def basis_of_short_vectors(self, show_lengths=False, safe_flag=True):
"""
Return a basis for `ZZ^n` made of vectors with minimal lengths Q(`v`).
The safe_flag allows us to select whether we want a copy of the
output, or the original output. By default safe_flag = True, so
we return a copy of the cached information. If this is set to
False, then the routine is much faster but the return values are
vulnerable to being corrupted by the user.
OUTPUT:
a list of vectors, and optionally a list of values for each vector.
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,3,5,7])
sage: Q.basis_of_short_vectors()
[(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)]
sage: Q.basis_of_short_vectors(True)
([(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)], [1, 3, 5, 7])
"""
## Try to use the cached results
try:
## Return the appropriate result
if show_lengths:
if safe_flag:
return deep_copy(self.__basis_of_short_vectors), deepcopy(self.__basis_of_short_vectors_lengths)
else:
return self.__basis_of_short_vectors, self.__basis_of_short_vectors_lengths
else:
if safe_flag:
return deepcopy(self.__basis_of_short_vectors)
else:
return deepcopy(self.__basis_of_short_vectors)
except Exception:
pass
## Set an upper bound for the number of vectors to consider
Max_number_of_vectors = 10000
## Generate a PARI matrix string for the associated Hessian matrix
M_str = str(gp(self.matrix()))
## Run through all possible minimal lengths to find a spanning set of vectors
n = self.dim()
# MS = MatrixSpace(QQ, n)
M1 = Matrix([[0]])
vec_len = 0
while M1.rank() < n:
## DIAGONSTIC
# print
# print "Starting with vec_len = ", vec_len
# print "M_str = ", M_str
vec_len += 1
gp_mat = gp.qfminim(M_str, vec_len, Max_number_of_vectors)[3].mattranspose()
number_of_vecs = ZZ(gp_mat.matsize()[1])
vector_list = []
for i in range(number_of_vecs):
# print "List at", i, ":", list(gp_mat[i+1,])
new_vec = vector([ZZ(x) for x in list(gp_mat[i + 1,])])
vector_list.append(new_vec)
## DIAGNOSTIC
# print "number_of_vecs = ", number_of_vecs
# print "vector_list = ", vector_list
## Make a matrix from the short vectors
if len(vector_list) > 0:
M1 = Matrix(vector_list)
## DIAGNOSTIC
# print "matrix of vectors = \n", M1
# print "rank of the matrix = ", M1.rank()
# print " vec_len = ", vec_len
# print M1
## Organize these vectors by length (and also introduce their negatives)
max_len = vec_len / 2
vector_list_by_length = [[] for _ in range(max_len + 1)]
for v in vector_list:
l = self(v)
vector_list_by_length[l].append(v)
vector_list_by_length[l].append(vector([-x for x in v]))
## Make a matrix from the column vectors (in order of ascending length).
sorted_list = []
for i in range(len(vector_list_by_length)):
for v in vector_list_by_length[i]:
sorted_list.append(v)
sorted_matrix = Matrix(sorted_list).transpose()
## Determine a basis of vectors of minimal length
pivots = sorted_matrix.pivots()
basis = [sorted_matrix.column(i) for i in pivots]
pivot_lengths = [self(v) for v in basis]
## DIAGNOSTIC
# print "basis = ", basis
# print "pivot_lengths = ", pivot_lengths
## Cache the result
self.__basis_of_short_vectors = basis
self.__basis_of_short_vectors_lenghts = pivot_lengths
## Return the appropriate result
if show_lengths:
return basis, pivot_lengths
else:
return basis
def basis_of_short_vectors(self, show_lengths=False, safe_flag=True):
"""
Return a basis for `ZZ^n` made of vectors with minimal lengths Q(`v`).
The safe_flag allows us to select whether we want a copy of the
output, or the original output. By default safe_flag = True, so
we return a copy of the cached information. If this is set to
False, then the routine is much faster but the return values are
vulnerable to being corrupted by the user.
OUTPUT:
a list of vectors, and optionally a list of values for each vector.
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,3,5,7])
sage: Q.basis_of_short_vectors()
[(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)]
sage: Q.basis_of_short_vectors(True)
([(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)], [1, 3, 5, 7])
"""
## Try to use the cached results
try:
## Return the appropriate result
if show_lengths:
if safe_flag:
return deep_copy(self.__basis_of_short_vectors), deepcopy(self.__basis_of_short_vectors_lengths)
else:
return self.__basis_of_short_vectors, self.__basis_of_short_vectors_lengths
else:
if safe_flag:
return deepcopy(self.__basis_of_short_vectors)
else:
return deepcopy(self.__basis_of_short_vectors)
except Exception:
pass
## Set an upper bound for the number of vectors to consider
Max_number_of_vectors = 10000
## Generate a PARI matrix string for the associated Hessian matrix
M_str = str(gp(self.matrix()))
## Run through all possible minimal lengths to find a spanning set of vectors
n = self.dim()
#MS = MatrixSpace(QQ, n)
M1 = Matrix([[0]])
vec_len = 0
while M1.rank() < n:
## DIAGONSTIC
#print
#print "Starting with vec_len = ", vec_len
#print "M_str = ", M_str
vec_len += 1
gp_mat = gp.qfminim(M_str, vec_len, Max_number_of_vectors)[3].mattranspose()
number_of_vecs = ZZ(gp_mat.matsize()[1])
vector_list = []
for i in range(number_of_vecs):
#print "List at", i, ":", list(gp_mat[i+1,])
new_vec = vector([ZZ(x) for x in list(gp_mat[i+1,])])
vector_list.append(new_vec)
## DIAGNOSTIC
#print "number_of_vecs = ", number_of_vecs
#print "vector_list = ", vector_list
## Make a matrix from the short vectors
if len(vector_list) > 0:
M1 = Matrix(vector_list)
## DIAGNOSTIC
#print "matrix of vectors = \n", M1
#print "rank of the matrix = ", M1.rank()
#print " vec_len = ", vec_len
#print M1
## Organize these vectors by length (and also introduce their negatives)
max_len = vec_len // 2
vector_list_by_length = [[] for _ in range(max_len + 1)]
for v in vector_list:
l = self(v)
vector_list_by_length[l].append(v)
vector_list_by_length[l].append(vector([-x for x in v]))
## Make a matrix from the column vectors (in order of ascending length).
sorted_list = []
for i in range(len(vector_list_by_length)):
for v in vector_list_by_length[i]:
sorted_list.append(v)
sorted_matrix = Matrix(sorted_list).transpose()
## Determine a basis of vectors of minimal length
pivots = sorted_matrix.pivots()
basis = [sorted_matrix.column(i) for i in pivots]
pivot_lengths = [self(v) for v in basis]
## DIAGNOSTIC
#print "basis = ", basis
#print "pivot_lengths = ", pivot_lengths
## Cache the result
self.__basis_of_short_vectors = basis
self.__basis_of_short_vectors_lenghts = pivot_lengths
## Return the appropriate result
if show_lengths:
return basis, pivot_lengths
else:
return basis
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. The list in output `[i]` indexes all vectors of
length `i`. If the up_to_sign_flag is set to True, then only one of
the vectors of the pair `[v, -v]` is listed.
Note: This processes the PARI/GP output to always give elements of type `ZZ`.
OUTPUT:
a list of lists of vectors.
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(40) #long time
The cases of ``len_bound < 2`` led to exception or infinite runtime before.
::
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: mat = matrix(2, [72, 12, 12, 120]) #long time
sage: len_bound = 22953421 #long time
sage: gp_mat = gp.qfminim(str(gp(mat)), 2 * len_bound - 2)[3] #long time
sage: rows = [ map(ZZ, str(gp_mat[i,])[1:-1].split(',')) for i in range(1, gp_mat.matsize()[1] + 1) ] #long time
sage: vec_list = map(vector, zip(*rows)) #long time
sage: eval_v_cython = cython_lambda( ", ".join( "int a{0}".format(i) for i in range(2) ), " + ".join( "{coeff} * a{i} * a{j}".format(coeff = mat[i,j], i = i, j = j) for i in range(2) for j in range(2) ) ) #long time
sage: any( eval_v_cython(*v) == 2 * 22955664 for v in vec_list ) # 22955664 > 22953421 = len_bound #long time
True
"""
if not self.is_positive_definite():
raise ValueError(
"Quadratic form must be positive definite in order to enumerate short vectors"
)
## Generate a PARI matrix string for the associated Hessian matrix
M_str = str(gp(self.matrix()))
if len_bound <= 0:
return list()
elif len_bound == 1:
return [[(vector([ZZ(0) for _ in range(self.dim())]))]]
## Generate the short vectors
gp_mat = gp.qfminim(M_str, 2 * len_bound - 2)[3]
## We read all n-th entries at once so that not too many sage[...] variables are
## used. This is important when to many vectors are returned.
rows = [
map(ZZ,
str(gp_mat[i, ])[1:-1].split(','))
for i in range(1,
gp_mat.matsize()[1] + 1)
]
vec_list = map(vector, zip(*rows))
if len(vec_list) > 500:
eval_v_cython = cython_lambda(
", ".join("int a{0}".format(i) for i in range(self.dim())),
" + ".join(
"{coeff} * a{i} * a{j}".format(coeff=self[i, j], i=i, j=j)
for i in range(self.dim()) for j in range(i, self.dim())))
eval_v = lambda v: eval_v_cython(*v)
else:
eval_v = self
## Sort the vectors into lists by their length
vec_sorted_list = [list() for i in range(len_bound)]
for v in vec_list:
v_evaluated = eval_v(v)
try:
vec_sorted_list[v_evaluated].append(v)
if not up_to_sign_flag:
vec_sorted_list[v_evaluated].append(-v)
except IndexError:
## We deal with a Pari but, that returns longer vectors that requested.
## E.g. : self.matrix() == matrix(2, [72, 12, 12, 120])
## len_bound = 22953421
## gives maximal length 22955664
pass
## Add the zero vector by hand
zero_vec = vector([ZZ(0) for _ in range(self.dim())])
vec_sorted_list[0].append(zero_vec)
## Return the sorted list
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. The list in output `[i]` indexes all vectors of
length `i`. If the up_to_sign_flag is set to True, then only one of
the vectors of the pair `[v, -v]` is listed.
Note: This processes the PARI/GP output to always give elements of type `ZZ`.
OUTPUT:
a list of lists of vectors.
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(40) #long time
The cases of ``len_bound < 2`` led to exception or infinite runtime before.
::
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: mat = matrix(2, [72, 12, 12, 120]) #long time
sage: len_bound = 22953421 #long time
sage: gp_mat = gp.qfminim(str(gp(mat)), 2 * len_bound - 2)[3] #long time
sage: rows = [ map(ZZ, str(gp_mat[i,])[1:-1].split(',')) for i in range(1, gp_mat.matsize()[1] + 1) ] #long time
sage: vec_list = map(vector, zip(*rows)) #long time
sage: eval_v_cython = cython_lambda( ", ".join( "int a{0}".format(i) for i in range(2) ), " + ".join( "{coeff} * a{i} * a{j}".format(coeff = mat[i,j], i = i, j = j) for i in range(2) for j in range(2) ) ) #long time
sage: any( eval_v_cython(*v) == 2 * 22955664 for v in vec_list ) # 22955664 > 22953421 = len_bound #long time
True
"""
if not self.is_positive_definite() :
raise ValueError( "Quadratic form must be positive definite in order to enumerate short vectors" )
## Generate a PARI matrix string for the associated Hessian matrix
M_str = str(gp(self.matrix()))
if len_bound <= 0 :
return list()
elif len_bound == 1 :
return [ [(vector([ZZ(0) for _ in range(self.dim())]))] ]
## Generate the short vectors
gp_mat = gp.qfminim(M_str, 2*len_bound - 2)[3]
## We read all n-th entries at once so that not too many sage[...] variables are
## used. This is important when to many vectors are returned.
rows = [ map(ZZ, str(gp_mat[i,])[1:-1].split(','))
for i in range(1, gp_mat.matsize()[1] + 1) ]
vec_list = map(vector, zip(*rows))
if len(vec_list) > 500 :
eval_v_cython = cython_lambda( ", ".join( "int a{0}".format(i) for i in range(self.dim()) ),
" + ".join( "{coeff} * a{i} * a{j}".format(coeff = self[i,j], i = i, j = j)
for i in range(self.dim()) for j in range(i, self.dim()) ) )
eval_v = lambda v: eval_v_cython(*v)
else :
eval_v = self
## Sort the vectors into lists by their length
vec_sorted_list = [list() for i in range(len_bound)]
for v in vec_list:
v_evaluated = eval_v(v)
try :
vec_sorted_list[v_evaluated].append(v)
if not up_to_sign_flag :
vec_sorted_list[v_evaluated].append(-v)
except IndexError :
## We deal with a Pari but, that returns longer vectors that requested.
## E.g. : self.matrix() == matrix(2, [72, 12, 12, 120])
## len_bound = 22953421
## gives maximal length 22955664
pass
## Add the zero vector by hand
zero_vec = vector([ZZ(0) for _ in range(self.dim())])
vec_sorted_list[0].append(zero_vec)
## Return the sorted list
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. The list in output `[i]` indexes all vectors of
length `i`. If the up_to_sign_flag is set to True, then only one of
the vectors of the pair `[v, -v]` is listed.
Note: This processes the PARI/GP output to always give elements of type `ZZ`.
OUTPUT:
a list of lists of vectors.
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)]]
"""
## Set an upper bound for the number of vectors to consider
Max_number_of_vectors = 10000
## Generate a PARI matrix string for the associated Hessian matrix
M_str = str(gp(self.matrix()))
## Generate the short vectors
gp_mat = gp.qfminim(M_str, 2 * len_bound - 2,
Max_number_of_vectors)[3].mattranspose()
number_of_rows = gp_mat.matsize()[1]
gp_mat_vector_list = [
vector([ZZ(x) for x in gp_mat[i + 1, ]]) for i in range(number_of_rows)
]
## Sort the vectors into lists by their length
vec_list = [[] for i in range(len_bound)]
for v in gp_mat_vector_list:
vec_list[self(v)].append(v)
if up_to_sign_flag == False:
vec_list[self(v)].append([-x for x in v])
## Add the zero vector by hand
zero_vec = vector([ZZ(0) for _ in range(self.dim())])
vec_list[0].append(zero_vec)
## Return the sorted list
return vec_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. The list in output `[i]` indexes all vectors of
length `i`. If the up_to_sign_flag is set to True, then only one of
the vectors of the pair `[v, -v]` is listed.
Note: This processes the PARI/GP output to always give elements of type `ZZ`.
OUTPUT:
a list of lists of vectors.
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)]]
"""
## Set an upper bound for the number of vectors to consider
Max_number_of_vectors = 10000
## Generate a PARI matrix string for the associated Hessian matrix
M_str = str(gp(self.matrix()))
## Generate the short vectors
gp_mat = gp.qfminim(M_str, 2*len_bound-2, Max_number_of_vectors)[3].mattranspose()
number_of_rows = gp_mat.matsize()[1]
gp_mat_vector_list = [vector([ZZ(x) for x in gp_mat[i+1,]]) for i in range(number_of_rows)]
## Sort the vectors into lists by their length
vec_list = [[] for i in range(len_bound)]
for v in gp_mat_vector_list:
vec_list[self(v)].append(v)
if up_to_sign_flag == False:
vec_list[self(v)].append([-x for x in v])
## Add the zero vector by hand
zero_vec = vector([ZZ(0) for _ in range(self.dim())])
vec_list[0].append(zero_vec)
## Return the sorted list
return vec_list
