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Documents authored by Solomon, Tomer


Document
From Donkeys to Kings in Tournaments

Authors: Amir Abboud, Tomer Grossman, Moni Naor, and Tomer Solomon

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)


Abstract
A tournament is an orientation of a complete graph. A vertex that can reach every other vertex within two steps is called a king. We study the complexity of finding k kings in a tournament graph. We show that the randomized query complexity of finding k ≤ 3 kings is O(n), and for the deterministic case it takes the same amount of queries (up to a constant) as finding a single king (the best known deterministic algorithm makes O(n^{3/2}) queries). On the other hand, we show that finding k ≥ 4 kings requires Ω(n²) queries, even in the randomized case. We consider the RAM model for k ≥ 4. We show an algorithm that finds k kings in time O(kn²), which is optimal for constant values of k. Alternatively, one can also find k ≥ 4 kings in time n^{ω} (the time for matrix multiplication). We provide evidence that this is optimal for large k by suggesting a fine-grained reduction from a variant of the triangle detection problem.

Cite as

Amir Abboud, Tomer Grossman, Moni Naor, and Tomer Solomon. From Donkeys to Kings in Tournaments. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 3:1-3:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{abboud_et_al:LIPIcs.ESA.2024.3,
  author =	{Abboud, Amir and Grossman, Tomer and Naor, Moni and Solomon, Tomer},
  title =	{{From Donkeys to Kings in Tournaments}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{3:1--3:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John 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.2024.3},
  URN =		{urn:nbn:de:0030-drops-210740},
  doi =		{10.4230/LIPIcs.ESA.2024.3},
  annote =	{Keywords: Tournament Graphs, Kings, Query Complexity, Fine Grained Complexity}
}
Document
Bootstrapping Homomorphic Encryption via Functional Encryption

Authors: Nir Bitansky and Tomer Solomon

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


Abstract
Homomorphic encryption is a central object in modern cryptography, with far-reaching applications. Constructions supporting homomorphic evaluation of arbitrary Boolean circuits have been known for over a decade, based on standard lattice assumptions. However, these constructions are leveled, meaning that they only support circuits up to some a-priori bounded depth. These leveled constructions can be bootstrapped into fully homomorphic ones, but this requires additional circular security assumptions, which are construction-dependent, and where reductions to standard lattice assumptions are no longer known. Alternative constructions are known based on indistinguishability obfuscation, which has been recently constructed under standard assumptions. However, this alternative requires subexponential hardness of the underlying primitives. We prove a new bootstrapping theorem based on functional encryption, which is known based on standard polynomial hardness assumptions. As a result we obtain the first fully homomorphic encryption scheme that avoids both circular security assumptions and super-polynomial hardness assumptions. The construction is secure against uniform adversaries, and can be made non-uniformly secure assuming a generalization of the time-hierarchy theorem, which follows for example from non-uniform ETH. At the heart of the construction is a new proof technique based on cryptographic puzzles and decomposable obfuscation. Unlike most cryptographic reductions, our security reduction does not fully treat the adversary as a black box, but rather makes explicit use of its running time (or circuit size).

Cite as

Nir Bitansky and Tomer Solomon. Bootstrapping Homomorphic Encryption via Functional Encryption. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 17:1-17:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


Copy BibTex To Clipboard

@InProceedings{bitansky_et_al:LIPIcs.ITCS.2023.17,
  author =	{Bitansky, Nir and Solomon, Tomer},
  title =	{{Bootstrapping Homomorphic Encryption via Functional Encryption}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{17:1--17:23},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.17},
  URN =		{urn:nbn:de:0030-drops-175200},
  doi =		{10.4230/LIPIcs.ITCS.2023.17},
  annote =	{Keywords: Fully Homomorphic Encryption, Polynomial Assumptions, Cryptographic Puzzles}
}
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