • Suppose . Then ) large for most • If is strong extractor then Ext(,) -close to uniform • Security of strong extractor = requirement for privacy ampl. [Lu02]! • Quantum eavesdropper: no such • Can still define, and [KRS’09] • [Renner’05] appropriate measure of extractable randomness • Usual definition of strong extractor no longer sufficient bits

  9. Example: the perfect matching extractor [GKKRW’07] Ext: {0,1}n x {0,1}2log n → {0,1}n/2 X: n-bit string Y: perfect matching chosen among n2 x1 x2 x3 x4 Ext Output is uniformly random xn-1 xn • 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!

  10. Summary of known constructions

  11. Outline • Privacy amplification and randomness extraction • A one-bit extractor • Trevisan’s construction

  12. A one-bit extractor • , seed , • Classical security proof • Given random Y, Eve can distinguish from uniform:she can predict a random k-XOR with advantage • Query Eve on every Y: recover string which agrees with k-XOR encoding of X in fraction of positions • List of all k-XORs is list-decodable encoding of Xnarrows X down to list of possibilities • Extractor is secure as long as • Proof based on reconstruction argument: recover X from Eve’s information impossible as long as large enough

  13. Quantum eavesdroppers • … cannot be repeated! • Unclear how to recover X from Eve’s state • Same problem arises in analysis of RAC • Thm[DV10,J11]: is strong extractor for any • [BRdW’07] proved weaker result in bounded storage model • Proof follows from [KT’06] • Argument constructive, based on Pretty-Good Measurement:Given seed y, Eve has to distinguish from PGM is almost-optimal. By linearity, equiv. to:measure using , get ,output • Reduces Eve to being classical

  14. Outline • Privacy amplification and randomness extraction • A one-bit extractor • Trevisan’s construction

  15. Trevisan’s construction (1) • How do we extract more bits? • Repeating m times works, but uses a lot of seed! • Idea: make more efficient use of the seed • Combinatorial design: subsets with small pairwise intersections. • Partition seed into overlapping sets, so bits can be re-used(Use to compute -th output.) • Ex [HR03]: for prime , where ranges over polynomials of degree get subsets of of size small pairwise intersection • Design can be pre-computed and stored y 1 0 0 1 0 1 0

  16. Trevisan’s construction (2) • Introduced in [T99]; breakthrough construction building on work on pseudo-random generators • Fix a design and one-bit extractor • Polyvalent: use any design; many possible one-bit extractors • Can focus on efficiency or optimality • Near-optimal in all parameters (seed&output length, efficiency) x + y 1 0 0 1 0 1 0

  17. Some parameters • Input length , seed length , output length , min-entropy • Construction based on k-XOR • , seed • Extracts bits from entropy • Locally computable • Optimal seed length • Extract bits from entropy • Optimal output length • Seed , extracts from any • Can also extract from weakly uniform seed • All constructions “efficient” (polynomial)

  18. Overview of security proof x: n bits + y: t bits 1 0 0 1 0 1 0 • By contradiction: assume eavesdropper E can distinguish output from uniform with success ɛ. • First step: using E, construct an eavesdropper E’ such that • E’ has access to the same side information as E • E’ has some additional classical information over m bits • E’ breaks the one-bit extractor with success prob. ½+ɛ/m Based on hybrid argument + properties of comb. design • Second step: such an E’ cannot exist! • We already know is secure against quantum eavesdroppers : log n bits

  19. 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]

  20. Open problems • Can we do even better? Extract many bits with a logarithmic seed? • Trevisan’s extractor only extracts , for any • Classical constructions exist, but based on different ideas. • Could all reasonable extractors be secure against quantum eavesdroppers? • Hidden matching is not, but really bad extractor • Could still have generic proof with small loss in parameters • How much information is there in a quantum state? • Similar questions asked in comm. compl., but in worst-case

  21. Thank you!

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