Discrete Gaussian Samplers for \(\ZZ[x]\)

This class realizes oracles which returns polynomials in \(\ZZ[x]\) where each coefficient is sampled independently with a probability proportional to \(\exp(-(x-c)²/(2σ²))\).

AUTHORS:

• Martin Albrecht, Robert Fitzpatrick, Daniel Cabracas, Florian Göpfert, Michael Schneider: initial version

EXAMPLES:

sage: from sage.stats.distributions.discrete_gaussian_polynomial import DiscreteGaussianDistributionPolynomialSampler
sage: sigma = 3.0; n=1000
sage: l = [DiscreteGaussianDistributionPolynomialSampler(ZZ['x'], 64, sigma)() for _ in range(n)]
sage: l = [vector(f).norm().n() for f in l]
sage: mean(l), sqrt(64)*sigma
(23.83..., 24.0...)

class sage.stats.distributions.discrete_gaussian_polynomial.DiscreteGaussianDistributionPolynomialSampler(P, n, sigma)

Bases: sage.structure.sage_object.SageObject

Discrete Gaussian sampler for polynomials.

EXAMPLES:

sage: from sage.stats.distributions.discrete_gaussian_polynomial import DiscreteGaussianDistributionPolynomialSampler
sage: DiscreteGaussianDistributionPolynomialSampler(ZZ['x'], 8, 3.0)()
3*x^7 + 3*x^6 - 3*x^5 - x^4 - 5*x^2 + 3
sage: gs = DiscreteGaussianDistributionPolynomialSampler(ZZ['x'], 8, 3.0)
sage: [gs() for _ in range(3)]
[4*x^7 + 4*x^6 - 4*x^5 + 2*x^4 + x^3 - 4*x + 7, -5*x^6 + 4*x^5 - 3*x^3 + 4*x^2 + x, 2*x^7 + 2*x^6 + 2*x^5 - x^4 - 2*x^2 + 3*x + 1]

__init__(P, n, sigma)

Construct a sampler for univariate polynomials of degree n-1 where coefficients are drawn independently with standard deviation sigma.

INPUT:

• P - a univariate polynomial ring over the Integers

• n - number of coefficients to be sampled

• sigma - coefficients \(x\) are accepted with probability proportional to \(\exp(-x²/(2σ²))\). If an object of type sage.stats.distributions.discrete_gaussian_integer.DiscreteGaussianDistributionIntegerSampler is passed, then this sampler is used to sample coefficients.

EXAMPLES:

sage: from sage.stats.distributions.discrete_gaussian_polynomial import DiscreteGaussianDistributionPolynomialSampler
sage: DiscreteGaussianDistributionPolynomialSampler(ZZ['x'], 8, 3.0)()
3*x^7 + 3*x^6 - 3*x^5 - x^4 - 5*x^2 + 3
sage: gs = DiscreteGaussianDistributionPolynomialSampler(ZZ['x'], 8, 3.0)
sage: [gs() for _ in range(3)]
[4*x^7 + 4*x^6 - 4*x^5 + 2*x^4 + x^3 - 4*x + 7, -5*x^6 + 4*x^5 - 3*x^3 + 4*x^2 + x, 2*x^7 + 2*x^6 + 2*x^5 - x^4 - 2*x^2 + 3*x + 1]

__call__()

Return a new sample.

EXAMPLES:

sage: from sage.stats.distributions.discrete_gaussian_polynomial import DiscreteGaussianDistributionPolynomialSampler
sage: sampler = DiscreteGaussianDistributionPolynomialSampler(ZZ['x'], 8, 12.0)
sage: sampler()
8*x^7 - 11*x^5 - 19*x^4 + 6*x^3 - 34*x^2 - 21*x + 9