Search Results

Documents authored by Belovs, Aleksandrs

Document
DOI: 10.4230/LIPIcs.TQC.2024.11
A Direct Reduction from the Polynomial to the Adversary Method

Authors: Aleksandrs Belovs

Published in: LIPIcs, Volume 310, 19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024)

Abstract
The polynomial and the adversary methods are the two main tools for proving lower bounds on query complexity of quantum algorithms. Both methods have found a large number of applications, some problems more suitable for one method, some for the other. It is known though that the adversary method, in its general negative-weighted version, is tight for bounded-error quantum algorithms, whereas the polynomial method is not. By the tightness of the former, for any polynomial lower bound, there ought to exist a corresponding adversary lower bound. However, direct reduction was not known. In this paper, we give a simple and direct reduction from the polynomial method (in the form of a dual polynomial) to the adversary method. This shows that any lower bound in the form of a dual polynomial is actually an adversary lower bound of a specific form.
Cite as

Aleksandrs Belovs. A Direct Reduction from the Polynomial to the Adversary Method. In 19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 310, pp. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{belovs:LIPIcs.TQC.2024.11,
  author =	{Belovs, Aleksandrs},
  title =	{{A Direct Reduction from the Polynomial to the Adversary Method}},
  booktitle =	{19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024)},
  pages =	{11:1--11:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-328-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{310},
  editor =	{Magniez, Fr\'{e}d\'{e}ric and Grilo, Alex Bredariol},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2024.11},
  URN =		{urn:nbn:de:0030-drops-206814},
  doi =		{10.4230/LIPIcs.TQC.2024.11},
  annote =	{Keywords: Polynomials, Quantum Adversary Bound}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2024.9
Quantum Algorithms for Hopcroft’s Problem

Authors: Vladimirs Andrejevs, Aleksandrs Belovs, and Jevgēnijs Vihrovs

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract
In this work we study quantum algorithms for Hopcroft’s problem which is a fundamental problem in computational geometry. Given n points and n lines in the plane, the task is to determine whether there is a point-line incidence. The classical complexity of this problem is well-studied, with the best known algorithm running in O(n^{4/3}) time, with matching lower bounds in some restricted settings. Our results are two different quantum algorithms with time complexity Õ(n^{5/6}). The first algorithm is based on partition trees and the quantum backtracking algorithm. The second algorithm uses a quantum walk together with a history-independent dynamic data structure for storing line arrangement which supports efficient point location queries. In the setting where the number of points and lines differ, the quantum walk-based algorithm is asymptotically faster. The quantum speedups for the aforementioned data structures may be useful for other geometric problems.
Cite as

Vladimirs Andrejevs, Aleksandrs Belovs, and Jevgēnijs Vihrovs. Quantum Algorithms for Hopcroft’s Problem. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{andrejevs_et_al:LIPIcs.MFCS.2024.9,
  author =	{Andrejevs, Vladimirs and Belovs, Aleksandrs and Vihrovs, Jevg\={e}nijs},
  title =	{{Quantum Algorithms for Hopcroft’s Problem}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{9:1--9:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.9},
  URN =		{urn:nbn:de:0030-drops-205653},
  doi =		{10.4230/LIPIcs.MFCS.2024.9},
  annote =	{Keywords: Quantum algorithms, Quantum walks, Computational Geometry}
}
Document
DOI: 10.4230/LIPIcs.CCC.2023.24
An Exponential Separation Between Quantum Query Complexity and the Polynomial Degree

Authors: Andris Ambainis and Aleksandrs Belovs

Published in: LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023)

Abstract
While it is known that there is at most a polynomial separation between quantum query complexity and the polynomial degree for total functions, the precise relationship between the two is not clear for partial functions. In this paper, we demonstrate an exponential separation between exact polynomial degree and approximate quantum query complexity for a partial Boolean function. For an unbounded alphabet size, we have a constant versus polynomial separation.
Cite as

Andris Ambainis and Aleksandrs Belovs. An Exponential Separation Between Quantum Query Complexity and the Polynomial Degree. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 24:1-24:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

Copy BibTex To Clipboard

@InProceedings{ambainis_et_al:LIPIcs.CCC.2023.24,
  author =	{Ambainis, Andris and Belovs, Aleksandrs},
  title =	{{An Exponential Separation Between Quantum Query Complexity and the Polynomial Degree}},
  booktitle =	{38th Computational Complexity Conference (CCC 2023)},
  pages =	{24:1--24:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-282-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{264},
  editor =	{Ta-Shma, Amnon},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.24},
  URN =		{urn:nbn:de:0030-drops-182943},
  doi =		{10.4230/LIPIcs.CCC.2023.24},
  annote =	{Keywords: Polynomials, Quantum Adversary Bound, Separations in Query Complexity}
}
Document
DOI: 10.4230/LIPIcs.TQC.2020.10
Quantum Coupon Collector

Authors: Srinivasan Arunachalam, Aleksandrs Belovs, Andrew M. Childs, Robin Kothari, Ansis Rosmanis, and Ronald de Wolf

Published in: LIPIcs, Volume 158, 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020)

Abstract
We study how efficiently a k-element set S⊆[n] can be learned from a uniform superposition |S> of its elements. One can think of |S>=∑_{i∈S}|i>/√|S| as the quantum version of a uniformly random sample over S, as in the classical analysis of the "coupon collector problem." We show that if k is close to n, then we can learn S using asymptotically fewer quantum samples than random samples. In particular, if there are n-k=O(1) missing elements then O(k) copies of |S> suffice, in contrast to the Θ(k log k) random samples needed by a classical coupon collector. On the other hand, if n-k=Ω(k), then Ω(k log k) quantum samples are necessary. More generally, we give tight bounds on the number of quantum samples needed for every k and n, and we give efficient quantum learning algorithms. We also give tight bounds in the model where we can additionally reflect through |S>. Finally, we relate coupon collection to a known example separating proper and improper PAC learning that turns out to show no separation in the quantum case.
Cite as

Srinivasan Arunachalam, Aleksandrs Belovs, Andrew M. Childs, Robin Kothari, Ansis Rosmanis, and Ronald de Wolf. Quantum Coupon Collector. In 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 158, pp. 10:1-10:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

Copy BibTex To Clipboard

@InProceedings{arunachalam_et_al:LIPIcs.TQC.2020.10,
  author =	{Arunachalam, Srinivasan and Belovs, Aleksandrs and Childs, Andrew M. and Kothari, Robin and Rosmanis, Ansis and de Wolf, Ronald},
  title =	{{Quantum Coupon Collector}},
  booktitle =	{15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020)},
  pages =	{10:1--10:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-146-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{158},
  editor =	{Flammia, Steven T.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2020.10},
  URN =		{urn:nbn:de:0030-drops-120692},
  doi =		{10.4230/LIPIcs.TQC.2020.10},
  annote =	{Keywords: Quantum algorithms, Adversary method, Coupon collector, Quantum learning theory}
}
Document
DOI: 10.4230/LIPIcs.ESA.2019.16
Quantum Algorithms for Classical Probability Distributions

Authors: Aleksandrs Belovs

Published in: LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

Abstract
We study quantum algorithms working on classical probability distributions. We formulate four different models for accessing a classical probability distribution on a quantum computer, which are derived from previous work on the topic, and study their mutual relationships. Additionally, we prove that quantum query complexity of distinguishing two probability distributions is given by their inverse Hellinger distance, which gives a quadratic improvement over classical query complexity for any pair of distributions. The results are obtained by using the adversary method for state-generating input oracles and for distinguishing probability distributions on input strings.
Cite as

Aleksandrs Belovs. Quantum Algorithms for Classical Probability Distributions. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 16:1-16:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

Copy BibTex To Clipboard

@InProceedings{belovs:LIPIcs.ESA.2019.16,
  author =	{Belovs, Aleksandrs},
  title =	{{Quantum Algorithms for Classical Probability Distributions}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{16:1--16:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-124-5},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{144},
  editor =	{Bender, Michael A. and Svensson, Ola and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.16},
  URN =		{urn:nbn:de:0030-drops-111370},
  doi =		{10.4230/LIPIcs.ESA.2019.16},
  annote =	{Keywords: quantum query complexity, quantum adversary method, distinguishing probability distributions, Hellinger distance}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2018.31
Adaptive Lower Bound for Testing Monotonicity on the Line

Authors: Aleksandrs Belovs

Published in: LIPIcs, Volume 116, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)

Abstract
In the property testing model, the task is to distinguish objects possessing some property from the objects that are far from it. One of such properties is monotonicity, when the objects are functions from one poset to another. This is an active area of research. In this paper we study query complexity of epsilon-testing monotonicity of a function f : [n]->[r]. All our lower bounds are for adaptive two-sided testers. - We prove a nearly tight lower bound for this problem in terms of r. The bound is Omega((log r)/(log log r)) when epsilon = 1/2. No previous satisfactory lower bound in terms of r was known. - We completely characterise query complexity of this problem in terms of n for smaller values of epsilon. The complexity is Theta(epsilon^{-1} log (epsilon n)). Apart from giving the lower bound, this improves on the best known upper bound. Finally, we give an alternative proof of the Omega(epsilon^{-1}d log n - epsilon^{-1}log epsilon^{-1}) lower bound for testing monotonicity on the hypergrid [n]^d due to Chakrabarty and Seshadhri (RANDOM'13).
Cite as

Aleksandrs Belovs. Adaptive Lower Bound for Testing Monotonicity on the Line. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 31:1-31:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

Copy BibTex To Clipboard

@InProceedings{belovs:LIPIcs.APPROX-RANDOM.2018.31,
  author =	{Belovs, Aleksandrs},
  title =	{{Adaptive Lower Bound for Testing Monotonicity on the Line}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
  pages =	{31:1--31:10},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-085-9},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{116},
  editor =	{Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.31},
  URN =		{urn:nbn:de:0030-drops-94350},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2018.31},
  annote =	{Keywords: property testing, monotonicity on the line, monotonicity on the hypergrid}
}
Document
DOI: 10.4230/LIPIcs.TQC.2018.3
Quantum Lower Bounds for Tripartite Versions of the Hidden Shift and the Set Equality Problems

Authors: Aleksandrs Belovs and Ansis Rosmanis

Published in: LIPIcs, Volume 111, 13th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2018)

Abstract
In this paper, we study quantum query complexity of the following rather natural tripartite generalisations (in the spirit of the 3-sum problem) of the hidden shift and the set equality problems, which we call the 3-shift-sum and the 3-matching-sum problems. The 3-shift-sum problem is as follows: given a table of 3 x n elements, is it possible to circularly shift its rows so that the sum of the elements in each column becomes zero? It is promised that, if this is not the case, then no 3 elements in the table sum up to zero. The 3-matching-sum problem is defined similarly, but it is allowed to arbitrarily permute elements within each row. For these problems, we prove lower bounds of Omega(n^{1/3}) and Omega(sqrt n), respectively. The second lower bound is tight. The lower bounds are proven by a novel application of the dual learning graph framework and by using representation-theoretic tools from [Belovs, 2018].
Cite as

Aleksandrs Belovs and Ansis Rosmanis. Quantum Lower Bounds for Tripartite Versions of the Hidden Shift and the Set Equality Problems. In 13th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 111, pp. 3:1-3:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

Copy BibTex To Clipboard

@InProceedings{belovs_et_al:LIPIcs.TQC.2018.3,
  author =	{Belovs, Aleksandrs and Rosmanis, Ansis},
  title =	{{Quantum Lower Bounds for Tripartite Versions of the Hidden Shift and the Set Equality Problems}},
  booktitle =	{13th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2018)},
  pages =	{3:1--3:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-080-4},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{111},
  editor =	{Jeffery, Stacey},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2018.3},
  URN =		{urn:nbn:de:0030-drops-92501},
  doi =		{10.4230/LIPIcs.TQC.2018.3},
  annote =	{Keywords: Adversary Bound, Dual Learning Graphs, Quantum Query Complexity, Representation Theory}
}
Document
DOI: 10.4230/LIPIcs.TQC.2017.3
Provably Secure Key Establishment Against Quantum Adversaries

Authors: Aleksandrs Belovs, Gilles Brassard, Peter Høyer, Marc Kaplan, Sophie Laplante, and Louis Salvail

Published in: LIPIcs, Volume 73, 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017)

Abstract
At Crypto 2011, some of us had proposed a family of cryptographic protocols for key establishment capable of protecting quantum and classical legitimate parties unconditionally against a quantum eavesdropper in the query complexity model. Unfortunately, our security proofs were unsatisfactory from a cryptographically meaningful perspective because they were sound only in a worst-case scenario. Here, we extend our results and prove that for any \eps > 0, there is a classical protocol that allows the legitimate parties to establish a common key after O(N) expected queries to a random oracle, yet any quantum eavesdropper will have a vanishing probability of learning their key after O(N^(1.5-\eps)) queries to the same oracle. The vanishing probability applies to a typical run of the protocol. If we allow the legitimate parties to use a quantum computer as well, their advantage over the quantum eavesdropper becomes arbitrarily close to the quadratic advantage that classical legitimate parties enjoyed over classical eavesdroppers in the seminal 1974 work of Ralph Merkle. Along the way, we develop new tools to give lower bounds on the number of quantum queries required to distinguish two probability distributions. This method in itself could have multiple applications in cryptography. We use it here to study average-case quantum query complexity, for which we develop a new composition theorem of independent interest.
Cite as

Aleksandrs Belovs, Gilles Brassard, Peter Høyer, Marc Kaplan, Sophie Laplante, and Louis Salvail. Provably Secure Key Establishment Against Quantum Adversaries. In 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 73, pp. 3:1-3:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

Copy BibTex To Clipboard

@InProceedings{belovs_et_al:LIPIcs.TQC.2017.3,
  author =	{Belovs, Aleksandrs and Brassard, Gilles and H{\o}yer, Peter and Kaplan, Marc and Laplante, Sophie and Salvail, Louis},
  title =	{{Provably Secure Key Establishment Against Quantum Adversaries}},
  booktitle =	{12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017)},
  pages =	{3:1--3:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-034-7},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{73},
  editor =	{Wilde, Mark M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2017.3},
  URN =		{urn:nbn:de:0030-drops-85816},
  doi =		{10.4230/LIPIcs.TQC.2017.3},
  annote =	{Keywords: Merkle puzzles, Key establishment schemes, Quantum cryptography, Adversary method, Average-case analysis}
}
Document
DOI: 10.4230/LIPIcs.CCC.2017.30
On the Polynomial Parity Argument Complexity of the Combinatorial Nullstellensatz

Authors: Aleksandrs Belovs, Gábor Ivanyos, Youming Qiao, Miklos Santha, and Siyi Yang

Published in: LIPIcs, Volume 79, 32nd Computational Complexity Conference (CCC 2017)

Abstract
The complexity class PPA consists of NP-search problems which are reducible to the parity principle in undirected graphs. It contains a wide variety of interesting problems from graph theory, combinatorics, algebra and number theory, but only a few of these are known to be complete in the class. Before this work, the known complete problems were all discretizations or combinatorial analogues of topological fixed point theorems. Here we prove the PPA-completeness of two problems of radically different style. They are PPA-Circuit CNSS and PPA-Circuit Chevalley, related respectively to the Combinatorial Nullstellensatz and to the Chevalley-Warning Theorem over the two elements field GF(2). The input of these problems contain PPA-circuits which are arithmetic circuits with special symmetric properties that assure that the polynomials computed by them have always an even number of zeros. In the proof of the result we relate the multilinear degree of the polynomials to the parity of the maximal parse subcircuits that compute monomials with maximal multilinear degree, and we show that the maximal parse subcircuits of a PPA-circuit can be paired in polynomial time.
Cite as

Aleksandrs Belovs, Gábor Ivanyos, Youming Qiao, Miklos Santha, and Siyi Yang. On the Polynomial Parity Argument Complexity of the Combinatorial Nullstellensatz. In 32nd Computational Complexity Conference (CCC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 79, pp. 30:1-30:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

Copy BibTex To Clipboard

@InProceedings{belovs_et_al:LIPIcs.CCC.2017.30,
  author =	{Belovs, Aleksandrs and Ivanyos, G\'{a}bor and Qiao, Youming and Santha, Miklos and Yang, Siyi},
  title =	{{On the Polynomial Parity Argument Complexity of the Combinatorial Nullstellensatz}},
  booktitle =	{32nd Computational Complexity Conference (CCC 2017)},
  pages =	{30:1--30:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-040-8},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{79},
  editor =	{O'Donnell, Ryan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2017.30},
  URN =		{urn:nbn:de:0030-drops-75260},
  doi =		{10.4230/LIPIcs.CCC.2017.30},
  annote =	{Keywords: Chevalley-Warning Theorem, Combinatorail Nullstellensatz, Polynomial Parity Argument, arithmetic circuit}
}
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact