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**Published in:** LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)

One-way functions (OWFs) are central objects of study in cryptography and computational complexity theory. In a seminal work, Liu and Pass (FOCS 2020) proved that the average-case hardness of computing time-bounded Kolmogorov complexity is equivalent to the existence of OWFs. It remained an open problem to establish such an equivalence for the average-case hardness of some natural NP-complete problem. In this paper, we make progress on this question by studying a conditional variant of the Minimum KT-complexity Problem (MKTP), which we call McKTP, as follows.
1) First, we prove that if McKTP is average-case hard on a polynomial fraction of its instances, then there exist OWFs.
2) Then, we observe that McKTP is NP-complete under polynomial-time randomized reductions.
3) Finally, we prove that the existence of OWFs implies the nontrivial average-case hardness of McKTP. Thus the existence of OWFs is inextricably linked to the average-case hardness of this NP-complete problem. In fact, building on recently-announced results of Ren and Santhanam [Rahul Ilango et al., 2021], we show that McKTP is hard-on-average if and only if there are logspace-computable OWFs.

Eric Allender, Mahdi Cheraghchi, Dimitrios Myrisiotis, Harsha Tirumala, and Ilya Volkovich. One-Way Functions and a Conditional Variant of MKTP. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 7:1-7:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{allender_et_al:LIPIcs.FSTTCS.2021.7, author = {Allender, Eric and Cheraghchi, Mahdi and Myrisiotis, Dimitrios and Tirumala, Harsha and Volkovich, Ilya}, title = {{One-Way Functions and a Conditional Variant of MKTP}}, booktitle = {41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)}, pages = {7:1--7:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-215-0}, ISSN = {1868-8969}, year = {2021}, volume = {213}, editor = {Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.7}, URN = {urn:nbn:de:0030-drops-155181}, doi = {10.4230/LIPIcs.FSTTCS.2021.7}, annote = {Keywords: Kolmogorov complexity, KT Complexity, Minimum KT-complexity Problem, MKTP, Conditional KT Complexity, Minimum Conditional KT-complexity Problem, McKTP, one-way functions, OWFs, average-case hardness, pseudorandom generators, PRGs, pseudorandom functions, PRFs, distinguishers, learning algorithms, NP-completeness, reductions} }

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**Published in:** LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

For a size parameter s: ℕ → ℕ, the Minimum Circuit Size Problem (denoted by MCSP[s(n)]) is the problem of deciding whether the minimum circuit size of a given function f : {0,1}ⁿ → {0,1} (represented by a string of length N : = 2ⁿ) is at most a threshold s(n). A recent line of work exhibited "hardness magnification" phenomena for MCSP: A very weak lower bound for MCSP implies a breakthrough result in complexity theory. For example, McKay, Murray, and Williams (STOC 2019) implicitly showed that, for some constant μ₁ > 0, if MCSP[2^{μ₁⋅ n}] cannot be computed by a one-tape Turing machine (with an additional one-way read-only input tape) running in time N^{1.01}, then P≠NP.
In this paper, we present the following new lower bounds against one-tape Turing machines and branching programs:
1) A randomized two-sided error one-tape Turing machine (with an additional one-way read-only input tape) cannot compute MCSP[2^{μ₂⋅n}] in time N^{1.99}, for some constant μ₂ > μ₁.
2) A non-deterministic (or parity) branching program of size o(N^{1.5}/log N) cannot compute MKTP, which is a time-bounded Kolmogorov complexity analogue of MCSP. This is shown by directly applying the Nečiporuk method to MKTP, which previously appeared to be difficult.
3) The size of any non-deterministic, co-non-deterministic, or parity branching program computing MCSP is at least N^{1.5-o(1)}. These results are the first non-trivial lower bounds for MCSP and MKTP against one-tape Turing machines and non-deterministic branching programs, and essentially match the best-known lower bounds for any explicit functions against these computational models.
The first result is based on recent constructions of pseudorandom generators for read-once oblivious branching programs (ROBPs) and combinatorial rectangles (Forbes and Kelley, FOCS 2018; Viola 2019). En route, we obtain several related results:
1) There exists a (local) hitting set generator with seed length Õ(√N) secure against read-once polynomial-size non-deterministic branching programs on N-bit inputs.
2) Any read-once co-non-deterministic branching program computing MCSP must have size at least 2^Ω̃(N).

Mahdi Cheraghchi, Shuichi Hirahara, Dimitrios Myrisiotis, and Yuichi Yoshida. One-Tape Turing Machine and Branching Program Lower Bounds for MCSP. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 23:1-23:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{cheraghchi_et_al:LIPIcs.STACS.2021.23, author = {Cheraghchi, Mahdi and Hirahara, Shuichi and Myrisiotis, Dimitrios and Yoshida, Yuichi}, title = {{One-Tape Turing Machine and Branching Program Lower Bounds for MCSP}}, booktitle = {38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)}, pages = {23:1--23:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-180-1}, ISSN = {1868-8969}, year = {2021}, volume = {187}, editor = {Bl\"{a}ser, Markus and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.23}, URN = {urn:nbn:de:0030-drops-136681}, doi = {10.4230/LIPIcs.STACS.2021.23}, annote = {Keywords: Minimum Circuit Size Problem, Kolmogorov Complexity, One-Tape Turing Machines, Branching Programs, Lower Bounds, Pseudorandom Generators, Hitting Set Generators} }

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**Published in:** LIPIcs, Volume 163, 1st Conference on Information-Theoretic Cryptography (ITC 2020)

Leakage-resilient secret sharing has mostly been studied in the compartmentalized models, where a leakage oracle can arbitrarily leak bounded number of bits from all shares, provided that the oracle only has access to a bounded number of shares when the leakage is taking place. We start a systematic study of leakage-resilient secret sharing against global leakage, where the leakage oracle can access the full set of shares simultaneously, but the access is restricted to a special class of leakage functions. More concretely, the adversary can corrupt several players and obtain their shares, as well as applying a leakage function from a specific class to the full share vector. We explicitly construct such leakage-resilient secret sharing with respect to affine leakage functions and low-degree multi-variate polynomial leakage functions, respectively. For affine leakage functions, we obtain schemes with threshold access structure that are leakage-resilient as long as there is a substantial difference between the total amount of information obtained by the adversary, through corrupting individual players and leaking from the full share vector, and the amount that the reconstruction algorithm requires for reconstructing the secret. Furthermore, if we assume the adversary is non-adaptive, we can even make the secret length asymptotically equal to the difference, as the share length grows. Specifically, we have a threshold scheme with parameters similar to Shamir’s scheme and is leakage-resilient against affine leakage. For multi-variate polynomial leakage functions with degree bigger than one, our constructions here only yield ramp schemes that are leakage-resilient against such leakage. Finally, as a result of independent interest, we show that our approach to leakage-resilient secret sharing also yields a competitive scheme compared with the state-of-the-art construction in the compartmentalized models.

Fuchun Lin, Mahdi Cheraghchi, Venkatesan Guruswami, Reihaneh Safavi-Naini, and Huaxiong Wang. Leakage-Resilient Secret Sharing in Non-Compartmentalized Models. In 1st Conference on Information-Theoretic Cryptography (ITC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 163, pp. 7:1-7:24, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{lin_et_al:LIPIcs.ITC.2020.7, author = {Lin, Fuchun and Cheraghchi, Mahdi and Guruswami, Venkatesan and Safavi-Naini, Reihaneh and Wang, Huaxiong}, title = {{Leakage-Resilient Secret Sharing in Non-Compartmentalized Models}}, booktitle = {1st Conference on Information-Theoretic Cryptography (ITC 2020)}, pages = {7:1--7:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-151-1}, ISSN = {1868-8969}, year = {2020}, volume = {163}, editor = {Tauman Kalai, Yael and Smith, Adam D. and Wichs, Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2020.7}, URN = {urn:nbn:de:0030-drops-121124}, doi = {10.4230/LIPIcs.ITC.2020.7}, annote = {Keywords: Leakage-resilient cryptography, Secret sharing scheme, Randomness extractor} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

The Minimum Circuit Size Problem (MCSP) asks if a given truth table of a Boolean function f can be computed by a Boolean circuit of size at most theta, for a given parameter theta. We improve several circuit lower bounds for MCSP, using pseudorandom generators (PRGs) that are local; a PRG is called local if its output bit strings, when viewed as the truth table of a Boolean function, can be computed by a Boolean circuit of small size. We get new and improved lower bounds for MCSP that almost match the best-known lower bounds against several circuit models. Specifically, we show that computing MCSP, on functions with a truth table of length N, requires
- N^{3-o(1)}-size de Morgan formulas, improving the recent N^{2-o(1)} lower bound by Hirahara and Santhanam (CCC, 2017),
- N^{2-o(1)}-size formulas over an arbitrary basis or general branching programs (no non-trivial lower bound was known for MCSP against these models), and
- 2^{Omega (N^{1/(d+2.01)})}-size depth-d AC^0 circuits, improving the superpolynomial lower bound by Allender et al. (SICOMP, 2006).
The AC^0 lower bound stated above matches the best-known AC^0 lower bound (for PARITY) up to a small additive constant in the depth. Also, for the special case of depth-2 circuits (i.e., CNFs or DNFs), we get an almost optimal lower bound of 2^{N^{1-o(1)}} for MCSP.

Mahdi Cheraghchi, Valentine Kabanets, Zhenjian Lu, and Dimitrios Myrisiotis. Circuit Lower Bounds for MCSP from Local Pseudorandom Generators. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 39:1-39:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{cheraghchi_et_al:LIPIcs.ICALP.2019.39, author = {Cheraghchi, Mahdi and Kabanets, Valentine and Lu, Zhenjian and Myrisiotis, Dimitrios}, title = {{Circuit Lower Bounds for MCSP from Local Pseudorandom Generators}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {39:1--39:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.39}, URN = {urn:nbn:de:0030-drops-106156}, doi = {10.4230/LIPIcs.ICALP.2019.39}, annote = {Keywords: minimum circuit size problem (MCSP), circuit lower bounds, pseudorandom generators (PRGs), local PRGs, de Morgan formulas, branching programs, constant depth circuits} }

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

Shamir's celebrated secret sharing scheme provides an efficient method for encoding a secret of arbitrary length l among any N <= 2^l players such that for a threshold parameter t, (i) the knowledge of any t shares does not reveal any information about the secret and, (ii) any choice of t+1 shares fully reveals the secret. It is known that any such threshold secret sharing scheme necessarily requires shares of length l, and in this sense Shamir's scheme is optimal. The more general notion of ramp schemes requires the reconstruction of secret from any t+g shares, for a positive integer gap parameter g. Ramp secret sharing scheme necessarily requires shares of length l/g. Other than the bound related to secret length l, the share lengths of ramp schemes can not go below a quantity that depends only on the gap ratio g/N.
In this work, we study secret sharing in the extremal case of bit-long shares and arbitrarily small gap ratio g/N, where standard ramp secret sharing becomes impossible. We show, however, that a slightly relaxed but equally effective notion of semantic security for the secret, and negligible reconstruction error probability, eliminate the impossibility. Moreover, we provide explicit constructions of such schemes. One of the consequences of our relaxation is that, unlike standard ramp schemes with perfect secrecy, adaptive and non-adaptive adversaries need different analysis and construction. For non-adaptive adversaries, we explicitly construct secret sharing schemes that provide secrecy against any tau fraction of observed shares, and reconstruction from any rho fraction of shares, for any choices of 0 <= tau < rho <= 1. Our construction achieves secret length N(rho-tau-o(1)), which we show to be optimal. For adaptive adversaries, we construct explicit schemes attaining a secret length Omega(N(rho-tau)). We discuss our results and open questions.

Fuchun Lin, Mahdi Cheraghchi, Venkatesan Guruswami, Reihaneh Safavi-Naini, and Huaxiong Wang. Secret Sharing with Binary Shares. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 53:1-53:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{lin_et_al:LIPIcs.ITCS.2019.53, author = {Lin, Fuchun and Cheraghchi, Mahdi and Guruswami, Venkatesan and Safavi-Naini, Reihaneh and Wang, Huaxiong}, title = {{Secret Sharing with Binary Shares}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {53:1--53:20}, 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.2019.53}, URN = {urn:nbn:de:0030-drops-101461}, doi = {10.4230/LIPIcs.ITCS.2019.53}, annote = {Keywords: Secret sharing scheme, Wiretap channel} }

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**Published in:** LIPIcs, Volume 65, 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)

Testing membership in lattices is of practical relevance, with applications to integer programming, error detection in lattice-based communication and cryptography. In this work, we initiate a systematic study of local testing for membership in lattices, complementing and building upon the extensive body of work on locally testable codes. In particular, we formally define the notion of local tests for lattices and present the following:
1. We show that in order to achieve low query complexity, it is sufficient to design one-sided non-adaptive canonical tests. This result is akin to, and based on an analogous result for error-correcting codes due to Ben-Sasson et al. (SIAM J. Computing, 35(1):1-21).
2. We demonstrate upper and lower bounds on the query complexity of local testing for membership in code formula lattices. We instantiate our results for code formula lattices constructed from Reed-Muller codes to obtain nearly-matching upper and lower bounds on the query complexity of testing such lattices.
3. We contrast lattice testing from code testing by showing lower bounds on the query complexity of testing low-dimensional lattices. This illustrates large lower bounds on the query complexity of testing membership in knapsack lattices. On the other hand, we show that knapsack lattices with bounded coefficients have low-query testers if the inputs are promised to lie in the span of the lattice.

Karthekeyan Chandrasekaran, Mahdi Cheraghchi, Venkata Gandikota, and Elena Grigorescu. Local Testing for Membership in Lattices. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 65, pp. 46:1-46:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chandrasekaran_et_al:LIPIcs.FSTTCS.2016.46, author = {Chandrasekaran, Karthekeyan and Cheraghchi, Mahdi and Gandikota, Venkata and Grigorescu, Elena}, title = {{Local Testing for Membership in Lattices}}, booktitle = {36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)}, pages = {46:1--46:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-027-9}, ISSN = {1868-8969}, year = {2016}, volume = {65}, editor = {Lal, Akash and Akshay, S. and Saurabh, Saket and Sen, Sandeep}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2016.46}, URN = {urn:nbn:de:0030-drops-68818}, doi = {10.4230/LIPIcs.FSTTCS.2016.46}, annote = {Keywords: Lattices, Property Testing, Locally Testable Codes, Complexity Theory} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

AC^0 o MOD_2 circuits are AC^0 circuits augmented with a layer of parity gates just above the input layer. We study AC^0 o MOD2 circuit lower bounds for computing the Boolean Inner Product functions. Recent works by Servedio and Viola (ECCC TR12-144) and Akavia et al. (ITCS 2014) have highlighted this problem as a frontier problem in circuit complexity that arose both as a first step towards solving natural special cases of the matrix rigidity problem and as a candidate for constructing pseudorandom generators of minimal complexity. We give the first superlinear lower bound for the Boolean Inner Product function against AC^0 o MOD2 of depth four or greater. Specifically, we prove a superlinear lower bound for circuits of arbitrary constant depth, and an ~Omega(n^2) lower bound for the special case of depth-4 AC^0 o MOD_2. Our proof of the depth-4 lower bound employs a new "moment-matching" inequality for bounded, nonnegative integer-valued random variables that may be of independent interest: we prove an optimal bound on the maximum difference between two discrete distributions’ values at 0, given that their first d moments match.

Mahdi Cheraghchi, Elena Grigorescu, Brendan Juba, Karl Wimmer, and Ning Xie. AC^0 o MOD_2 Lower Bounds for the Boolean Inner Product. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 35:1-35:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{cheraghchi_et_al:LIPIcs.ICALP.2016.35, author = {Cheraghchi, Mahdi and Grigorescu, Elena and Juba, Brendan and Wimmer, Karl and Xie, Ning}, title = {{AC^0 o MOD\underline2 Lower Bounds for the Boolean Inner Product}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {35:1--35:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.35}, URN = {urn:nbn:de:0030-drops-63150}, doi = {10.4230/LIPIcs.ICALP.2016.35}, annote = {Keywords: Boolean analysis, circuit complexity, lower bounds} }

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**Published in:** LIPIcs, Volume 3, 26th International Symposium on Theoretical Aspects of Computer Science (2009)

We consider the problem of uniform sampling of points on an algebraic variety. Specifically, we develop a randomized algorithm that, given a small set of multivariate polynomials over a sufficiently large finite field, produces a common zero of the polynomials almost uniformly at random. The statistical distance between the output distribution of the algorithm and the uniform distribution on the set of common zeros is polynomially small in the field size, and the running time of the algorithm is polynomial in the description of the polynomials and their degrees provided that the number of the polynomials is a constant.

Mahdi Cheraghchi and Amin Shokrollahi. Almost-Uniform Sampling of Points on High-Dimensional Algebraic Varieties. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 277-288, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2009)

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@InProceedings{cheraghchi_et_al:LIPIcs.STACS.2009.1817, author = {Cheraghchi, Mahdi and Shokrollahi, Amin}, title = {{Almost-Uniform Sampling of Points on High-Dimensional Algebraic Varieties}}, booktitle = {26th International Symposium on Theoretical Aspects of Computer Science}, pages = {277--288}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-09-5}, ISSN = {1868-8969}, year = {2009}, volume = {3}, editor = {Albers, Susanne and Marion, Jean-Yves}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1817}, URN = {urn:nbn:de:0030-drops-18174}, doi = {10.4230/LIPIcs.STACS.2009.1817}, annote = {Keywords: Uniform sampling, Algebraic varieties, Randomized algorithms, Computational complexity} }

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