Found 2 Possible Name Variants:

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Invited Talk

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

We survey recent progress and many open questions in the field of sampling high-dimensional distributions, with specific focus on sampling with non-Euclidean metrics.

Yin Tat Lee and Santosh S. Vempala. The Manifold Joys of Sampling (Invited Talk). In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 4:1-4:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{lee_et_al:LIPIcs.ICALP.2022.4, author = {Lee, Yin Tat and Vempala, Santosh S.}, title = {{The Manifold Joys of Sampling}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {4:1--4:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.4}, URN = {urn:nbn:de:0030-drops-163459}, doi = {10.4230/LIPIcs.ICALP.2022.4}, annote = {Keywords: Sampling, Diffusion, Optimization, High Dimension} }

Document

**Published in:** LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

The Gibbs Sampler is a general method for sampling high-dimensional distributions, dating back to 1971. In each step of the Gibbs Sampler, we pick a random coordinate and re-sample that coordinate from the distribution induced by fixing all the other coordinates. While it has become widely used over the past half-century, guarantees of efficient convergence have been elusive. We show that for a convex body K in ℝⁿ with diameter D, the mixing time of the Coordinate Hit-and-Run (CHAR) algorithm on K is polynomial in n and D. We also give a lower bound on the mixing rate of CHAR, showing that it is strictly worse than hit-and-run and the ball walk in the worst case.

Aditi Laddha and Santosh S. Vempala. Convergence of Gibbs Sampling: Coordinate Hit-And-Run Mixes Fast. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 51:1-51:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{laddha_et_al:LIPIcs.SoCG.2021.51, author = {Laddha, Aditi and Vempala, Santosh S.}, title = {{Convergence of Gibbs Sampling: Coordinate Hit-And-Run Mixes Fast}}, booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)}, pages = {51:1--51:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-184-9}, ISSN = {1868-8969}, year = {2021}, volume = {189}, editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.51}, URN = {urn:nbn:de:0030-drops-138503}, doi = {10.4230/LIPIcs.SoCG.2021.51}, annote = {Keywords: Gibbs Sampler, Coordinate Hit and run, Mixing time of Markov Chain} }

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RANDOM

**Published in:** LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

We study Hamiltonian Monte Carlo (HMC) for sampling from a strongly logconcave density proportional to e^{-f} where f:R^d -> R is mu-strongly convex and L-smooth (the condition number is kappa = L/mu). We show that the relaxation time (inverse of the spectral gap) of ideal HMC is O(kappa), improving on the previous best bound of O(kappa^{1.5}); we complement this with an example where the relaxation time is Omega(kappa). When implemented using a nearly optimal ODE solver, HMC returns an epsilon-approximate point in 2-Wasserstein distance using O~((kappa d)^{0.5} epsilon^{-1}) gradient evaluations per step and O~((kappa d)^{1.5}epsilon^{-1}) total time.

Zongchen Chen and Santosh S. Vempala. Optimal Convergence Rate of Hamiltonian Monte Carlo for Strongly Logconcave Distributions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 64:1-64:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chen_et_al:LIPIcs.APPROX-RANDOM.2019.64, author = {Chen, Zongchen and Vempala, Santosh S.}, title = {{Optimal Convergence Rate of Hamiltonian Monte Carlo for Strongly Logconcave Distributions}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)}, pages = {64:1--64:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-125-2}, ISSN = {1868-8969}, year = {2019}, volume = {145}, editor = {Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.64}, URN = {urn:nbn:de:0030-drops-112790}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.64}, annote = {Keywords: logconcave distribution, sampling, Hamiltonian Monte Carlo, spectral gap, strong convexity} }

Document

**Published in:** LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

It has been recently shown via simulations [Dasgupta et al., 2017] that random projection followed by a cap operation (setting to one the k largest elements of a vector and everything else to zero), a map believed to be an important part of the insect olfactory system, has strong locality sensitivity properties. We calculate the asymptotic law whereby the overlap in the input vectors is conserved, verifying mathematically this empirical finding. We then focus on the far more complex homologous operation in the mammalian brain, the creation through successive projections and caps of an assembly (roughly, a set of excitatory neurons representing a memory or concept) in the presence of recurrent synapses and plasticity. After providing a careful definition of assemblies, we prove that the operation of assembly projection converges with high probability, over the randomness of synaptic connectivity, even if plasticity is relatively small (previous proofs relied on high plasticity). We also show that assembly projection has itself some locality preservation properties. Finally, we propose a large repertoire of assembly operations, including associate, merge, reciprocal project, and append, each of them both biologically plausible and consistent with what we know from experiments, and show that this computational system is capable of simulating, again with high probability, arbitrary computation in a quite natural way. We hope that this novel way of looking at brain computation, open-ended and based on reasonably mainstream ideas in neuroscience, may prove an attractive entry point for computer scientists to work on understanding the brain.

Christos H. Papadimitriou and Santosh S. Vempala. Random Projection in the Brain and Computation with Assemblies of Neurons. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 57:1-57:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{papadimitriou_et_al:LIPIcs.ITCS.2019.57, author = {Papadimitriou, Christos H. and Vempala, Santosh S.}, title = {{Random Projection in the Brain and Computation with Assemblies of Neurons}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {57:1--57:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-095-8}, ISSN = {1868-8969}, year = {2019}, volume = {124}, editor = {Blum, Avrim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.57}, URN = {urn:nbn:de:0030-drops-101506}, doi = {10.4230/LIPIcs.ITCS.2019.57}, annote = {Keywords: Brain computation, random projection, assemblies, plasticity, memory, association} }

Document

**Published in:** LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

In a recent experiment, a cell in the human medial temporal lobe (MTL) encoding one sensory stimulus starts to also respond to a second stimulus following a combined experience associating the two. We develop a theoretical model predicting that an assembly of cells with exceptionally high synaptic intraconnectivity can emerge, in response to a particular sensory experience, to encode and abstract that experience. We also show that two such assemblies are modified to increase their intersection after a sensory event that associates the two corresponding stimuli. The main technical tools employed are random graph theory, and Bernoulli approximations. Assembly creation must overcome a computational challenge akin to the Densest K-Subgraph problem, namely selecting, from a large population of randomly and sparsely interconnected cells, a subset with exceptionally high density of interconnections. We identify three mechanisms that help achieve this feat in our model: (1) a simple two-stage randomized algorithm, and (2) the "triangle completion bias" in synaptic connectivity and a "birthday paradox", while (3) the strength of these connections is enhanced through Hebbian plasticity.

Robert Legenstein, Wolfgang Maass, Christos H. Papadimitriou, and Santosh S. Vempala. Long Term Memory and the Densest K-Subgraph Problem. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 57:1-57:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{legenstein_et_al:LIPIcs.ITCS.2018.57, author = {Legenstein, Robert and Maass, Wolfgang and Papadimitriou, Christos H. and Vempala, Santosh S.}, title = {{Long Term Memory and the Densest K-Subgraph Problem}}, booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)}, pages = {57:1--57:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-060-6}, ISSN = {1868-8969}, year = {2018}, volume = {94}, editor = {Karlin, Anna R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.57}, URN = {urn:nbn:de:0030-drops-83593}, doi = {10.4230/LIPIcs.ITCS.2018.57}, annote = {Keywords: Brain computation, long term memory, assemblies, association} }

Document

Complete Volume

**Published in:** LIPIcs, Volume 81, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)

LIPIcs, Volume 81, APPROX/RANDOM'17, Complete Volume

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@Proceedings{jansen_et_al:LIPIcs.APPROX-RANDOM.2017, title = {{LIPIcs, Volume 81, APPROX/RANDOM'17, Complete Volume}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-044-6}, ISSN = {1868-8969}, year = {2017}, volume = {81}, editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017}, URN = {urn:nbn:de:0030-drops-77101}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2017}, annote = {Keywords: Network Architecture and Design, Coding and Information Theory, Error Control Codes, Modes of Computation: Online computation, Complexity Measures and Classes, Analysis of Algorithms and Problem Complexity, Numerical Algorithms and Problems, Nonnumerical Algorithms and Problems} }

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Front Matter

**Published in:** LIPIcs, Volume 81, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)

Frontmatter, Table of Contents, Preface, Organization, External Reviewers, List of Authors

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, p. 0:i, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{jansen_et_al:LIPIcs.APPROX-RANDOM.2017.0, author = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.}, title = {{Frontmatter, Table of Contents, Preface, Organization, External Reviewers, List of Authors}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)}, pages = {0:i--0:i}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-044-6}, ISSN = {1868-8969}, year = {2017}, volume = {81}, editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.0}, URN = {urn:nbn:de:0030-drops-75493}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.0}, annote = {Keywords: Frontmatter, Table of Contents, Preface, Organization, External Reviewers, List of Authors} }

Document

**Published in:** LIPIcs, Volume 18, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)

The classic k-d tree data structure continues to be widely used in spite of its vulnerability to the so-called curse of dimensionality. Here we provide a rigorous explanation: for randomly rotated data, a k-d tree adapts to the intrinsic dimension of the data and is not affected by the ambient dimension, thus keeping the data structure efficient for objects such as low-dimensional manifolds and sparse data.
The main insight of the analysis can be used as an algorithmic pre-processing step to realize the same benefit: rotate the data randomly; then build a k-d tree. Our work can be seen as a refinement of Random Projection trees [Dasgupta 2008], which also adapt to intrinsic dimension but incur higher traversal costs as the resulting cells are polyhedra and not cuboids. Using k-d trees after a random rotation results in cells that are cuboids, thus preserving the traversal efficiency of standard k-d trees.

Santosh S. Vempala. Randomly-oriented k-d Trees Adapt to Intrinsic Dimension. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 18, pp. 48-57, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{vempala:LIPIcs.FSTTCS.2012.48, author = {Vempala, Santosh S.}, title = {{Randomly-oriented k-d Trees Adapt to Intrinsic Dimension}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)}, pages = {48--57}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-47-7}, ISSN = {1868-8969}, year = {2012}, volume = {18}, editor = {D'Souza, Deepak and Radhakrishnan, Jaikumar and Telikepalli, Kavitha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2012.48}, URN = {urn:nbn:de:0030-drops-38470}, doi = {10.4230/LIPIcs.FSTTCS.2012.48}, annote = {Keywords: Data structures, Nearest Neighbors, Intrinsic Dimension, k-d Tree} }

Document

**Published in:** LIPIcs, Volume 8, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)

This article is a survey of developments in algorithmic convex geometry over the past decade. These include algorithms for sampling, optimization, integration, rounding and learning, as well as mathematical tools such as isoperimetric and concentration inequalities. Several open problems and conjectures are discussed on the way.

Santosh S. Vempala. Recent Progress and Open Problems in Algorithmic Convex Geometry. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010). Leibniz International Proceedings in Informatics (LIPIcs), Volume 8, pp. 42-64, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{vempala:LIPIcs.FSTTCS.2010.42, author = {Vempala, Santosh S.}, title = {{Recent Progress and Open Problems in Algorithmic Convex Geometry}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)}, pages = {42--64}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-23-1}, ISSN = {1868-8969}, year = {2010}, volume = {8}, editor = {Lodaya, Kamal and Mahajan, Meena}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2010.42}, URN = {urn:nbn:de:0030-drops-28529}, doi = {10.4230/LIPIcs.FSTTCS.2010.42}, annote = {Keywords: convex geometry, geometric inequalities, algorithms, sampling, optimization, integration, rounding, learning} }

Document

RANDOM

**Published in:** LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)

We study a unified approach and algorithm for constructive discrepancy minimization based on a stochastic process. By varying the parameters of the process, one can recover various state-of-the-art results. We demonstrate the flexibility of the method by deriving a discrepancy bound for smoothed instances, which interpolates between known bounds for worst-case and random instances.

Nikhil Bansal, Aditi Laddha, and Santosh Vempala. A Unified Approach to Discrepancy Minimization. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 1:1-1:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bansal_et_al:LIPIcs.APPROX/RANDOM.2022.1, author = {Bansal, Nikhil and Laddha, Aditi and Vempala, Santosh}, title = {{A Unified Approach to Discrepancy Minimization}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)}, pages = {1:1--1:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-249-5}, ISSN = {1868-8969}, year = {2022}, volume = {245}, editor = {Chakrabarti, Amit and Swamy, Chaitanya}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.1}, URN = {urn:nbn:de:0030-drops-171238}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2022.1}, annote = {Keywords: Discrepancy theory, smoothed analysis} }

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Invited Paper

**Published in:** LIPIcs, Volume 122, 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)

While the design of algorithms is traditionally a discrete endeavour, in recent years many advances have come from continuous perspectives. Typically, a continuous process, deterministic or randomized, is designed and shown to have desirable properties, such as approaching an optimal solution or a target distribution, and an algorithm is derived from this by appropriate discretization. We will discuss examples of this for optimization (gradient descent, interior-point method) and sampling (Brownian motion, Hamiltonian Monte Carlo), with applications to learning. In some interesting and rather general settings, the current fastest methods have been obtained via this approach.

Santosh Vempala. Continuous Algorithms (Invited Paper). In 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 122, p. 4:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{vempala:LIPIcs.FSTTCS.2018.4, author = {Vempala, Santosh}, title = {{Continuous Algorithms}}, booktitle = {38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)}, pages = {4:1--4:1}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-093-4}, ISSN = {1868-8969}, year = {2018}, volume = {122}, editor = {Ganguly, Sumit and Pandya, Paritosh}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2018.4}, URN = {urn:nbn:de:0030-drops-99037}, doi = {10.4230/LIPIcs.FSTTCS.2018.4}, annote = {Keywords: Algorithms} }

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**Published in:** LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

An interesting challenge for the cryptography community is to design authentication protocols that are so simple that a human can execute them
without relying on a fully trusted computer. We propose several candidate authentication protocols for a setting in which the human user can
only receive assistance from a semi-trusted computer - a computer that stores information and performs computations correctly
but does not provide confidentiality. Our schemes use a semi-trusted computer to store and display public challenges C_i\in[n]^k.
The human user memorizes a random secret mapping \sigma:[n]\rightarrow \mathbb{Z}_d and authenticates by computing responses f(\sigma(C_i)) to
a sequence of public challenges where f:\mathbb{Z}_d^k\rightarrow \mathbb{Z}_d is a function that is easy for the human to evaluate. We prove
that any statistical adversary needs to sample m=\tilde{\Omega}\paren{n^{s(f)}} challenge-response pairs to recover \sigma, for a security
parameter s(f) that depends on two key properties of f. Our lower bound generalizes recent results of Feldman et al. [Feldman'15]
who proved analogous results for the special case d=2. To obtain our results, we apply the general hypercontractivity theorem [O'Donnell'14]
to lower bound the statistical dimension of the distribution over challenge-response pairs induced by f and \sigma.
Our statistical dimension lower bounds apply to arbitrary functions f:\mathbb{Z}_d^k\rightarrow \mathbb{Z}_d (not just to functions that
are easy for a human to evaluate). As an application, we propose a family of human computable password
functions f_{k_1,k_2} in which the user needs to perform 2k_1+2k_2+1 primitive operations (e.g., adding two digits or remembering a
secret value \sigma(i)), and we show that s(f) = \min{k_1+1, (k_2+1)/2}. For these schemes, we prove that forging passwords is
equivalent to recovering the secret mapping. Thus, our human computable password schemes can maintain strong security guarantees even after
an adversary has observed the user login to many different accounts.

Jeremiah Blocki, Manuel Blum, Anupam Datta, and Santosh Vempala. Towards Human Computable Passwords. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 10:1-10:47, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{blocki_et_al:LIPIcs.ITCS.2017.10, author = {Blocki, Jeremiah and Blum, Manuel and Datta, Anupam and Vempala, Santosh}, title = {{Towards Human Computable Passwords}}, booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)}, pages = {10:1--10:47}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-029-3}, ISSN = {1868-8969}, year = {2017}, volume = {67}, editor = {Papadimitriou, Christos H.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.10}, URN = {urn:nbn:de:0030-drops-81847}, doi = {10.4230/LIPIcs.ITCS.2017.10}, annote = {Keywords: Passwords, Cognitive Authentication, Human Computation, Planted Constraint Satisfaction Problem, Statistical Dimension} }

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