LatticeTester::NormaLaminated Class Reference

This Normalizer class implements approximate upper bounds on the length of the shortest nonzero vector in a lattice. More...

Inheritance diagram for LatticeTester::NormaLaminated:

Public Member Functions
	NormaLaminated (double logm, int64_t k, int64_t maxDim, NormType norm=L2NORM)
	This constructor assumes that the rescaled primal lattice has scaling factor \(m\) and order \(k\), so its density is \(m^{k-t}\) for \(t\geq k\), and cannot exceed 1 for projections in \(s < k\) dimensions.
	NormaLaminated (double logDensity, int64_t maxDim, NormType norm=L2NORM)
	Constructs a NormaLaminated for up to maxDim dimensions, by assuming that the log density is logDensity in all dimensions and the lattice was not rescaled.
	NormaLaminated (int64_t maxDim, NormType norm=L2NORM)
	Constructs a NormaLaminated for up to maxDim dimensions, without computing the bounds.
double	getGamma (int64_t j) const
	Returns the value of the bound on the Hermite's constant \(\gamma_j\) in dimension \(j\).
Public Member Functions inherited from LatticeTester::Normalizer
	Normalizer (double logm, int64_t k, int64_t maxDim, NormType norm=L2NORM)
	This is the preferred constructor in all the subclasses.
	Normalizer (double logDensity, int64_t maxDim, NormType norm=L2NORM)
	This old (legacy) constructor assumes that the lattice is not rescaled and that the (log)density is the same in all dimensions, which is practically never true when rescaling.
	Normalizer (int64_t maxDim, NormType norm=L2NORM)
	This constructor creates a Normalizer object without computing any bounds.
virtual	~Normalizer ()
	Destructor.
void	computeBounds (double logm, int64_t k)
	This method computes the bounds that this normalizer will return, by assuming that the primal lattice was rescaled by the factor \(m\) and has order \(k\), so its density is \(m^{k-t}\) for \(t\geq k\), and cannot exceed \(m^s\) for projections in \(s < k\) dimensions.
virtual void	computeBounds (double logDensity)
	This method computes bounds by assuming that the log density is logDensity for all dimensions up to the maximal dimension maxDim, which is rarely true.
std::string	toString () const
	Returns a string that describes this object.
NormType	getNorm () const
	Returns the norm associated with this object.
int64_t	getMaxDim () const
	Returns the maximal dimension for this object.
virtual double *	getBounds () const
	Returns the array that contains the bounds on the lengths of the shortest nonzero vectors.
virtual double	getBound (int64_t j) const
	Returns the bound on the length of the shortest nonzero vector in j dimensions.

Additional Inherited Members
Static Public Attributes inherited from LatticeTester::Normalizer
static const int64_t	MAX_DIM = 48
	The maximum dimension of the lattices for which this class can compute a bound.
Protected Attributes inherited from LatticeTester::Normalizer
std::string	m_name
	Name of the normalizer.
NormType	m_norm
	Norm type associated with this object.
int64_t	m_maxDim
	Maximum dimension.
double *	m_bounds
	Contains the bounds on the length of the shortest nonzero vector in the lattice in each dimension up to maxDim.

Detailed Description

This Normalizer class implements approximate upper bounds on the length of the shortest nonzero vector in a lattice.

These are not strict upper bounds, because the Hermite constants \(\gamma_s\) here are approximated by lower bounds given by the values that correspond to the laminated lattices [2]. These lattices are not always the densest lattices available, but they are intuitive constructions dense lattices. In [2], Table 6.1 gives the determinant \(\lambda_s\) that can be used to recover the center density \(\delta_s = \lambda_s^{-1/2}\) of the densest laminated lattice in dimension \(s\). From there, we get

\[ \gamma_s = 4 \delta_s^{2/s}. \]

For the L1NORM, the bounds must be multiplied by s in s dimensions.

Constructor & Destructor Documentation

NormaLaminated() [1/3]

LatticeTester::NormaLaminated::NormaLaminated	(	double	logm,
		int64_t	k,
		int64_t	maxDim,
		NormType	norm = L2NORM )

This constructor assumes that the rescaled primal lattice has scaling factor \(m\) and order \(k\), so its density is \(m^{k-t}\) for \(t\geq k\), and cannot exceed 1 for projections in \(s < k\) dimensions.

NormaLaminated() [2/3]

LatticeTester::NormaLaminated::NormaLaminated	(	double	logDensity,
		int64_t	maxDim,
		NormType	norm = L2NORM )

Constructs a NormaLaminated for up to maxDim dimensions, by assuming that the log density is logDensity in all dimensions and the lattice was not rescaled.

Restriction: maxDim \( \le 48\).

NormaLaminated() [3/3]

LatticeTester::NormaLaminated::NormaLaminated	(	int64_t	maxDim,
		NormType	norm = L2NORM )

Constructs a NormaLaminated for up to maxDim dimensions, without computing the bounds.

Restriction: maxDim \( \le 48\).

Member Function Documentation

getGamma()

double LatticeTester::NormaLaminated::getGamma ( int64_t j ) const

inlinevirtual

Returns the value of the bound on the Hermite's constant \(\gamma_j\) in dimension \(j\).

Reimplemented from LatticeTester::Normalizer.

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The documentation for this class was generated from the following file:

• include/latticetester/NormaLaminated.h