All methods to sample the Gaussian distribution on an n-dimensional lattice use as a building block a procedure to sample the 1-dimensional Gaussian distribution on the integers. Algorithms for 1-dimensional discrete gaussian sampling include the inverse method (12), the discrete ziggurat (3), lazy rejection sampling (2), the exact method (4), the binary method (1), the Knuth-Yao algorithm used in (5,6), and the convolution method (7). The survey (20) covers most of these methods, with the exception of the most recent (7).

**Lattice Signatures and Bimodal Gaussians**

(*Ducas, Durmus, Lepoint & Lyubashevsky*- Crypto 2013)

Binary gaussian sampler with probabilities proportional to*exp(-ln(2) x*^{2})**Faster Gaussian Lattice Sampling Using Lazy Floating-Point Arithmetic**

(*Ducas & Nguyen*Asiacrypt 2012

Lazy Rejection Sampling**Discrete Ziggurat: A Time-Memory Trade-Off for Sampling from a Gaussian Distribution over the Integers**

(*Buchmann, Cabarcas, Gopfert, Hulsing & Weiden*- SAC 2013)

“For large standard deviations, the Ziggurat algorithm outperforms all existing methods”**Sampling Exactly from the Normal Distribution**

(*Karney*- ToMS 2016)

“It can easily be adapted to sample exactly from the discrete normal distribution”**High Precision Discrete Gaussian Sampling on FPGAs**

(*Roy, Vercauteren & Verbauwhede*- SAC 2013) Employs the Knuth-Yao algorithm.**Sampling from discrete Gaussians for lattice-based cryptography on a constrained device**

(*Dwarakanath & Galbraith*- AAECC 2014)**Gaussian Sampling over the Integers: Efficient, Generic, Constant-Time**

(*Micciancio & Walter*- CRYPTO 2017)

Additional references on 1-dimensional Gaussian Sampling include

**Gaussian Sampling Precision in Lattice Cryptography**

*Saarinen*- ePrint 2015/953**Towards efficient discrete Gaussian sampling for lattice-based cryptography**(*Du & Bai*- FLP 2015)**Discrete Gaussian sampling for low-power devices**

(*More & Katti*- PACRIM 2015)**Compact and Side Channel Secure Discrete Gaussian Sampling**

(*Roy, Reparaz, Vercauteren & Verbauwhede*- ePrint 2014/591)

**An Efficient and Parallel Gaussian Sampler for Lattices**

(*Peikert*- Crypto 2010)

Uses inverse method to sample the discrete gaussian**Faster Gaussian Sampling for Trapdoor Lattices with Arbitrary Modulus**

(*Genise & Micciancio*- )

MCMC techniques have also been considered to sample discrete gaussians on lattices:

**On the Geometric Ergodicity of Metropolis-Hastings Algorithms for Lattice Gaussian Sampling**

(*Wang & Ling*- ISIT 2015, ISIT 2016)**Markov Chain Monte Carlo Algorithms for Lattice Gaussian Sampling**

(*Wang, Ling & Hanrot*- ISIT 2014)

**Fast Fourier Orthogonalization**

(*Ducas & Prest*- ePrint 2015/1014)**Quadratic Time, Linear Space Algorithms for Gram-Schmidt Orthogonalization and Gaussian Sampling in Structured Lattices**

(*Lyubashevsky & Prest*- Eurocrypt 2015)**A Hybrid Gaussian Sampler for Lattices over Rings**

(*Ducas & Prest*- ePrint 2015/660) Superseded by FFO

**Simple Lattice Trapdoor Sampling from a Broad Class of Distributions**

(*Lyubashevsky & Wichs*- PKC)

**Gaussian Sampling in Lattice Based Cryptography**

(*Follath*- TMMP 2014)**GLITCH: A Discrete Gaussian Testing Suite For Lattice-Based Cryptography**

(*Howe & O’Neill*- SECRYPT 2017)**An Investigation of Sources of Randomness Within Discrete Gaussian Sampling**

(*Brannigan, Smyth, Oder, Valencia, O’Sullivan, Güneysu & Regazzoni*- )**Implementation and Evaluation of Improved Gaussian Sampling for Lattice Trapdoors**

(*Gür, Polyakov, Rohloff, Ryan & Savaş*- )