# Discrete Gaussian Sampling Reduces to CVP and SVP

@article{StephensDavidowitz2015DiscreteGS, title={Discrete Gaussian Sampling Reduces to CVP and SVP}, author={Noah Stephens-Davidowitz}, journal={ArXiv}, year={2015}, volume={abs/1506.07490} }

The discrete Gaussian $D_{L- t, s}$ is the distribution that assigns to each vector $x$ in a shifted lattice $L - t$ probability proportional to $e^{-\pi \|x\|^2/s^2}$. It has long been an important tool in the study of lattices. More recently, algorithms for discrete Gaussian sampling (DGS) have found many applications in computer science. In particular, polynomial-time algorithms for DGS with very high parameters $s$ have found many uses in cryptography and in reductions between lattice…

## 36 Citations

### Solving the Shortest Vector Problem in 2n Time Using Discrete Gaussian Sampling: Extended Abstract

- Computer Science, MathematicsSTOC
- 2015

The SVP result follows from a natural reduction from SVP to DGS, and a more refined algorithm for DGS above the so-called smoothing parameter of the lattice, which can generate 2n/2 discrete Gaussian samples in just 1.93-approximate decision SVP.

### Improved (Provable) Algorithms for the Shortest Vector Problem via Bounded Distance Decoding

- Computer ScienceSTACS
- 2021

New algorithms that improve the state-of-the-art for provable classical/quantum algorithms for SVP are presented, including a new algorithm that provides a smooth tradeoff between time complexity and memory requirement.

### Solving the Closest Vector Problem in 2^n Time -- The Discrete Gaussian Strikes Again!

- Computer Science, Mathematics2015 IEEE 56th Annual Symposium on Foundations of Computer Science
- 2015

A 2n+o(n)-time and space randomized algorithm for solving the exact Closest Vector Problem (CVP) on n-dimensional Euclidean lattices and it is shown that the approximate closest vectors to a target vector t can be grouped into “lower-dimensional clusters,” and the discrete Gaussian sampling algorithm can be used to solve this variant of approximate CVP.

### Search-to-Decision Reductions for Lattice Problems with Approximation Factors (Slightly) Greater Than One

- Computer Science, MathematicsAPPROX-RANDOM
- 2016

The first dimension-preserving search-to-decision reductions for approximate SVP and CVP are shown and they generalize the known equivalences of the search and decision versions of these problems in the exact case when $\gamma = 1".

### Lattice Gaussian Sampling by Markov Chain Monte Carlo: Convergence Rate and Decoding Complexity

- Computer ScienceArXiv
- 2017

Decoding by MCMC-based lattice Gaussian sampling is investigated in full details, revealing a flexible trade-off between the decoding performance and complexity.

### Approximate $\mathrm {CVP}_{}$ in Time 20.802 n - Now in Any Norm!

- Mathematics, Computer ScienceIPCO
- 2022

We show that a constant factor approximation of the shortest and closest lattice vector problem in any norm can be computed in time 20.802n . This contrasts the corresponding 2n time, (gap)SETH based…

### Dimension-Preserving Reductions Between SVP and CVP in Different p-Norms

- Computer Science, MathematicsSODA
- 2021

The techniques combine those from the recent breakthrough work of Eisenbrand and Venzin [EV20] (which showed how to adapt the current fastest known algorithm for these problems in the l2 norm to all lp norms) together with sparsification-based techniques.

### Deterministic Sampling Decoding: Where Sphere Decoding Meets Lattice Gaussian Distribution

- Computer ScienceArXiv
- 2019

The regularized SD (RSD) algorithm based on Klein's sampling probability is proposed, which achieves a better decoding trade-off than the equivalent SD by fully utilizing the regularization terms.

### Just how hard are rotations of ℤn? Algorithms and cryptography with the simplest lattice

- Computer Science, MathematicsIACR Cryptol. ePrint Arch.
- 2021

The “provably hard” distribution of bases described above is studied and a threshold phenomenon in which “basis reduction algorithms on Z n nearly always ﬁnd a shortest non-zero vector once they have found a vector with length less than √ n/ 2 is observed.

### Lattice-Reduction-Aided Gibbs Algorithm for Lattice Gaussian Sampling: Convergence Enhancement and Decoding Optimization

- Computer ScienceIEEE Transactions on Signal Processing
- 2019

A startup mechanism is proposed for Gibbs sampler decoding, where decoding complexity can be reduced without performance loss and the recycling Gibbs sampling that exploits the potential of samples is also considered to improve the decoding performance in lattice decoding.

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The SVP result follows from a natural reduction from SVP to DGS, and a more refined algorithm for DGS above the so-called smoothing parameter of the lattice, which can generate 2n/2 discrete Gaussian samples in just 1.93-approximate decision SVP.

### Solving the Closest Vector Problem in 2^n Time -- The Discrete Gaussian Strikes Again!

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- 2015

A 2n+o(n)-time and space randomized algorithm for solving the exact Closest Vector Problem (CVP) on n-dimensional Euclidean lattices and it is shown that the approximate closest vectors to a target vector t can be grouped into “lower-dimensional clusters,” and the discrete Gaussian sampling algorithm can be used to solve this variant of approximate CVP.

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