(Gap/S)ETH hardness of SVP

Aggarwal, Divesh; Stephens-Davidowitz, Noah

doi:10.1145/3188745.3188840

Cited by 28 publications

(79 citation statements)

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“…[59] showed that for almost all p ≥ 1, CVP in the p norm cannot be solved in 2 n(1−ε) of time under the strong exponential time hypothesis. A similar difficulty result has also been obtained for SVP in the p norm [60].…”

Section: Hardness Resultssupporting

confidence: 79%

Faster Provable Sieving Algorithms for the Shortest Vector Problem and the Closest Vector Problem on Lattices in ℓp Norm

Mukhopadhyay

2021

Algorithms

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In this work, we give provable sieving algorithms for the Shortest Vector Problem (SVP) and the Closest Vector Problem (CVP) on lattices in ℓp norm (1≤p≤∞). The running time we obtain is better than existing provable sieving algorithms. We give a new linear sieving procedure that works for all ℓp norm (1≤p≤∞). The main idea is to divide the space into hypercubes such that each vector can be mapped efficiently to a sub-region. We achieve a time complexity of 22.751n+o(n), which is much less than the 23.849n+o(n) complexity of the previous best algorithm. We also introduce a mixed sieving procedure, where a point is mapped to a hypercube within a ball and then a quadratic sieve is performed within each hypercube. This improves the running time, especially in the ℓ2 norm, where we achieve a time complexity of 22.25n+o(n), while the List Sieve Birthday algorithm has a running time of 22.465n+o(n). We adopt our sieving techniques to approximation algorithms for SVP and CVP in ℓp norm (1≤p≤∞) and show that our algorithm has a running time of 22.001n+o(n), while previous algorithms have a time complexity of 23.169n+o(n).

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Section: Hardness Resultssupporting

confidence: 79%

Faster Provable Sieving Algorithms for the Shortest Vector Problem and the Closest Vector Problem on Lattices in ℓp Norm

Mukhopadhyay

2021

Algorithms

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“…Our third hardness result for BDD, presented in Theorem 1.3, uses ideas from the other reductions together with our new framework. Finally, our improved result for SVP, presented in Theorem 1.4, answers a natural question left open by [AS18,ABGS21].…”

Section: Our Resultsmentioning

confidence: 69%

“…At a technical level, our improved results for BDD follow from improved analysis of the techniques used in [AS18] and [BP20] together with our new framework. Aggarwal and Stephens-Davidowitz [AS18] presented three main results on the fine-grained hardness of SVP, summarized in Items 1 to 3 in its abstract. Bennett and Peikert [BP20] showed new hardness results for BDD by refining and adapting the analysis used to show [AS18], Item 1.…”

Section: Our Resultsmentioning

confidence: 99%

“…Aggarwal and Stephens-Davidowitz [AS18] presented three main results on the fine-grained hardness of SVP, summarized in Items 1 to 3 in its abstract. Bennett and Peikert [BP20] showed new hardness results for BDD by refining and adapting the analysis used to show [AS18], Item 1. Analogously, in this work we obtain two of our hardness results for BDD by refining and adapting the analysis used to show [AS18], Items 2 and 3.…”

Section: Our Resultsmentioning

confidence: 99%

“…Analogously, in this work we obtain two of our hardness results for BDD by refining and adapting the analysis used to show [AS18], Items 2 and 3. Specifically, Theorem 1.1 corresponds to [AS18], Item 3 and Theorem 1.2 to [AS18], Item 2. Our third hardness result for BDD, presented in Theorem 1.3, uses ideas from the other reductions together with our new framework.…”

Section: Our Resultsmentioning

confidence: 99%

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