Improved (Provable) Algorithms for the Shortest Vector Problem via Bounded Distance Decoding
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| Název: | Improved (Provable) Algorithms for the Shortest Vector Problem via Bounded Distance Decoding |
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| Autoři: | Divesh Aggarwal and Yanlin Chen and Rajendra Kumar and Yixin Shen, Aggarwal, Divesh, Chen, Yanlin, Kumar, Rajendra, Shen, Yixin |
| Informace o vydavateli: | Schloss Dagstuhl – Leibniz-Zentrum für Informatik 2021 |
| Druh dokumentu: | Electronic Resource |
| Abstrakt: | The most important computational problem on lattices is the Shortest Vector Problem (SVP). In this paper, we present new algorithms that improve the state-of-the-art for provable classical/quantum algorithms for SVP. We present the following results. 1) A new algorithm for SVP that provides a smooth tradeoff between time complexity and memory requirement. For any positive integer 4 ≤ q ≤ √n, our algorithm takes q^{13n+o(n)} time and requires poly(n)⋅ q^{16n/q²} memory. This tradeoff which ranges from enumeration (q = √n) to sieving (q constant), is a consequence of a new time-memory tradeoff for Discrete Gaussian sampling above the smoothing parameter. 2) A quantum algorithm that runs in time 2^{0.9533n+o(n)} and requires 2^{0.5n+o(n)} classical memory and poly(n) qubits. This improves over the previously fastest classical (which is also the fastest quantum) algorithm due to [Divesh Aggarwal et al., 2015] that has a time and space complexity 2^{n+o(n)}. 3) A classical algorithm for SVP that runs in time 2^{1.741n+o(n)} time and 2^{0.5n+o(n)} space. This improves over an algorithm of [Yanlin Chen et al., 2018] that has the same space complexity. The time complexity of our classical and quantum algorithms are expressed using a quantity related to the kissing number of a lattice. A known upper bound of this quantity is 2^{0.402n}, but in practice for most lattices, it can be much smaller and even 2^o(n). In that case, our classical algorithm runs in time 2^{1.292n} and our quantum algorithm runs in time 2^{0.750n}. |
| Témata: | Lattices, Shortest Vector Problem, Discrete Gaussian Sampling, Time-Space Tradeoff, Quantum computation, Bounded distance decoding, InProceedings, Text, doc-type:ResearchArticle, publishedVersion |
| DOI: | 10.4230.LIPIcs.STACS.2021.4 |
| URL: | Is Part Of LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021) |
| Dostupnost: | Open access content. Open access content https://creativecommons.org/licenses/by/4.0/legalcode |
| Poznámka: | application/pdf English |
| Other Numbers: | DEDAG oai:drops-oai.dagstuhl.de:13649 doi:10.4230/LIPIcs.STACS.2021.4 urn:nbn:de:0030-drops-136494 1358728562 |
| Přispívající zdroj: | SCHLOSS DAGSTUHL LEIBNIZ ZENTRUM GMBH From OAIster®, provided by the OCLC Cooperative. |
| Přístupové číslo: | edsoai.on1358728562 |
| Databáze: | OAIster |
Nájsť tento článok vo Web of Science | Abstrakt: | The most important computational problem on lattices is the Shortest Vector Problem (SVP). In this paper, we present new algorithms that improve the state-of-the-art for provable classical/quantum algorithms for SVP. We present the following results. 1) A new algorithm for SVP that provides a smooth tradeoff between time complexity and memory requirement. For any positive integer 4 ≤ q ≤ √n, our algorithm takes q^{13n+o(n)} time and requires poly(n)⋅ q^{16n/q²} memory. This tradeoff which ranges from enumeration (q = √n) to sieving (q constant), is a consequence of a new time-memory tradeoff for Discrete Gaussian sampling above the smoothing parameter. 2) A quantum algorithm that runs in time 2^{0.9533n+o(n)} and requires 2^{0.5n+o(n)} classical memory and poly(n) qubits. This improves over the previously fastest classical (which is also the fastest quantum) algorithm due to [Divesh Aggarwal et al., 2015] that has a time and space complexity 2^{n+o(n)}. 3) A classical algorithm for SVP that runs in time 2^{1.741n+o(n)} time and 2^{0.5n+o(n)} space. This improves over an algorithm of [Yanlin Chen et al., 2018] that has the same space complexity. The time complexity of our classical and quantum algorithms are expressed using a quantity related to the kissing number of a lattice. A known upper bound of this quantity is 2^{0.402n}, but in practice for most lattices, it can be much smaller and even 2^o(n). In that case, our classical algorithm runs in time 2^{1.292n} and our quantum algorithm runs in time 2^{0.750n}. |
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| DOI: | 10.4230.LIPIcs.STACS.2021.4 |