2 Search Results for "Schwartzbach, Nikolaj I."


Document
PPP-Completeness and Extremal Combinatorics

Authors: Romain Bourneuf, Lukáš Folwarczný, Pavel Hubáček, Alon Rosen, and Nikolaj I. Schwartzbach

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)


Abstract
Many classical theorems in combinatorics establish the emergence of substructures within sufficiently large collections of objects. Well-known examples are Ramsey’s theorem on monochromatic subgraphs and the Erdős-Rado sunflower lemma. Implicit versions of the corresponding total search problems are known to be PWPP-hard under randomized reductions in the case of Ramsey’s theorem and PWPP-hard in the case of the sunflower lemma; here "implicit” means that the collection is represented by a poly-sized circuit inducing an exponentially large number of objects. We show that several other well-known theorems from extremal combinatorics - including Erdős-Ko-Rado, Sperner, and Cayley’s formula – give rise to complete problems for PWPP and PPP. This is in contrast to the Ramsey and Erdős-Rado problems, for which establishing inclusion in PWPP has remained elusive. Besides significantly expanding the set of problems that are complete for PWPP and PPP, our work identifies some key properties of combinatorial proofs of existence that can give rise to completeness for these classes. Our completeness results rely on efficient encodings for which finding collisions allows extracting the desired substructure. These encodings are made possible by the tightness of the bounds for the problems at hand (tighter than what is known for Ramsey’s theorem and the sunflower lemma). Previous techniques for proving bounds in TFNP invariably made use of structured algorithms. Such algorithms are not known to exist for the theorems considered in this work, as their proofs "from the book" are non-constructive.

Cite as

Romain Bourneuf, Lukáš Folwarczný, Pavel Hubáček, Alon Rosen, and Nikolaj I. Schwartzbach. PPP-Completeness and Extremal Combinatorics. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 22:1-22:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{bourneuf_et_al:LIPIcs.ITCS.2023.22,
  author =	{Bourneuf, Romain and Folwarczn\'{y}, Luk\'{a}\v{s} and Hub\'{a}\v{c}ek, Pavel and Rosen, Alon and Schwartzbach, Nikolaj I.},
  title =	{{PPP-Completeness and Extremal Combinatorics}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{22:1--22:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.22},
  URN =		{urn:nbn:de:0030-drops-175255},
  doi =		{10.4230/LIPIcs.ITCS.2023.22},
  annote =	{Keywords: total search problems, extremal combinatorics, PPP-completeness}
}
Document
More Communication Lower Bounds for Information-Theoretic MPC

Authors: Ivan Bjerre Damgård, Boyang Li, and Nikolaj Ignatieff Schwartzbach

Published in: LIPIcs, Volume 199, 2nd Conference on Information-Theoretic Cryptography (ITC 2021)


Abstract
We prove two classes of lower bounds on the communication complexity of information-theoretically secure multiparty computation. The first lower bound applies to perfect passive secure multiparty computation in the standard model with n = 2t+1 parties of which t are corrupted. We show a lower bound that applies to secure evaluation of any function, assuming that each party can choose to learn or not learn the output. Specifically, we show that there is a function H^* such that for any protocol that evaluates y_i = b_i ⋅ f(x₁,...,x_n) with perfect passive security (where b_i is a private boolean input), the total communication must be at least 1/2 ∑_{i = 1}ⁿ H_f^*(x_i) bits of information. The second lower bound applies to the perfect maliciously secure setting with n = 3t+1 parties. We show that for any n and all large enough S, there exists a reactive functionality F_S taking an S-bit string as input (and with short output) such that any protocol implementing F_S with perfect malicious security must communicate Ω(nS) bits. Since the functionalities we study can be implemented with linear size circuits, the result can equivalently be stated as follows: for any n and all large enough g ∈ ℕ there exists a reactive functionality F_C doing computation specified by a Boolean circuit C with g gates, where any perfectly secure protocol implementing F_C must communicate Ω(n g) bits. The results easily extends to constructing similar functionalities defined over any fixed finite field. Using known techniques, we also show an upper bound that matches the lower bound up to a constant factor (existing upper bounds are a factor lg n off for Boolean circuits). Both results also extend to the case where the threshold t is suboptimal. Namely if n = kt+s the bound is weakened by a factor O(s), which corresponds to known optimizations via packed secret-sharing.

Cite as

Ivan Bjerre Damgård, Boyang Li, and Nikolaj Ignatieff Schwartzbach. More Communication Lower Bounds for Information-Theoretic MPC. In 2nd Conference on Information-Theoretic Cryptography (ITC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 199, pp. 2:1-2:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


Copy BibTex To Clipboard

@InProceedings{damgard_et_al:LIPIcs.ITC.2021.2,
  author =	{Damg\r{a}rd, Ivan Bjerre and Li, Boyang and Schwartzbach, Nikolaj Ignatieff},
  title =	{{More Communication Lower Bounds for Information-Theoretic MPC}},
  booktitle =	{2nd Conference on Information-Theoretic Cryptography (ITC 2021)},
  pages =	{2:1--2:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-197-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{199},
  editor =	{Tessaro, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2021.2},
  URN =		{urn:nbn:de:0030-drops-143211},
  doi =		{10.4230/LIPIcs.ITC.2021.2},
  annote =	{Keywords: Multiparty Computation, Lower bounds}
}
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