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Micciancio, Daniele

doi:10.4086/toc.2012.v008a022

Cited by 32 publications

(6 citation statements)

References 31 publications

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“…Ideas appearing in the recent NP-hardness proofs of the minimum distance problem in linear codes [11,6] might be useful. Finally, we mention that a significant step towards derandomization was recently made by Micciancio [23]: he strengthened our results by showing reductions with only one-sided error.…”

Section: Open Questionssupporting

confidence: 70%

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Haviv¹,

Regev²

2012

Theory of Comput.

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We show that unless NP ⊆ RTIME(2 poly(log n) ), there is no polynomial-time algorithm approximating the Shortest Vector Problem (SVP) on n-dimensional lattices in the p norm (1 ≤ p < ∞) to within a factor of 2 (log n) 1−ε for any ε > 0. This improves the previous best factor of 2 (log n) 1/2−ε under the same complexity assumption due to Khot (J. ACM, 2005). Under the stronger assumption NP RSUBEXP, we obtain a hardness factor of n c/ log log n for some c > 0.Our proof starts with Khot's SVP instances that are hard to approximate to within some constant. To boost the hardness factor we simply apply the standard tensor product of lattices. The main novelty is in the analysis, where we show that the lattices of Khot behave nicely under tensorization. At the heart of the analysis is a certain matrix inequality which was first used in the context of lattices by de Shalit and Parzanchevski (2006).

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Section: Open Questionssupporting

confidence: 70%

Untitled

Haviv¹,

Regev²

2012

Theory of Comput.

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“…Further, assuming NP RTIME(2 poly(log(n)) ) there is no polynomial-time algorithm approximates SVP to within factor of 2 log 1 2 −ε (n) , which is almost polynomial in n. This result is way more stronger than Micciancio (1998) and Khot (2003). Later, Micciancio (2012) proposed another proof for the NP-hardness of approximating SVP. Micciancio (2012) used the same technique as the one used by Khot (2004), which is called the BCH code, in a different manner.…”

Section: Introductionmentioning

confidence: 99%

“…Later, Micciancio (2012) proposed another proof for the NP-hardness of approximating SVP. Micciancio (2012) used the same technique as the one used by Khot (2004), which is called the BCH code, in a different manner. He had also proved the NP-hardness of SVP for any constant approximation factor, moreover he proved that approximating SVP for subpolynomial factors n 1 O(log log n) is NP-hard assuming that NP is not contained by subexponential time.…”

Section: Introductionmentioning

confidence: 99%

Improved lower bound for the complexity of unique shortest vector problem

Jin,

Xue

2023

Cybersecurity

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Unique shortest vector problem (uSVP) plays an important role in lattice based cryptography. Many cryptographic schemes based their security on it. For the cofidence of those applications, it is essential to clarify the complexity of uSVP with different parameters. However, proving the NP-hardness of uSVP appears quite hard. To the state of the art, we are even not able to prove the NP-hardness of uSVP with constant parameters. In this work, we gave a lower bound for the hardness of uSVP with constant parameters, i.e. we proved that uSVP is at least as hard as gap shortest vector problem (GapSVP) with gap of $$O(\sqrt{n/\log (n)})$$ O ( n / log ( n ) ) , which is in $$NP \cap coAM$$ N P ∩ c o A M . Unlike previous works, our reduction works for paramters in a bigger range, especially when the constant hidden by the big-O in GapSVP is smaller than 1. Graphical abstract

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“…On the negative side, SVP in 2 is NP-hard (under randomized reductions) to solve exactly, or even to approximate to within any constant factor [1,9,34,29]. Many more hardness results are known for other p norms and under stronger complexity assumptions than P = NP [17,14,43,22,35]. CVP is NP-hard to approximate to within n c/ log log n factors for some constant c > 0 [4,15], where n is the dimension of the lattice.…”

Section: Introductionmentioning

confidence: 99%