def short_vector_list_up_to_length(self, len_bound, up_to_sign_flag=False):
"""
Return a list of lists of short vectors `v`, sorted by length, with
Q(`v`) < len_bound.
INPUT:
- ``len_bound`` -- bound for the length of the vectors.
- ``up_to_sign_flag`` -- (default: ``False``) if set to True, then
only one of the vectors of the pair `[v, -v]` is listed.
OUTPUT:
A list of lists of vectors such that entry `[i]` contains all
vectors of length `i`.
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,3,5,7])
sage: Q.short_vector_list_up_to_length(3)
[[(0, 0, 0, 0)], [(1, 0, 0, 0), (-1, 0, 0, 0)], []]
sage: Q.short_vector_list_up_to_length(4)
[[(0, 0, 0, 0)],
[(1, 0, 0, 0), (-1, 0, 0, 0)],
[],
[(0, 1, 0, 0), (0, -1, 0, 0)]]
sage: Q.short_vector_list_up_to_length(5)
[[(0, 0, 0, 0)],
[(1, 0, 0, 0), (-1, 0, 0, 0)],
[],
[(0, 1, 0, 0), (0, -1, 0, 0)],
[(1, 1, 0, 0),
(-1, -1, 0, 0),
(-1, 1, 0, 0),
(1, -1, 0, 0),
(2, 0, 0, 0),
(-2, 0, 0, 0)]]
sage: Q.short_vector_list_up_to_length(5, True)
[[(0, 0, 0, 0)],
[(1, 0, 0, 0)],
[],
[(0, 1, 0, 0)],
[(1, 1, 0, 0), (-1, 1, 0, 0), (2, 0, 0, 0)]]
sage: Q = QuadraticForm(matrix(6, [2, 1, 1, 1, -1, -1, 1, 2, 1, 1, -1, -1, 1, 1, 2, 0, -1, -1, 1, 1, 0, 2, 0, -1, -1, -1, -1, 0, 2, 1, -1, -1, -1, -1, 1, 2]))
sage: vs = Q.short_vector_list_up_to_length(8)
sage: [len(vs[i]) for i in range(len(vs))]
[1, 72, 270, 720, 936, 2160, 2214, 3600]
sage: vs = Q.short_vector_list_up_to_length(30) # long time (28s on sage.math, 2014)
sage: [len(vs[i]) for i in range(len(vs))] # long time
[1, 72, 270, 720, 936, 2160, 2214, 3600, 4590, 6552, 5184, 10800, 9360, 12240, 13500, 17712, 14760, 25920, 19710, 26064, 28080, 36000, 25920, 47520, 37638, 43272, 45900, 59040, 46800, 75600]
The cases of ``len_bound < 2`` led to exception or infinite runtime before.
::
sage: Q.short_vector_list_up_to_length(-1)
[]
sage: Q.short_vector_list_up_to_length(0)
[]
sage: Q.short_vector_list_up_to_length(1)
[[(0, 0, 0, 0, 0, 0)]]
In the case of quadratic forms that are not positive definite an error is raised.
::
sage: QuadraticForm(matrix(2, [2, 0, 0, -2])).short_vector_list_up_to_length(3)
Traceback (most recent call last):
...
ValueError: Quadratic form must be positive definite in order to enumerate short vectors
Sometimes, PARI does not compute short vectors correctly. It returns too long vectors.
::
sage: Q = QuadraticForm(matrix(2, [72, 12, 12, 120]))
sage: len_bound_pari = 2*22953421 - 2; len_bound_pari
45906840
sage: vs = list(Q._pari_().qfminim(len_bound_pari)[2]) # long time (18s on sage.math, 2014)
sage: v = vs[0]; v # long time
[-65, 623]~
sage: v.Vec() * Q._pari_() * v # long time
45907800
"""
if not self.is_positive_definite():
raise ValueError("Quadratic form must be positive definite in order to enumerate short vectors")
if len_bound <= 0:
return []
# Free module in which the vectors live
V = FreeModule(ZZ, self.dim())
# Adjust length for PARI. We need to subtract 1 because PARI returns
# returns vectors of length less than or equal to b, but we want
# strictly less. We need to double because the matrix is doubled.
len_bound_pari = 2 * (len_bound - 1)
# Call PARI's qfminim()
parilist = self._pari_().qfminim(len_bound_pari)[2].Vec()
# List of lengths
parilens = pari(r"(M,v) -> vector(#v, i, (v[i]~ * M * v[i])\2)")(self, parilist)
# Sort the vectors into lists by their length
vec_sorted_list = [list() for i in range(len_bound)]
for i in range(len(parilist)):
length = ZZ(parilens[i])
# PARI can sometimes return longer vectors than requested.
# E.g. : self.matrix() == matrix(2, [72, 12, 12, 120])
# len_bound = 22953421
# gives maximal length 22955664
if length < len_bound:
v = parilist[i]
sagevec = V(list(parilist[i]))
vec_sorted_list[length].append(sagevec)
if not up_to_sign_flag:
vec_sorted_list[length].append(-sagevec)
# Add the zero vector by hand
vec_sorted_list[0].append(V.zero_vector())
return vec_sorted_list
def short_vector_list_up_to_length(self, len_bound, up_to_sign_flag=False):
"""
Return a list of lists of short vectors `v`, sorted by length, with
Q(`v`) < len_bound.
INPUT:
- ``len_bound`` -- bound for the length of the vectors.
- ``up_to_sign_flag`` -- (default: ``False``) if set to True, then
only one of the vectors of the pair `[v, -v]` is listed.
OUTPUT:
A list of lists of vectors such that entry `[i]` contains all
vectors of length `i`.
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,3,5,7])
sage: Q.short_vector_list_up_to_length(3)
[[(0, 0, 0, 0)], [(1, 0, 0, 0), (-1, 0, 0, 0)], []]
sage: Q.short_vector_list_up_to_length(4)
[[(0, 0, 0, 0)],
[(1, 0, 0, 0), (-1, 0, 0, 0)],
[],
[(0, 1, 0, 0), (0, -1, 0, 0)]]
sage: Q.short_vector_list_up_to_length(5)
[[(0, 0, 0, 0)],
[(1, 0, 0, 0), (-1, 0, 0, 0)],
[],
[(0, 1, 0, 0), (0, -1, 0, 0)],
[(1, 1, 0, 0),
(-1, -1, 0, 0),
(-1, 1, 0, 0),
(1, -1, 0, 0),
(2, 0, 0, 0),
(-2, 0, 0, 0)]]
sage: Q.short_vector_list_up_to_length(5, True)
[[(0, 0, 0, 0)],
[(1, 0, 0, 0)],
[],
[(0, 1, 0, 0)],
[(1, 1, 0, 0), (-1, 1, 0, 0), (2, 0, 0, 0)]]
sage: Q = QuadraticForm(matrix(6, [2, 1, 1, 1, -1, -1, 1, 2, 1, 1, -1, -1, 1, 1, 2, 0, -1, -1, 1, 1, 0, 2, 0, -1, -1, -1, -1, 0, 2, 1, -1, -1, -1, -1, 1, 2]))
sage: vs = Q.short_vector_list_up_to_length(8)
sage: [len(vs[i]) for i in range(len(vs))]
[1, 72, 270, 720, 936, 2160, 2214, 3600]
sage: vs = Q.short_vector_list_up_to_length(30) # long time (28s on sage.math, 2014)
sage: [len(vs[i]) for i in range(len(vs))] # long time
[1, 72, 270, 720, 936, 2160, 2214, 3600, 4590, 6552, 5184, 10800, 9360, 12240, 13500, 17712, 14760, 25920, 19710, 26064, 28080, 36000, 25920, 47520, 37638, 43272, 45900, 59040, 46800, 75600]
The cases of ``len_bound < 2`` led to exception or infinite runtime before.
::
sage: Q.short_vector_list_up_to_length(-1)
[]
sage: Q.short_vector_list_up_to_length(0)
[]
sage: Q.short_vector_list_up_to_length(1)
[[(0, 0, 0, 0, 0, 0)]]
In the case of quadratic forms that are not positive definite an error is raised.
::
sage: QuadraticForm(matrix(2, [2, 0, 0, -2])).short_vector_list_up_to_length(3)
Traceback (most recent call last):
...
ValueError: Quadratic form must be positive definite in order to enumerate short vectors
Sometimes, PARI does not compute short vectors correctly. It returns too long vectors.
::
sage: Q = QuadraticForm(matrix(2, [72, 12, 12, 120]))
sage: len_bound_pari = 2*22953421 - 2; len_bound_pari
45906840
sage: vs = list(Q._pari_().qfminim(len_bound_pari)[2]) # long time (18s on sage.math, 2014)
sage: v = vs[0]; v # long time
[-65, 623]~
sage: v.Vec() * Q._pari_() * v # long time
45907800
"""
if not self.is_positive_definite() :
raise ValueError( "Quadratic form must be positive definite in order to enumerate short vectors" )
if len_bound <= 0:
return []
# Free module in which the vectors live
V = FreeModule(ZZ, self.dim())
# Adjust length for PARI. We need to subtract 1 because PARI returns
# returns vectors of length less than or equal to b, but we want
# strictly less. We need to double because the matrix is doubled.
len_bound_pari = 2*(len_bound - 1)
# Call PARI's qfminim()
parilist = self._pari_().qfminim(len_bound_pari)[2].Vec()
# List of lengths
parilens = pari(r"(M,v) -> vector(#v, i, (v[i]~ * M * v[i])\2)")(self, parilist)
# Sort the vectors into lists by their length
vec_sorted_list = [list() for i in range(len_bound)]
for i in range(len(parilist)):
length = ZZ(parilens[i])
# PARI can sometimes return longer vectors than requested.
# E.g. : self.matrix() == matrix(2, [72, 12, 12, 120])
# len_bound = 22953421
# gives maximal length 22955664
if length < len_bound:
v = parilist[i]
sagevec = V(list(parilist[i]))
vec_sorted_list[length].append(sagevec)
if not up_to_sign_flag :
vec_sorted_list[length].append(-sagevec)
# Add the zero vector by hand
vec_sorted_list[0].append(V.zero_vector())
return vec_sorted_list
