On the Lattice Smoothing Parameter Problem

@article{Chung2013OnTL,
  title={On the Lattice Smoothing Parameter Problem},
  author={Kai-Min Chung and Daniel Dadush and Feng-Hao Liu and Chris Peikert},
  journal={2013 IEEE Conference on Computational Complexity},
  year={2013},
  pages={230-241}
}
The smoothing parameter ηε(L) of a Euclidean lattice L, introduced by Micciancio and Regev (FOCS'04; SICOMP'07), is (informally) the smallest amount of Gaussian noise that “smooths out” the discrete structure of L (up to error ε). It plays a central role in the best known worst-case/average-case reductions for lattice problems, a wealth of lattice-based cryptographic constructions, and (implicitly) the tightest known transference theorems for fundamental lattice quantities. In this work we… 
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References

SHOWING 1-10 OF 41 REFERENCES
Limits on the Hardness of Lattice Problems in ℓp Norms
  • Chris Peikert
  • Computer Science, Mathematics
    Twenty-Second Annual IEEE Conference on Computational Complexity (CCC'07)
  • 2007
TLDR
The results improve prior approximation factors for ℓp norms by up to up to $$\sqrt{n}$$ factors, and provide some evidence that lattice problems in ™p norms (for p > 2) may not be substantially harder than they are in the ™2 norm.
On the Limits of Nonapproximability of Lattice Problems
We show simple constant-round interactive proof systems for problems capturing the approximability, to within a factor of n, of optimization problems in integer lattices, specifically, the closest…
Worst-case to average-case reductions based on Gaussian measures
TLDR
It is shown that solving modular linear equation on the average is at least as hard as approximating several lattice problems in the worst case within a factor almost linear in the rank of the lattice, and it is proved that the distribution that one obtains after adding Gaussian noise to the lattices has the following interesting property.
New lattice-based cryptographic constructions
  • O. Regev
  • Mathematics, Computer Science
    JACM
  • 2004
TLDR
A new public key cryptosystem whose security guarantee is considerably stronger than previous results is provided, and a family of collision resistant hash functions with an improved security guarantee in terms of the unique shortest vector problem is proposed.
Lattice problems in NP ∩ coNP
We show that the problems of approximating the shortest and closest vector in a lattice to within a factor of &nradic; lie in NP intersect coNP. The result (almost) subsumes the three…
Generating Hard Instances of Lattice Problems
  • M. Ajtai
  • Mathematics, Computer Science
    Electron. Colloquium Comput. Complex.
  • 1996
We give a random class of lattices in Z n so that, if there is a probabilistic polynomial time algorithm which nds a short vector in a random lattice with a probability of at least 1 2 then there is…
An Efficient and Parallel Gaussian Sampler for Lattices
TLDR
To the knowledge, this is the first algorithm and rigorous analysis demonstrating the security of a perturbation-like technique and a new Gaussian sampling algorithm for lattices that is efficient and highly parallelizable.
The complexity of the covering radius problem
TLDR
The computational complexity of the covering radius problem for lattices, and approximation versions of the problem for both lattices and linear codes are studied, and it is proved that the problem is NP-hard for any constant approximation factor, it is Π2- hard for some constant approximation factors, and that it is unlikely to be Π1-hardfor approximation factors larger than 2.
Enumerative Lattice Algorithms in any Norm Via M-ellipsoid Coverings
TLDR
A novel algorithm for enumerating lattice points in any convex body known as the M-ellipsoid is given, and an expected O(f*(n))^n-time algorithm for Integer Programming, where f*( n) denotes the optimal bound in the so-calledflatnesstheorem, which is conjectured to be f* (n) = O(n).
Trapdoors for hard lattices and new cryptographic constructions
TLDR
A new notion of trapdoor function with preimage sampling, simple and efficient "hash-and-sign" digital signature schemes, and identity-based encryption are included.
...
...