# Discrete Gaussians over the Integers

A discrete Gaussian distribution on the Integers is a distribution where the integer $x$ is sampled with probability proportional to $exp(-(x-c)²/(2σ²))$. It is denoted by $D_{σ,c}$ where σ is the width parameter (close to the standard deviation) and $c$ is the center.

This library samples from this distributions.

## Algorithms

• DGS_DISC_GAUSS_UNIFORM_TABLE - classical rejection sampling, sampling from the uniform distribution and accepted with probability proportional to $\exp(-(x-c)²/(2σ²))$ where $\exp(-(x-c)²/(2σ²))$ is precomputed and stored in a table. Any real-valued c is supported.

• DGS_DISC_GAUSS_UNIFORM_LOGTABLE - samples are drawn from a uniform distribution and accepted with probability proportional to $\exp(-(x-c)²/(2σ²))$ where $\exp(-(x-c)²/(2σ²))$ is computed using logarithmically many calls to Bernoulli distributions. Only integer-valued $c$ are supported.

• DGS_DISC_GAUSS_UNIFORM_ONLINE - samples are drawn from a uniform distribution and accepted with probability proportional to $\exp(-(x-c)²/(2σ²))$ where $\exp(-(x-c)²/(2σ²))$ is computed in each invocation. Typically this is very slow. Any real-valued $c$ is accepted.

• DGS_DISC_SIGMA2_LOGTABLE - samples are drawn from an easily samplable distribution with $σ = k·σ₂$ where $σ₂ := \sqrt{1/(2\log 2)}$ and accepted with probability proportional to $\exp(-(x-c)²/(2σ²))$ where $\exp(-(x-c)²/(2σ²))$ is computed using logarithmically many calls to Bernoulli distributions (but no calls to $\exp$). Note that this sampler adjusts sigma to match $σ₂·k$ for some integer $k$. Only integer-valued $c$ are supported.

Algorithm 2-4 are described in:

Léo Ducas, Alain Durmus, Tancrède Lepoint and Vadim Lyubashevsky. Lattice Signatures and Bimodal Gaussians; in Advances in Cryptology – CRYPTO 2013; Lecture Notes in Computer Science Volume 8042, 2013, pp 40-56 (PDF)

## Precisions

• mp - multi-precision using MPFR, cf. dgs_gauss_mp.c

• dp - double precision using machine doubles, cf. dgs_gauss_dp.c.

For readers unfamiliar with the implemented algorithms it makes sense to start with dgs_gauss_dp.c which implements the same algorithms as dgs_gauss_mp.c but should be easier to read.

## Typical Usage

dgs_disc_gauss_dp_t *D = dgs_disc_gauss_dp_init(<sigma>, <c>, <tau>, <algorithm>);
D->call(D); // as often as needed
dgs_disc_gauss_dp_clear(D);