Robust Randomness Amplifiers: Upper and Lower Bounds

Coudron, Matthew; Vidick, Thomas; Yuen, Henry

doi:10.1007/978-3-642-40328-6_33

Cited by 10 publications

(16 citation statements)

References 20 publications

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“…A general version of the spot-checking protocol is shown in Figure 1. (In [7], this protocol was showed to be secure against classical adversaries. )…”

Section: Central Resultsmentioning

confidence: 99%

“…(Without this requirement, the amount of seed needed to generate the random inputs would have to be at least linearly proportional to the number of extractable bits in the output.) This motivates a protocol which we import from [7], referred to here as the spot-checking protocol. A general version of the spot-checking protocol is shown in Figure 1.…”

Section: Central Resultsmentioning

confidence: 99%

“…Following a method used in [7], if we let q = (log 2 N)/N in Protocol R gen , then it will require only O G (log 3 N) bits of initial seed to approximate the input distribution TA, and O G (log 2 N) bits of initial seed to perform randomness extraction on X once the protocol concludes [9]. Since the number of final random bits is Θ(N), exponential randomness expansion is achieved.…”

Section: Central Resultsmentioning

confidence: 99%

See 2 more Smart Citations

Universal Security for Randomness Expansion from the Spot-Checking Protocol

Miller¹,

Shi²

2017

SIAM J. Comput.

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Colbeck (Thesis, 2006) proposed using Bell inequality violations to generate certified random numbers. While full quantum-security proofs have been given, it remains a major open problem to identify the broadest class of Bell inequalities and lowest performance requirements to achieve such security. In this paper, working within the broad class of spot-checking protocols, we prove exactly which Bell inequality violations can be used to achieve full security. Our result greatly improves the known noise tolerance for secure randomness expansion: for the commonly used CHSH game, full security was only known with a noise tolerance of 1.5%, and we improve this to 10.3%. We also generalize our results beyond Bell inequalities and give the first security proof for randomness expansion based on Kochen-Specker inequalities. The central technical contribution of the paper is a new uncertainty principle for the Schatten norm, which is based on the uniform convexity inequality of Ball, Carlen, and Lieb (Inventiones mathematicae, 115:463-482, 1994).

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“…A general version of the spot-checking protocol is shown in Figure 1. (In [7], this protocol was showed to be secure against classical adversaries. )…”

Section: Central Resultsmentioning

confidence: 99%

Section: Central Resultsmentioning

confidence: 99%

Section: Central Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Universal Security for Randomness Expansion from the Spot-Checking Protocol

Miller¹,

Shi²

2017

SIAM J. Comput.

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“…By taking γ = Θ( log n n ) for instance we have used a polylogarithmic (in n) number of random bits and generated a linear number of bits n, thus achieving exponential randomness expansion. We refer the reader to [34], [37] for further discussions on the way to generate the random bits needed for the protocol.…”

Section: Chsh-based Di-re Protocolmentioning

confidence: 99%

Entropy Accumulation With Improved Second-Order Term

Dupuis

Fawzi

2019

IEEE Trans. Inform. Theory

111

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The entropy accumulation theorem [1] states that the smooth min-entropy of an n-partite system A = (A1, . . . , An) is lower-bounded by the sum of the von Neumann entropies of suitably chosen conditional states up to corrections that are sublinear in n. This theorem is particularly suited to proving the security of quantum cryptographic protocols, and in particular so-called device-independent protocols for randomness expansion and key distribution, where the devices can be built and preprogrammed by a malicious supplier [2]. However, while the bounds provided by this theorem are optimal in the first order, the second-order term is bounded more crudely, in such a way that the bounds deteriorate significantly when the theorem is applied directly to protocols where parameter estimation is done by sampling a small fraction of the positions, as is done in most QKD protocols. The objective of this paper is to improve this second-order sublinear term and remedy this problem. On the way, we prove various bounds on the divergence variance, which might be of independent interest.

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“…The precise formalization of DIRE protocols is rather involved, and we direct the interested reader to the flourishing collection of works on the topic [CK11, PAM10, VV12, FGS13, AMP12, UZZ + 13, GL + 13, MS14, CSW14]. In particular the precise formulation of the model is a focus of [CVY13].…”

Section: Device-independent Randomness Expansion and Weak Cross-talkmentioning

confidence: 99%

Interactive Proofs with Approximately Commuting Provers

Coudron

Vidick

2015

Automata, Languages, and Programming

Self Cite

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The class MIP * of promise problems that can be decided through an interactive proof system with multiple entangled provers provides a complexity-theoretic framework for the exploration of the nonlocal properties of entanglement. Very little is known in terms of the power of this class. The only proposed approach for establishing upper bounds is based on a hierarchy of semidefinite programs introduced independently by Pironio et al. and Doherty et al. in 2006. This hierarchy converges to a value, the fieldtheoretic value, that is only known to coincide with the provers' maximum success probability in a given proof system under a plausible but difficult mathematical conjecture, Connes' embedding conjecture.No bounds on the rate of convergence are known.We introduce a rounding scheme for the hierarchy, establishing that any solution to its N-th level can be mapped to a strategy for the provers in which measurement operators associated with distinct provers have pairwise commutator bounded by O(ℓ 2 / √ N) in operator norm, where ℓ is the number of possible answers per prover.Our rounding scheme motivates the introduction of a variant of quantum multiprover interactive proof systems, called MIP * δ , in which the soundness property is required to hold against provers allowed to operate on the same Hilbert space as long as the commutator of operations performed by distinct provers has norm at most δ. Our rounding scheme implies the upper bound MIP * δ ⊆ DTIME(exp(exp(poly)/δ 2 )).In terms of lower bounds we establish that MIP * 2 − poly , with completeness 1 and soundness 1 − 2 − poly , contains NEXP. The relationship of MIP * δ to MIP * has connections with the mathematical literature on approximate commutation. Our rounding scheme gives an elementary proof that the Strong Kirchberg Conjecture implies that MIP * is computable. We also discuss applications to device-independent cryptography.

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Robust Randomness Amplifiers: Upper and Lower Bounds

Cited by 10 publications

References 20 publications

Universal Security for Randomness Expansion from the Spot-Checking Protocol

Universal Security for Randomness Expansion from the Spot-Checking Protocol

Entropy Accumulation With Improved Second-Order Term

Interactive Proofs with Approximately Commuting Provers

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