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Sharp indistinguishability bounds from non-uniform approximations [article]

Christopher Williamson
2021 arXiv   pre-print
Upper bounds on the statistical distance have typically been obtained by providing uniform low-degree polynomial approximations to certain higher-degree polynomials; the sharpness and wider applicability  ...  gave a tight bound for all k in the case of distinguishing with the OR function.  ...  In Section 7 we see that the bound 2 −k+1 is insufficient to get a non-trivial indistinguishability upper bound for k close to n. Hardness of discrete monomial approximations Corollary 5.1.  ... 
arXiv:2103.07842v1 fatcat:rvxlox6i3rcjvfcdg55z5m5koy

Approximate Bounded Indistinguishability

Andrej Bogdanov, Christopher Williamson
unpublished
We prove bounds tight up to constants when f is the OR function, and tight up to logarithmic factors when f is a read-once uniform AND • OR formula, extending previous works that address the perfect indistinguishability  ...  As a secondary contribution, we derive new threshold weight lower bounds for bounded depth AND-OR formulas. 1998 ACM Subject Classification F.0 [Theory of Computation] General  ...  We thank Mark Bun for telling us about the work of Servedio, Tan, and Thaler and for his advice on polynomial approximations, and the ICALP 2017 reviewers for several helpful suggestions. A.  ... 
fatcat:7kcs5ydnnbd6nkz52wvujblt7i

Sharp Indistinguishability Bounds from Non-Uniform Approximations

Christopher Williamson, Petra Berenbrink, Benjamin Monmege
2022
Upper bounds on the statistical distance have typically been obtained by providing uniform low-degree polynomial approximations to certain higher-degree polynomials.  ...  Krause and Simon [Krause and Simon, 2000] gave improved upper and lower bounds for general k and n when distinguishing with the OR function, but these bounds are exponentially far apart when k = Ω(n).  ...  In Section 7 we see that the bound 2 −k+1 is insufficient to get a non-trivial indistinguishability upper bound for k close to n. Hardness of discrete monomial approximations ▶ Corollary 9.  ... 
doi:10.4230/lipics.stacs.2022.59 fatcat:ieti7m5n2bcibm2sdoobih4myu

Bounded Indistinguishability and the Complexity of Recovering Secrets [chapter]

Andrej Bogdanov, Yuval Ishai, Emanuele Viola, Christopher Williamson
2016 Lecture Notes in Computer Science  
Motivated by cryptographic applications, we study the notion of bounded indistinguishability, a natural relaxation of the well studied notion of bounded independence.  ...  When Σ = {0, 1}, we observe that whether f is fooled is closely related to its approximate degree. For larger alphabets Σ, we obtain several positive and negative results.  ...  Emanuele thanks Daniel Wichs for asking whether bounded indistinguishability fools AND, and Mark Bun and Justin Thaler for many discussions about the approximate degree literature.  ... 
doi:10.1007/978-3-662-53015-3_21 fatcat:3whk27geizdgnnjd2ny5dybaz4

Approximate Degree, Secret Sharing, and Concentration Phenomena

Andrej Bogdanov, Nikhil S. Mande, Justin Thaler, Christopher Williamson, Michael Wagner
2019 International Workshop on Approximation Algorithms for Combinatorial Optimization  
These upper and lower bounds were also previously only known in the case f = AND.  ...  A sound and complete certificate for approximate degree being at least k is a pair of probability distributions, also known as a dual polynomial, that are perfectly k-wise indistinguishable, but are distinguishable  ...  Owing to orthogonality and boundedness of the Chebyshev polynomials, this is a lower bound on the approximate degree of AND K .  ... 
doi:10.4230/lipics.approx-random.2019.71 dblp:conf/approx/BogdanovMTW19 fatcat:uomc4z6h3zcdtboyl6b3c7lv5e

Secrecy Without Perfect Randomness: Cryptography with (Bounded) Weak Sources [chapter]

Michael Backes, Aniket Kate, Sebastian Meiser, Tim Ruffing
2015 Lecture Notes in Computer Science  
In this paper, we show that the loss of security induced by using a weak source can be meaningfully quantified if the source is bounded, e.g., for the well-studied Santha-Vazirani (SV) sources.  ...  More concretely, we prove that indistinguishability with uniform randomness implies differential indistinguishability with weak randomness.  ...  2 α (∆ + δ) + δ)-differentially indistinguishable for A and any δ-approximate (α, β)-bounded weak source.  ... 
doi:10.1007/978-3-319-28166-7_33 fatcat:ahpzefndhngv7ikuf3yle572dq

Tight Lower Bound for Comparison-Based Quantile Summaries [article]

Graham Cormode, Pavel Veselý
2020 arXiv   pre-print
We prove that this space bound is optimal by showing a matching lower bound.  ...  That is, an ε-approximate quantile summary first processes a stream of items and then, given any quantile query 0<ϕ< 1, returns an item from the stream, which is a ϕ'-quantile for some ϕ' = ϕ±ε.  ...  We show that our construction from Section 4 can be used to improve the lower bound for εapproximate biased quantile summaries by a further log εN factor.  ... 
arXiv:1905.03838v2 fatcat:5zddf3bpwvelnptmtets4gwa5q

Nonasymptotic Mean-Field Games

Hamidou Tembine
2014 IEEE Transactions on Cybernetics  
For such large systems, the basic idea consists to approximate large games by a stylized game model with a continuum of players. The approach has been shown to be useful in some applications.  ...  However, the stylized game model with continuum of decision-makers is rarely observed in practice and the approximation proposed in the asymptotic regime is meaningless for networked systems with few entities  ...  We show that the indistinguishability property provides an accurate error bound for any system size.  ... 
doi:10.1109/tcyb.2014.2315171 pmid:24759999 fatcat:k5o3vds5yveqtfvhxtakhdoydm

Nonasymptotic Mean-Field Games

Hamidou Tembine
2014 IFAC Proceedings Volumes  
For such large systems, the basic idea consists to approximate large games by a stylized game model with a continuum of players. The approach has been shown to be useful in some applications.  ...  However, the stylized game model with continuum of decision-makers is rarely observed in practice and the approximation proposed in the asymptotic regime is meaningless for networked systems with few entities  ...  We show that the indistinguishability property provides an accurate error bound for any system size.  ... 
doi:10.3182/20140824-6-za-1003.01869 fatcat:ke7zkrlzebfptit76ydvx3q32a

Regularity, Boosting, and Efficiently Simulating Every High-Entropy Distribution

Luca Trevisan, Madhur Tulsiani, Salil Vadhan
2009 2009 24th Annual IEEE Conference on Computational Complexity  
Stated in a more abstract form (where we refer to indistinguishability by arbitrary families of distinguishers rather than bounded-size circuits), our result implies (a) the Weak Szemerédi Regularity Lemma  ...  We show that every high-entropy distribution is indistinguishable from an efficiently samplable distribution of the same entropy.  ...  In each case, we have Ex |h(x) − h (x)| = O( ), and so h and h are indistinguishable by arbitrary bounded functions.  ... 
doi:10.1109/ccc.2009.41 dblp:conf/coco/TrevisanTV09 fatcat:x6e2bipzizgpxeivqip3etip6i

A Tight Lower Bound for Comparison-Based Quantile Summaries

Graham Cormode, Pavel Veselý
2020 Proceedings of the 39th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems  
In Section 3, we introduce indistinguishable streams, and in Section 4 we describe our construction.  ...  We prove that this space bound is optimal by showing a matching lower bound.  ...  We are not aware of any lower bound which holds for a known universe of bounded size, apart from the trivial bound Ω( 1 ε ).  ... 
doi:10.1145/3375395.3387650 dblp:conf/pods/CormodeV20 fatcat:uoqzl6vkknb2jpyfqtmomgihii

Topological quantum error correction in the Kitaev honeycomb model

Yi-Chan Lee, Courtney G Brell, Steven T Flammia
2017 Journal of Statistical Mechanics: Theory and Experiment  
We first prove explicit exponential bounds on the approximate degeneracy, local indistinguishability, and correctability of the code space.  ...  These bounds are tighter than can be achieved using known general properties of topological phases.  ...  We first prove explicit exponential bounds on the approximate degeneracy, local indistinguishability, and correctability of the code space.  ... 
doi:10.1088/1742-5468/aa7ee2 fatcat:3rudcgmcivgr7owy2jsgrrcgde

Classically simulating near-term partially-distinguishable and lossy boson sampling

Alexandra Emily Moylett, Raúl García-Patrón, Jelmer J Renema, Peter Turner
2019 Quantum Science and Technology  
Boson Sampling is the problem of sampling from the same distribution as indistinguishable single photons at the output of a linear optical interferometer.  ...  In this work, we develop a scheme for classical simulation of Boson Sampling under uniform distinguishability and loss, based on the idea of sampling from distributions where at most k photons are indistinguishable  ...  V, we improve this bound to a worst-case error bound, by computing an upper bound for the trace distance between our approximation and the model. In Sec.  ... 
doi:10.1088/2058-9565/ab5555 fatcat:joa2nt2psjd7bflqfbucejpjpe

Optimal Clock Synchronization with Signatures [article]

Christoph Lenzen, Julian Loss
2022 arXiv   pre-print
any skew bound (STOC '84, PODC '85).  ...  We prove matching upper and lower bounds of Θ(u+(ϑ-1)d) on the skew for the resilience range from ⌈ n/3⌉ to ⌈ n/2⌉-1.  ...  Since a lower bound of ( − 1) follows from a simple indistinguishability argument even in absence of faults, we hence establish an asymptotically tight bound of Θ( ˜ + ( − 1) ) on the skew that can be  ... 
arXiv:2203.02553v1 fatcat:allmmb5twvadrnb4kuzzlno7ju

Achievable Information Rates for Molecular Communication with Distinct Molecules

Andrew W. Eckford
2007 Proceedings of the 2nd International Conference on Bio-Inspired Models of Network Information and Computing Systems  
Results are provided which indicate large gains in information rate over the case where the released molecules are indistinguishable from each other.  ...  However, the tightness of the bound is governed by a term related to the Kullback-Leibler distance between the approximation and the true distribution, so better approximations will likely lead to better  ...  In fact, g(y |x) and g(y) need not be close approximations of f Y|X (y|x) and f Y (y), as the bound is valid for any g(y|x) and g(y) that satisfy the given properties.  ... 
doi:10.4108/icst.bionetics2007.2415 dblp:conf/bionetics/Eckford07 fatcat:l55fwhlv6vg3jnailm6urdohqa
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