def __init__(self, projection_direction, height = 1.1):
"""
Initializes the projection.
EXAMPLES::
sage: from sage.geometry.polyhedron.plot import ProjectionFuncSchlegel
sage: proj = ProjectionFuncSchlegel([2,2,2])
sage: proj.__init__([2,2,2])
sage: proj(vector([1.1,1.1,1.11]))[0]
0.0302...
sage: TestSuite(proj).run(skip='_test_pickling')
"""
self.projection_dir = vector(RDF, projection_direction)
if norm(self.projection_dir).is_zero():
raise ValueError, "projection direction must be a non-zero vector."
self.dim = self.projection_dir.degree()
spcenter = height * self.projection_dir/norm(self.projection_dir)
self.height = height
v = vector(RDF, [0.0]*(self.dim-1) + [self.height]) - spcenter
polediff = matrix(RDF,v).transpose()
denom = (polediff.transpose()*polediff)[0][0]
if denom.is_zero():
self.house = identity_matrix(RDF,self.dim)
else:
self.house = identity_matrix(RDF,self.dim) \
- 2*polediff*polediff.transpose()/denom #Householder reflector
def mutate(self, k, mutating_F=True):
r"""
mutate seed
bla bla ba
"""
n = self.parent().rk
if k not in xrange(n):
raise ValueError('Cannot mutate in direction ' + str(k) + '.')
# store mutation path
if self._path != [] and self._path[-1] == k:
self._path.pop()
else:
self._path.append(k)
# find sign of k-th c-vector
# Will this be used enough that it should be a built-in function?
if any(x > 0 for x in self._C.column(k)):
eps = +1
else:
eps = -1
# store the g-vector to be mutated in case we are mutating F-polynomials also
old_g_vector = self.g_vector(k)
# G-matrix
J = identity_matrix(n)
for j in xrange(n):
J[j,k] += max(0, -eps*self._B[j,k])
J[k,k] = -1
self._G = self._G*J
# g-vector path list
g_vector = self.g_vector(k)
if g_vector not in self.parent().g_vectors_so_far():
self.parent()._path_dict[g_vector] = copy(self._path)
# F-polynomials
if mutating_F:
self.parent()._F_poly_dict[g_vector] = self._mutated_F(k, old_g_vector)
# C-matrix
J = identity_matrix(n)
for j in xrange(n):
J[k,j] += max(0, eps*self._B[k,j])
J[k,k] = -1
self._C = self._C*J
# B-matrix
self._B.mutate(k)
# exchange relation
if self.parent()._store_exchange_relations:
ex_pair = frozenset([g_vector,old_g_vector])
if ex_pair not in self.parent()._exchange_relations:
coefficient = self.coefficient(k).lift()
variables = zip(self.g_vectors(), self.b_matrix().column(k))
Mp = [ (g,p) for (g,p) in variables if p > 0 ]
Mm = [ (g,-p) for (g,p) in variables if p < 0 ]
self.parent()._exchange_relations[ex_pair] = ( (Mp,coefficient.numerator()), (Mm,coefficient.denominator()) )
def __init__(self, projection_point):
"""
Create a stereographic projection function.
INPUT:
- ``projection_point`` -- a list of coordinates in the
appropriate dimension, which is the point projected from.
EXAMPLES::
sage: from sage.geometry.polyhedron.plot import ProjectionFuncStereographic
sage: proj = ProjectionFuncStereographic([1.0,1.0])
sage: proj.__init__([1.0,1.0])
sage: proj.house
[-0.7071067811... 0.7071067811...]
[ 0.7071067811... 0.7071067811...]
sage: TestSuite(proj).run(skip='_test_pickling')
"""
self.projection_point = vector(projection_point)
self.dim = self.projection_point.degree()
pproj = vector(RDF,self.projection_point)
self.psize = norm(pproj)
if (self.psize).is_zero():
raise ValueError, "projection direction must be a non-zero vector."
v = vector(RDF, [0.0]*(self.dim-1) + [self.psize]) - pproj
polediff = matrix(RDF,v).transpose()
denom = RDF((polediff.transpose()*polediff)[0][0])
if denom.is_zero():
self.house = identity_matrix(RDF,self.dim)
else:
self.house = identity_matrix(RDF,self.dim) \
- 2*polediff*polediff.transpose()/denom # Householder reflector
def mutate(self, sequence, inplace=True, mutate_F=True):
if not isinstance(mutate_F, bool):
raise ValueError('mutate_F must be boolean.')
if not isinstance(inplace, bool):
raise ValueError('inplace must be boolean.')
if inplace:
seed = self.current_seed
else:
seed = copy(self.current_seed)
if sequence in xrange(seed._n):
seq = iter([sequence])
else:
seq = iter(sequence)
for k in seq:
if k not in xrange(seed._n):
raise ValueError('Cannot mutate in direction' + str(k) + '.')
# G-matrix
J = identity_matrix(seed._n)
if any(x > 0 for x in seed._C.column(k)):
eps = +1
else:
eps = -1
for j in xrange(seed._n):
J[j,k] += max(0, -eps*seed._B[j,k])
J[k,k] = -1
seed._G = seed._G*J
# F-polynomials
if mutating_F:
gvect = tuple(seed._G.column(k))
if not self._F_dict.has_key(gvect):
self._F_dict.setdefault(gvect,self._mutated_F(k))
seed._F[k] = self._F_dict[gvect]
# C-matrix
J = identity_matrix(seed._n)
if any(x > 0 for x in seed._C.column(k)):
eps = +1
else:
eps = -1
for j in xrange(seed._n):
J[k,j] += max(0, eps*seed._B[k,j])
J[k,k] = -1
seed._C = seed._C*J
# B-matrix
seed._B.mutate(k)
if not inplace:
return seed
def _extend1(self):
"Increases the matrix size by 1"
#save current self
self.iv0 = copy(self)
self.iv0.abel_raw0 = lambda z: self.iv0.abel0poly(z-self.iv0.x0)
N = self.N
A = self.A
#Carleman matrix without 0-th row:
Ct = A + identity_matrix(self.R,N).submatrix(1,0,N-1,N-1)
AI = self.AI
#assert AI*self.A == identity_matrix(N-1)
if isinstance(self.f,FormalPowerSeries):
coeffs = [ self.f[n] for n in xrange(0,N) ]
else:
x = self.f.args()[0]
coeffs = taylor(self.f.substitute({x:x+self.x0sym})-self.x0sym,x,0,N).polynomial(self.R)
self.Ct = matrix(self.R,N,N)
self.Ct.set_block(0,0,Ct)
self.Ct[0,N-1] = coeffs[N-1]
for m in range(1,N-1):
self.Ct[m,N-1] = psmul_at(self.Ct[0],self.Ct[m-1],N-1)
self.Ct[N-1] = psmul(self.Ct[0],self.Ct[N-2])
print "C extended"
self.A = self.Ct - identity_matrix(self.R,N+1).submatrix(1,0,N,N)
av=self.A.column(N-1)[:N-1]
ah=self.A.row(N-1)[:N-1]
a_n=self.A[N-1,N-1]
# print 'A:';print A
# print 'A\''; print self.A
# print 'ah:',ah
# print 'av:',av
# print 'a_n:',a_n
AI0 = matrix(self.R,N,N)
AI0.set_block(0,0,self.AI)
horiz = matrix(self.R,1,N)
horiz.set_block(0,0,(ah*AI).transpose().transpose())
horiz[0,N-1] = -1
vert = matrix(self.R,N,1)
vert.set_block(0,0,(AI*av).transpose())
vert[N-1,0] = -1
self.N += 1
self.AI = AI0 + vert*horiz/(a_n-ah*AI*av)
def burau_matrix(self, var='t'):
"""
Return the Burau matrix of the braid.
INPUT:
- ``var`` -- string (default: ``'t'``). The name of the
variable in the entries of the matrix.
OUTPUT:
The Burau matrix of the braid. It is a matrix whose entries
are Laurent polynomials in the variable ``var``.
EXAMPLES::
sage: B = BraidGroup(4)
sage: B.inject_variables()
Defining s0, s1, s2
sage: b=s0*s1/s2/s1
sage: b.burau_matrix()
[ -t + 1 0 -t^2 + t t^2]
[ 1 0 0 0]
[ 0 0 1 0]
[ 0 t^-2 t^-1 - t^-2 1 - t^-1]
sage: s2.burau_matrix('x')
[ 1 0 0 0]
[ 0 1 0 0]
[ 0 0 -x + 1 x]
[ 0 0 1 0]
REFERENCES:
http://en.wikipedia.org/wiki/Burau_representation
"""
R = LaurentPolynomialRing(IntegerRing(), var)
t = R.gen()
M = identity_matrix(R, self.strands())
for i in self.Tietze():
A = identity_matrix(R, self.strands())
if i>0:
A[i-1, i-1] = 1-t
A[i, i] = 0
A[i, i-1] = 1
A[i-1, i] = t
if i<0:
A[-1-i, -1-i] = 0
A[-i, -i] = 1-t**(-1)
A[-1-i, -i] = 1
A[-i, -1-i] = t**(-1)
M=M*A
return M
def mutate_initial(self, k):
r"""
Mutate ``self`` in direction `k` at the initial cluster.
INPUT:
- ``k`` -- integer in between 0 and ``self.rk``
"""
n = self.rk
if k not in xrange(n):
raise ValueError('Cannot mutate in direction %s, please try a value between 0 and %s.'%(str(k),str(n-1)))
#modify self._path_dict using Nakanishi-Zelevinsky (4.1) and self._F_poly_dict using CA-IV (6.21)
new_path_dict = dict()
new_F_dict = dict()
new_path_dict[tuple(identity_matrix(n).column(k))] = []
new_F_dict[tuple(identity_matrix(n).column(k))] = self._U(1)
poly_ring = PolynomialRing(ZZ,'u')
h_subs_tuple = tuple([poly_ring.gen(0)**(-1) if j==k else poly_ring.gen(0)**max(-self._B0[k][j],0) for j in xrange(n)])
F_subs_tuple = tuple([self._U.gen(k)**(-1) if j==k else self._U.gen(j)*self._U.gen(k)**max(-self._B0[k][j],0)*(1+self._U.gen(k))**(self._B0[k][j]) for j in xrange(n)])
for g_vect in self._path_dict:
#compute new path
path = self._path_dict[g_vect]
if g_vect == tuple(identity_matrix(n).column(k)):
new_path = [k]
elif path != []:
if path[0] != k:
new_path = [k] + path
else:
new_path = path[1:]
else:
new_path = []
#compute new g-vector
new_g_vect = vector(g_vect) - 2*g_vect[k]*identity_matrix(n).column(k)
for i in xrange(n):
new_g_vect += max(sign(g_vect[k])*self._B0[i,k],0)*g_vect[k]*identity_matrix(n).column(i)
new_path_dict[tuple(new_g_vect)] = new_path
#compute new F-polynomial
h = 0
trop = tropical_evaluation(self._F_poly_dict[g_vect](h_subs_tuple))
if trop != 1:
h = trop.denominator().exponents()[0]-trop.numerator().exponents()[0]
new_F_dict[tuple(new_g_vect)] = self._F_poly_dict[g_vect](F_subs_tuple)*self._U.gen(k)**h*(self._U.gen(k)+1)**g_vect[k]
self._path_dict = new_path_dict
self._F_poly_dict = new_F_dict
self._B0.mutate(k)
def initial_seed(self):
r"""
Return the initial seed
"""
n = self.rk
I = identity_matrix(n)
return ClusterAlgebraSeed(self._B0, I, I, self)
def invariant_form(self):
"""
Return the hermitian form preserved by the unitary
group.
OUTPUT:
A square matrix describing the bilinear form
EXAMPLES::
sage: SU4 = SU(4,QQ)
sage: SU4.invariant_form()
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
"""
if self._invariant_form is not None:
return self._invariant_form
from sage.matrix.constructor import identity_matrix
m = identity_matrix(self.base_ring(), self.degree())
m.set_immutable()
return m
def initial_pair(self):
"""
Return an initial double description pair.
Picks an initial set of rays by selecting a basis. This is
probably the most efficient way to select the initial set.
INPUT:
- ``pair_class`` -- subclass of
:class:`DoubleDescriptionPair`.
OUTPUT:
A pair consisting of a :class:`DoubleDescriptionPair` instance
and the tuple of remaining unused inequalities.
EXAMPLES::
sage: A = matrix([(-1, 1), (-1, 2), (1/2, -1/2), (1/2, 2)])
sage: from sage.geometry.polyhedron.double_description import Problem
sage: DD, remaining = Problem(A).initial_pair()
sage: DD.verify()
sage: remaining
[(1/2, -1/2), (1/2, 2)]
"""
pivot_rows = self.A_matrix().pivot_rows()
A0 = [self.A()[pivot] for pivot in pivot_rows]
Ac = [self.A()[i] for i in range(len(self.A())) if i not in pivot_rows]
from sage.matrix.constructor import identity_matrix, matrix
I = identity_matrix(self.base_ring(), self.dim())
R = matrix(self.base_ring(), A0).solve_right(I)
return self.pair_class(self, A0, R.columns()), list(Ac)
def initial_seed(self):
r"""
Return the initial seed
"""
n = self.rk
I = identity_matrix(n)
return ClusterAlgebraSeed(self._B0, I, I, self)
def ARP(self):
from sage.matrix.constructor import identity_matrix
A1 = matrix(3, [1,1,1, 0,1,0, 0,0,1])
A2 = matrix(3, [1,0,0, 1,1,1, 0,0,1])
A3 = matrix(3, [1,0,0, 0,1,0, 1,1,1])
P12 = matrix(3, [1,0,1, 1,1,1, 0,0,1])
P13 = matrix(3, [1,1,0, 0,1,0, 1,1,1])
P23 = matrix(3, [1,0,0, 1,1,0, 1,1,1])
P21 = matrix(3, [1,1,1, 0,1,1, 0,0,1])
P31 = matrix(3, [1,1,1, 0,1,0, 0,1,1])
P32 = matrix(3, [1,0,0, 1,1,1, 1,0,1])
gens = (A1, A2, A3, P23, P32, P13, P31, P12, P21)
alphabet = [1, 2, 3, 123, 132, 213, 231, 312, 321]
gens = dict(zip(alphabet, gens))
cone = {}
cone[123] = H23 = matrix(3, [1,0,1, 0,1,0, 0,0,1])
cone[132] = H32 = matrix(3, [1,1,0, 0,1,0, 0,0,1])
cone[213] = H13 = matrix(3, [1,0,0, 0,1,1, 0,0,1])
cone[231] = H31 = matrix(3, [1,0,0, 1,1,0, 0,0,1])
cone[312] = H12 = matrix(3, [1,0,0, 0,1,0, 0,1,1])
cone[321] = H21 = matrix(3, [1,0,0, 0,1,0, 1,0,1])
cone[1] = cone[2] = cone[3] = identity_matrix(3)
from .language import languages
return MatrixCocycle(gens, cone, language=languages.ARP())
def calculate_generators(self):
"""
If generators haven't already been computed, calculate generators
for this homspace. If they have been computed, do nothing.
EXAMPLES::
sage: E = End(J0(11))
sage: E.calculate_generators()
"""
if self._gens is not None:
return
if (self.domain() == self.codomain()) and (self.domain().dimension() == 1):
self._gens = tuple([ identity_matrix(ZZ,2) ])
return
phi = self.domain()._isogeny_to_product_of_powers()
psi = self.codomain()._isogeny_to_product_of_powers()
H_simple = phi.codomain().Hom(psi.codomain())
im_gens = H_simple._calculate_product_gens()
M = phi.matrix()
Mt = psi.complementary_isogeny().matrix()
R = ZZ**(4*self.domain().dimension()*self.codomain().dimension())
gens = R.submodule([ (M*self._get_matrix(g)*Mt).list()
for g in im_gens ]).saturation().basis()
self._gens = tuple([ self._get_matrix(g) for g in gens ])
def calculate_generators(self):
"""
If generators haven't already been computed, calculate generators
for this homspace. If they have been computed, do nothing.
EXAMPLES::
sage: E = End(J0(11))
sage: E.calculate_generators()
"""
if self._gens is not None:
return
if (self.domain() == self.codomain()) and (self.domain().dimension()
== 1):
self._gens = tuple([identity_matrix(ZZ, 2)])
return
phi = self.domain()._isogeny_to_product_of_powers()
psi = self.codomain()._isogeny_to_product_of_powers()
H_simple = phi.codomain().Hom(psi.codomain())
im_gens = H_simple._calculate_product_gens()
M = phi.matrix()
Mt = psi.complementary_isogeny().matrix()
R = ZZ**(4 * self.domain().dimension() * self.codomain().dimension())
gens = R.submodule([(M * self._get_matrix(g) * Mt).list()
for g in im_gens]).saturation().basis()
self._gens = tuple([self._get_matrix(g) for g in gens])
def initial_pair(self):
"""
Return an initial double description pair.
Picks an initial set of rays by selecting a basis. This is
probably the most efficient way to select the initial set.
INPUT:
- ``pair_class`` -- subclass of
:class:`DoubleDescriptionPair`.
OUTPUT:
A pair consisting of a :class:`DoubleDescriptionPair` instance
and the tuple of remaining unused inequalities.
EXAMPLES::
sage: A = matrix([(-1, 1), (-1, 2), (1/2, -1/2), (1/2, 2)])
sage: from sage.geometry.polyhedron.double_description import Problem
sage: DD, remaining = Problem(A).initial_pair()
sage: DD.verify()
sage: remaining
[(1/2, -1/2), (1/2, 2)]
"""
pivot_rows = self.A_matrix().pivot_rows()
A0 = [self.A()[pivot] for pivot in pivot_rows]
Ac = [self.A()[i] for i in range(len(self.A())) if i not in pivot_rows]
from sage.matrix.constructor import identity_matrix, matrix
I = identity_matrix(self.base_ring(), self.dim())
R = matrix(self.base_ring(), A0).solve_right(I)
return self.pair_class(self, A0, R.columns()), list(Ac)
def __init__(self, similarity_surface):
if similarity_surface.underlying_surface().is_mutable():
if similarity_surface.is_finite():
self._ss=similarity_surface.copy()
else:
raise ValueError("Can not construct MinimalTranslationCover of a surface that is mutable and infinite.")
else:
self._ss = similarity_surface
# We are finite if and only if self._ss is a finite RationalConeSurface.
if not self._ss.is_finite():
finite = False
else:
try:
from flatsurf.geometry.rational_cone_surface import RationalConeSurface
ss_copy = self._ss.reposition_polygons(relabel=True)
rcs = RationalConeSurface(ss_copy)
rcs._test_edge_matrix()
finite=True
except AssertionError:
# print("Warning: Could be indicating infinite surface falsely.")
finite=False
I = identity_matrix(self._ss.base_ring(),2)
I.set_immutable()
base_label=(self._ss.base_label(), I)
Surface.__init__(self, self._ss.base_ring(), base_label, finite=finite, mutable=False)
def calculate_voronoi_cell(basis, radius=None, verbose=False):
"""
Calculate the Voronoi cell of the lattice defined by basis
INPUT:
- ``basis`` -- embedded basis matrix of the lattice
- ``radius`` -- radius of basis vectors to consider
- ``verbose`` -- whether to print debug information
OUTPUT:
A :class:`Polyhedron` instance.
EXAMPLES::
sage: from sage.modules.diamond_cutting import calculate_voronoi_cell
sage: V = calculate_voronoi_cell(matrix([[1, 0], [0, 1]]))
sage: V.volume()
1
"""
dim = basis.dimensions()
artificial_length = None
if dim[0] < dim[1]:
# introduce "artificial" basis points (representing infinity)
artificial_length = max(abs(v) for v in basis).ceil() * 2
additional_vectors = identity_matrix(dim[1]) * artificial_length
basis = basis.stack(additional_vectors)
# LLL-reduce to get quadratic matrix
basis = basis.LLL()
basis = matrix([v for v in basis if v])
dim = basis.dimensions()
if dim[0] != dim[1]:
raise ValueError("invalid matrix")
basis = basis / 2
ieqs = []
for v in basis:
ieqs.append(plane_inequality(v))
ieqs.append(plane_inequality(-v))
Q = Polyhedron(ieqs=ieqs)
# twice the length of longest vertex in Q is a safe choice
if radius is None:
radius = 2 * max(abs(v) ** 2 for v in basis)
V = diamond_cut(Q, basis, radius, verbose=verbose)
if artificial_length is not None:
# remove inequalities introduced by artificial basis points
H = V.Hrepresentation()
H = [v for v in H if all(not V._is_zero(v.A() * w / 2 - v.b() and
not V._is_zero(v.A() * (-w) / 2 - v.b())) for w in additional_vectors)]
V = Polyhedron(ieqs=H)
return V
def step_transition_matrix(step,
eps,
rows=None,
ctx=None,
split=0,
max_split=3):
dop = step.start.dop
order = dop.order()
if rows is None:
rows = order
z0, z1 = step
if order == 0:
logger.info("%s: trivial case", step)
return matrix(ZZ) # 0 by 0
elif z0.value == z1.value:
logger.info("%s: trivial case", step)
return identity_matrix(ZZ, order)[:rows]
elif z0.is_ordinary() and z1.is_ordinary():
logger.info("%s: ordinary case", step)
logger.debug("fraction of cvrad: %s/%s", step.length(),
z0.dist_to_sing())
fun = ordinary_step_transition_matrix
elif z0.is_regular() and z1.is_ordinary():
logger.info("%s: regular singular case (going out)", step)
logger.debug("fraction of cvrad: %s/%s", step.length(),
z0.dist_to_sing())
fun = singular_step_transition_matrix
elif z0.is_ordinary() and z1.is_regular():
logger.info("%s: regular singular case (going in)", step)
logger.debug("fraction of cvrad: %s/%s", step.length(),
z1.dist_to_sing())
fun = inverse_singular_step_transition_matrix
else:
raise TypeError(type(z0), type(z1))
try:
return fun(dop,
step,
eps,
rows,
ctx=ctx,
fail_fast=(split < max_split))
except (accuracy.PrecisionError, bounds.BoundPrecisionError):
# XXX it would be nicer to return something in this case...
if split >= max_split:
raise
logger.info("splitting step...")
s0, s1 = step.split()
m0 = step_transition_matrix(s0,
eps / 2,
rows=None,
ctx=ctx,
split=split + 1)
m1 = step_transition_matrix(s1,
eps / 2,
rows=rows,
ctx=ctx,
split=split + 1)
return m1 * m0
def matrix_power(self,t):
eigenvalues = self.eigenvalues
iprec = self.iprec
CM = self.CM
ev = [ e.n(iprec) for e in eigenvalues]
n = len(ev)
Char = [CM - ev[k] * identity_matrix(n) for k in range(n)]
#product till k-1
prodwo = n * [0]
prod = identity_matrix(n)
#if we were to start here with prod = IdentityMatrix
#prodwo[k]/sprodwo[k] would be the component projector of ev[k]
#component projector of ev[k] is a matrix Z such that
#CM * Z = ev[k] * Z and Z*Z=Z
#then f(CM)=sum_k f(ev[k])*Z[k]
#as we are only interested in the first line we can start
#left with (0,1,...) instead of the identity matrix
for k in range(n):
prodwo[k] = prod
for i in range(k+1,n):
prodwo[k] = prodwo[k] * Char[i]
if k == n:
break
prod = prod * Char[k]
sprodwo = n * [0]
for k in range(n):
if k == 0:
sprodwo[k] = ev[k] - ev[1]
start = 2
else:
sprodwo[k] = ev[k] - ev[0]
start = 1
for i in range(start,n):
if not i == k:
sprodwo[k] = sprodwo[k] * (ev[k]-ev[i])
return sum([ev[k]**t/sprodwo[k]*prodwo[k] for k in range(n)])
def reduce(self, t) :
## We compute the rational diagonal form of t. Whenever a zero entry occures we
## find a primitive isotropic vector and apply a base change, such that t finally
## has the form diag(0,0,...,P) where P is positive definite. P will then be a
## LLL reduced.
ot = t
t = matrix(QQ, t)
n = t.nrows()
u = identity_matrix(QQ, n)
for i in xrange(n) :
if t[i,i] < 0 :
return None
elif t[i,i] == 0 :
## get the isotropic vector
v = u.column(i)
v = v.denominator() * v
v = v / gcd(list(v))
u = self._gln_lift(v, 0)
t = u.transpose() * ot * u
ts = self.reduce(t.submatrix(1,1,n-1,n-1))
if ts is None :
return None
t.set_block(1, 1, ts[0])
t.set_immutable()
return (t,1)
else :
for j in xrange(i + 1, n) :
us = identity_matrix(QQ, n, n)
us[i,j] = -t[i,j]/t[i,i]
u = u * us
t.add_multiple_of_row(j, i, -t[j,i]/t[i,i])
t.add_multiple_of_column(j, i, -t[i,j]/t[i,i])
u = ot.LLL_gram()
t = u.transpose() * ot * u
t.set_immutable()
return (t, 1)
def _rank(self, K) :
if K is QQ or K in NumberFields() :
return len(_jacobi_forms_by_taylor_expansion_coords(self.__index, self.__weight, 0))
## This is the formula used by Poor and Yuen in Paramodular cusp forms
if self.__weight == 2 :
delta = len(self.__index.divisors()) // 2 - 1
else :
delta = 0
return sum( ModularForms(1, self.__weight + 2 * j).dimension() + j**2 // (4 * self.__index)
for j in xrange(self.__index + 1) ) \
+ delta
## This is the formula given by Skoruppa in
## Jacobi forms of critical weight and Weil representations
##FIXME: There is some mistake here
if self.__weight % 2 != 0 :
## Otherwise the space X(i**(n - 2 k)) is different
## See: Skoruppa, Jacobi forms of critical weight and Weil representations
raise NotImplementedError
m = self.__index
K = CyclotomicField(24 * m, 'zeta')
zeta = K.gen(0)
quadform = lambda x : 6 * x**2
bilinform = lambda x,y : quadform(x + y) - quadform(x) - quadform(y)
T = diagonal_matrix([zeta**quadform(i) for i in xrange(2*m)])
S = sum(zeta**(-quadform(x)) for x in xrange(2 * m)) / (2 * m) \
* matrix([[zeta**(-bilinform(j,i)) for j in xrange(2*m)] for i in xrange(2*m)])
subspace_matrix_1 = matrix( [ [1 if j == i or j == 2*m - i else 0 for j in xrange(m + 1) ]
for i in xrange(2*m)] )
subspace_matrix_2 = zero_matrix(ZZ, m + 1, 2*m)
subspace_matrix_2.set_block(0,0,identity_matrix(m+1))
T = subspace_matrix_2 * T * subspace_matrix_1
S = subspace_matrix_2 * S * subspace_matrix_1
sqrt3 = (zeta**(4*m) - zeta**(-4*m)) * zeta**(-6*m)
rank = (self.__weight - 1/2 - 1) / 2 * (m + 1) \
+ 1/8 * ( zeta**(3*m * (2*self.__weight - 1)) * S.trace()
+ zeta**(3*m * (1 - 2*self.__weight)) * S.trace().conjugate() ) \
+ 2/(3*sqrt3) * ( zeta**(4 * m * self.__weight) * (S*T).trace()
+ zeta**(-4 * m * self.__weight) * (S*T).trace().conjugate() ) \
- sum((j**2 % (m+1))/(m+1) -1/2 for j in range(0,m+1))
if self.__weight > 5 / 2 :
return rank
else :
raise NotImplementedError
raise NotImplementedError
def step_transition_matrix(dop, step, eps, rows=None, split=0, ctx=dctx):
r"""
TESTS::
sage: from ore_algebra.examples import fcc
sage: fcc.dop4.numerical_solution([0, 0, 0, 1], [0, 1], 1e-3)
[1...] + [+/- ...]*I
"""
order = dop.order()
if rows is None:
rows = order
z0, z1 = step
if order == 0:
logger.debug("%s: trivial case", step)
return matrix(ZZ) # 0 by 0
elif z0.value == z1.value:
logger.debug("%s: trivial case", step)
return identity_matrix(ZZ, order)[:rows]
elif z0.is_ordinary() and z1.is_ordinary():
logger.info("%s: ordinary case", step)
if z0.is_exact():
inverse = False
# XXX maybe also invert the step when z1 is much simpler than z0
else: # can happen with the very first step
step = Step(z1, z0, max_split=0)
inverse = True
elif z0.is_regular() and z1.is_ordinary():
logger.info("%s: regular singular case (going out)", step)
inverse = False
elif z0.is_ordinary() and z1.is_regular():
logger.info("%s: regular singular case (going in)", step)
step = Step(z1, z0)
inverse = True
eps /= 2
else:
raise ValueError(z0, z1)
try:
mat = regular_step_transition_matrix(dop,
step,
eps,
rows,
fail_fast=(step.max_split > 0),
effort=split,
ctx=ctx)
except (accuracy.PrecisionError, bounds.BoundPrecisionError):
if step.max_split == 0:
raise # XXX: can we return something?
logger.info("splitting step...")
s0, s1 = step.split()
m0 = step_transition_matrix(dop, s0, eps / 4, None, split + 1, ctx)
m1 = step_transition_matrix(dop, s1, eps / 4, rows, split + 1, ctx)
mat = m1 * m0
if inverse:
mat = ~mat
return mat
def fundamental_matrix_ordinary(dop, pt, eps, rows, maj, max_prec=100000):
eps_col = bounds.IR(eps)/bounds.IR(dop.order()).sqrt()
evpt = EvaluationPoint(pt, jet_order=rows)
inis = [
LogSeriesInitialValues(ZZ.zero(), ini, dop, check=False)
for ini in identity_matrix(dop.order())]
cols = [
series_sum(dop, ini, evpt, eps_col, maj=maj, max_prec=max_prec)
for ini in inis]
return matrix(cols).transpose()
def _normalisation_factor_zz(self, tau=3):
r"""
This function returns an approximation of `∑_{x ∈ \ZZ^n}
\exp(-|x|_2^2/(2σ²))`, i.e. the normalisation factor such that the sum
over all probabilities is 1 for `\ZZⁿ`.
If this ``self.B`` is not an identity matrix over `\ZZ` a
``NotImplementedError`` is raised.
INPUT:
- ``tau`` -- all vectors `v` with `|v|_∞ ≤ τ·σ` are enumerated
(default: ``3``).
EXAMPLES::
sage: from sage.stats.distributions.discrete_gaussian_lattice import DiscreteGaussianDistributionLatticeSampler
sage: n = 3; sigma = 1.0
sage: D = DiscreteGaussianDistributionLatticeSampler(ZZ^n, sigma)
sage: f = D.f
sage: c = D._normalisation_factor_zz(); c
15.528...
sage: from collections import defaultdict
sage: counter = defaultdict(Integer)
sage: m = 0
sage: def add_samples(i):
....: global counter, m
....: for _ in range(i):
....: counter[D()] += 1
....: m += 1
sage: v = vector(ZZ, n, (0, 0, 0))
sage: v.set_immutable()
sage: while v not in counter: add_samples(1000)
sage: while abs(m*f(v)*1.0/c/counter[v] - 1.0) >= 0.1: add_samples(1000)
sage: v = vector(ZZ, n, (-1, 2, 3))
sage: v.set_immutable()
sage: while v not in counter: add_samples(1000)
sage: while abs(m*f(v)*1.0/c/counter[v] - 1.0) >= 0.2: add_samples(1000) # long time
"""
if self.B != identity_matrix(ZZ, self.B.nrows()):
raise NotImplementedError(
"This function is only implemented when B is an identity matrix."
)
f = self.f
n = self.B.ncols()
sigma = self._sigma
return sum(
f(x)
for x in _iter_vectors(n, -ceil(tau * sigma), ceil(tau * sigma)))
def homogeneous_components(self):
deg_matrix = block_matrix([[identity_matrix(self.parent().rk),-self.parent()._B0]])
components = dict()
x = self.lift()
monomials = x.monomials()
for m in monomials:
g_vect = tuple(deg_matrix*vector(m.exponents()[0]))
if g_vect in components:
components[g_vect] += self.parent().retract(x.monomial_coefficient(m)*m)
else:
components[g_vect] = self.parent().retract(x.monomial_coefficient(m)*m)
return components
def cone_points_iter(self):
"""
Iterate over the open torus orbits and yield distinct points.
OUTPUT:
For each open torus orbit (cone): A triple consisting of the
cone, the nonzero homogeneous coordinates in that orbit (list
of integers), and the nonzero log coordinates of distinct
points as a cokernel.
EXAMPLES::
sage: fan = NormalFan(ReflexivePolytope(2, 0))
sage: X = ToricVariety(fan, base_ring=GF(7))
sage: point_set = X.point_set()
sage: ffe = point_set._finite_field_enumerator()
sage: cpi = ffe.cone_points_iter()
sage: cone, nonzero_points, cokernel = list(cpi)[5]
sage: cone
1-d cone of Rational polyhedral fan in 2-d lattice N
sage: cone.ambient_ray_indices()
(2,)
sage: nonzero_points
[0, 1]
sage: cokernel
Finitely generated module V/W over Integer Ring with invariants (2)
sage: list(cokernel)
[(0), (1)]
sage: [p.lift() for p in cokernel]
[(0, 0), (0, 1)]
"""
from sage.matrix.constructor import matrix, block_matrix, identity_matrix
from sage.rings.all import ZZ
nrays = len(self.rays())
N = self.multiplicative_group_order()
# Want cokernel of the log rescalings in (ZZ/N)^(#rays). But
# ZZ/N is not a integral domain. Instead: work over ZZ
log_generators = self.rescaling_log_generators()
log_relations = block_matrix(2, 1, [
matrix(ZZ, len(log_generators), nrays, log_generators),
N * identity_matrix(ZZ, nrays)
])
for cone in self.cone_iter():
nrays = self.fan().nrays() + len(self.fan().virtual_rays())
nonzero_coordinates = [
i for i in range(nrays) if i not in cone.ambient_ray_indices()
]
log_relations_nonzero = log_relations.matrix_from_columns(
nonzero_coordinates)
image = log_relations_nonzero.image()
cokernel = image.ambient_module().quotient(image)
yield cone, nonzero_coordinates, cokernel
def lift(A, N):
r"""
Lift a matrix A from SL_m(Z/NZ) to SL_m(Z).
Follows Shimura, Lemma 1.38, p21.
"""
assert A.is_square()
assert A.determinant() != 0
m = A.nrows()
if m == 1:
return identity_matrix(1)
D, U, V = A.smith_form()
if U.determinant() == -1:
U = matrix(2, 2, [-1, 0, 0, 1]) * U
if V.determinant() == -1:
V = V * matrix(2, 2, [-1, 0, 0, 1])
D = U * A * V
assert U.determinant() == 1
assert V.determinant() == 1
a = [D[i, i] for i in range(m)]
b = prod(a[1:])
W = identity_matrix(m)
W[0, 0] = b
W[1, 0] = b - 1
W[0, 1] = 1
X = identity_matrix(m)
X[0, 1] = -a[1]
Ap = D.parent()(D)
Ap[0, 0] = 1
Ap[1, 0] = 1 - a[0]
Ap[1, 1] *= a[0]
assert (W * U * A * V * X).change_ring(Zmod(N)) == Ap.change_ring(Zmod(N))
Cp = diagonal_matrix(a[1:])
Cp[0, 0] *= a[0]
C = lift(Cp, N)
Cpp = block_diagonal_matrix(identity_matrix(1), C)
Cpp[1, 0] = 1 - a[0]
return (~U * ~W * Cpp * ~X * ~V).change_ring(ZZ)
def lift(A, N):
r"""
Lift a matrix A from SL_m(Z/NZ) to SL_m(Z).
Follows Shimura, Lemma 1.38, p21.
"""
assert A.is_square()
assert A.determinant() != 0
m = A.nrows()
if m == 1:
return identity_matrix(1)
D, U, V = A.smith_form()
if U.determinant() == -1 :
U = matrix(2,2,[-1,0,0,1])* U
if V.determinant() == -1 :
V = V *matrix(2,2,[-1,0,0,1])
D = U*A*V
assert U.determinant() == 1
assert V.determinant() == 1
a = [ D[i, i] for i in range(m) ]
b = prod(a[1:])
W = identity_matrix(m)
W[0, 0] = b
W[1, 0] = b-1
W[0, 1] = 1
X = identity_matrix(m)
X[0, 1] = -a[1]
Ap = D.parent()(D)
Ap[0, 0] = 1
Ap[1, 0] = 1-a[0]
Ap[1, 1] *= a[0]
assert (W*U*A*V*X).change_ring(Zmod(N)) == Ap.change_ring(Zmod(N))
Cp = diagonal_matrix(a[1:])
Cp[0, 0] *= a[0]
C = lift(Cp, N)
Cpp = block_diagonal_matrix(identity_matrix(1), C)
Cpp[1, 0] = 1-a[0]
return (~U * ~W * Cpp * ~X * ~V).change_ring(ZZ)
def cone_points_iter(self):
"""
Iterate over the open torus orbits and yield distinct points.
OUTPUT:
For each open torus orbit (cone): A triple consisting of the
cone, the nonzero homogeneous coordinates in that orbit (list
of integers), and the nonzero log coordinates of distinct
points as a cokernel.
EXAMPLES::
sage: fan = NormalFan(ReflexivePolytope(2, 0))
sage: X = ToricVariety(fan, base_ring=GF(7))
sage: point_set = X.point_set()
sage: ffe = point_set._finite_field_enumerator()
sage: cpi = ffe.cone_points_iter()
sage: cone, nonzero_points, cokernel = list(cpi)[5]
sage: cone
1-d cone of Rational polyhedral fan in 2-d lattice N
sage: cone.ambient_ray_indices()
(2,)
sage: nonzero_points
[0, 1]
sage: cokernel
Finitely generated module V/W over Integer Ring with invariants (2)
sage: list(cokernel)
[(0), (1)]
sage: [p.lift() for p in cokernel]
[(0, 0), (0, 1)]
"""
from sage.matrix.constructor import matrix, block_matrix, identity_matrix
from sage.rings.all import ZZ
nrays = len(self.rays())
N = self.multiplicative_group_order()
# Want cokernel of the log rescalings in (ZZ/N)^(#rays). But
# ZZ/N is not a integral domain. Instead: work over ZZ
log_generators = self.rescaling_log_generators()
log_relations = block_matrix(2, 1, [
matrix(ZZ, len(log_generators), nrays, log_generators),
N * identity_matrix(ZZ, nrays)])
for cone in self.cone_iter():
nrays = self.fan().nrays() + len(self.fan().virtual_rays())
nonzero_coordinates = [i for i in range(nrays)
if i not in cone.ambient_ray_indices()]
log_relations_nonzero = log_relations.matrix_from_columns(nonzero_coordinates)
image = log_relations_nonzero.image()
cokernel = image.ambient_module().quotient(image)
yield cone, nonzero_coordinates, cokernel
def analytic_continuation(ctx, ini=None, post=None):
"""
INPUT:
- ``ini`` (constant matrix, optional) - initial values, one column per
solution
- ``post`` (matrix of polynomial/rational functions, optional) - linear
combinations of the first Taylor coefficients to take, as a function of
the evaluation point
TESTS::
sage: from ore_algebra import DifferentialOperators
sage: _, x, Dx = DifferentialOperators()
sage: (Dx^2 + 2*x*Dx).numerical_solution([0, 2/sqrt(pi)], [0,i])
[+/- ...] + [1.65042575879754...]*I
"""
logger.info("path: %s", ctx.path)
eps1 = (ctx.eps / (1 + len(ctx.path))) >> 2 # TBI, +: move to ctx?
prec = utilities.prec_from_eps(eps1)
if ini is not None:
if not isinstance(ini, Matrix): # should this be here?
try:
ini = matrix(ctx.dop.order(), 1, list(ini))
except (TypeError, ValueError):
raise ValueError("incorrect initial values: {}".format(ini))
try:
ini = ini.change_ring(RealBallField(prec))
except (TypeError, ValueError):
ini = ini.change_ring(ComplexBallField(prec))
res = []
path_mat = identity_matrix(ZZ, ctx.dop.order())
def store_value_if_wanted(point):
if point.options.get('keep_value'):
value = path_mat
if ini is not None: value = value * ini
if post is not None: value = post(point.value) * value
res.append((point.value, value))
store_value_if_wanted(ctx.path.vert[0])
for step in ctx.path:
step_mat = step_transition_matrix(step, eps1, ctx=ctx)
path_mat = step_mat * path_mat
store_value_if_wanted(step.end)
cm = sage.structure.element.get_coercion_model()
OutputIntervals = cm.common_parent(
utilities.ball_field(ctx.eps, ctx.real()),
*[mat.base_ring() for pt, mat in res])
return [(pt, mat.change_ring(OutputIntervals)) for pt, mat in res]
def permutation_simplicial_action(r, u, n, w):
r"""
From a word in 'l','r' return the simplicial action on homology as
well as the obtained origami.
INPUT:
- ``r`` and ``u`` - permutations
- ``n`` - the degree of the permutations
- ``w`` - a string in 'l' and 'r' or a list of 2-tuples which are made of
the letter 'l' or 'r' and a positive integer
"""
if w is None:
w = []
elif isinstance(w, list):
w = flatten_word(w)
res = identity_matrix(2 * n)
for letter in reversed(w):
if letter == 'l':
u = u * ~r
m = identity_matrix(2 * n)
m[:n, n:] = u.matrix()
res = m * res
elif letter == 'r':
r = r * ~u
m = identity_matrix(2 * n)
m[n:, :n] = r.matrix()
res = m * res
else:
raise ValueError, "does not understand the letter %s" % str(letter)
return r, u, res
def permutation_simplicial_action(r,u,n,w):
r"""
From a word in 'l','r' return the simplicial action on homology as
well as the obtained origami.
INPUT:
- ``r`` and ``u`` - permutations
- ``n`` - the degree of the permutations
- ``w`` - a string in 'l' and 'r' or a list of 2-tuples which are made of
the letter 'l' or 'r' and a positive integer
"""
if w is None:
w = []
elif isinstance(w,list):
w = flatten_word(w)
res = identity_matrix(2*n)
for letter in reversed(w):
if letter == 'l':
u = u*~r
m = identity_matrix(2*n)
m[:n,n:] = u.matrix()
res = m * res
elif letter == 'r':
r = r*~u
m = identity_matrix(2*n)
m[n:,:n] = r.matrix()
res = m * res
else:
raise ValueError, "does not understand the letter %s" %str(letter)
return r,u,res
def homogeneous_components(self):
deg_matrix = block_matrix(
[[identity_matrix(self.parent().rk), -self.parent()._B0]])
components = dict()
x = self.lift()
monomials = x.monomials()
for m in monomials:
g_vect = tuple(deg_matrix * vector(m.exponents()[0]))
if g_vect in components:
components[g_vect] += self.parent().retract(
x.monomial_coefficient(m) * m)
else:
components[g_vect] = self.parent().retract(
x.monomial_coefficient(m) * m)
return components
def construct_from_dim_degree(dim, max_degree, base_ring, check):
"""
Construct a filtered vector space.
INPUT:
- ``dim`` -- integer. The dimension.
- ``max_degree`` -- integer or infinity. The maximal degree where
the vector subspace of the filtration is still the entire space.
EXAMPLES::
sage: V = FilteredVectorSpace(2, 5); V
QQ^2 >= 0
sage: V.get_degree(5)
Vector space of degree 2 and dimension 2 over Rational Field
Basis matrix:
[1 0]
[0 1]
sage: V.get_degree(6)
Vector space of degree 2 and dimension 0 over Rational Field
Basis matrix:
[]
sage: FilteredVectorSpace(2, oo)
QQ^2
sage: FilteredVectorSpace(2, -oo)
0 in QQ^2
TESTS::
sage: from sage.modules.filtered_vector_space import construct_from_dim_degree
sage: V = construct_from_dim_degree(2, 5, QQ, True); V
QQ^2 >= 0
"""
if dim not in ZZ:
raise ValueError('dimension must be an integer')
dim = ZZ(dim)
from sage.matrix.constructor import identity_matrix
generators = identity_matrix(base_ring, dim).columns()
filtration = dict()
if max_degree is None:
max_degree = infinity
filtration[normalize_degree(max_degree)] = range(dim)
return construct_from_generators_indices(generators, filtration, base_ring,
check)
def construct_from_dim_degree(dim, max_degree, base_ring, check):
"""
Construct a filtered vector space.
INPUT:
- ``dim`` -- integer. The dimension.
- ``max_degree`` -- integer or infinity. The maximal degree where
the vector subspace of the filtration is still the entire space.
EXAMPLES::
sage: V = FilteredVectorSpace(2, 5); V
QQ^2 >= 0
sage: V.get_degree(5)
Vector space of degree 2 and dimension 2 over Rational Field
Basis matrix:
[1 0]
[0 1]
sage: V.get_degree(6)
Vector space of degree 2 and dimension 0 over Rational Field
Basis matrix:
[]
sage: FilteredVectorSpace(2, oo)
QQ^2
sage: FilteredVectorSpace(2, -oo)
0 in QQ^2
TESTS::
sage: from sage.modules.filtered_vector_space import construct_from_dim_degree
sage: V = construct_from_dim_degree(2, 5, QQ, True); V
QQ^2 >= 0
"""
if dim not in ZZ:
raise ValueError('dimension must be an integer')
dim = ZZ(dim)
from sage.matrix.constructor import identity_matrix
generators = identity_matrix(base_ring, dim).columns()
filtration = dict()
if max_degree is None:
max_degree = infinity
filtration[normalize_degree(max_degree)] = range(dim)
return construct_from_generators_indices(generators, filtration, base_ring, check)
def gens(self):
"""
Return the generators.
OUTPUT:
A tuple of linear expressions, one for each linear variable.
EXAMPLES::
sage: from sage.geometry.linear_expression import LinearExpressionModule
sage: L = LinearExpressionModule(QQ, ('x', 'y', 'z'))
sage: L.gens()
(x + 0*y + 0*z + 0, 0*x + y + 0*z + 0, 0*x + 0*y + z + 0)
"""
from sage.matrix.constructor import identity_matrix
identity = identity_matrix(self.base_ring(), self.ngens())
return tuple(self(e, 0) for e in identity.rows())
def gens(self):
"""
Return the generators of ``self``.
OUTPUT:
A tuple of linear expressions, one for each linear variable.
EXAMPLES::
sage: from sage.geometry.linear_expression import LinearExpressionModule
sage: L = LinearExpressionModule(QQ, ('x', 'y', 'z'))
sage: L.gens()
(x + 0*y + 0*z + 0, 0*x + y + 0*z + 0, 0*x + 0*y + z + 0)
"""
from sage.matrix.constructor import identity_matrix
identity = identity_matrix(self.base_ring(), self.ngens())
return tuple(self(e, 0) for e in identity.rows())
def __init__(self, Prg, Rng, H = [], Settings = {'Iterations':150,'detail':True,'tryKKT':0, 'AutoMatrix':True, 'precision':7}):
f = Prg[0]
self.Prog = []
self.PolyRng = Rng
self.VARS = self.PolyRng.gens()
self.polynomial = f
self.Field = self.polynomial.base_ring()
self.number_of_variables = len(self.VARS)
self.total_degree = f.total_degree()
self.constant_term = f.constant_coefficient()
self.prg_size = 1
if len(Prg) > 1:
self.prg_size = len(Prg[1]) + 1
if H == []:
self.H = identity_matrix(self.Field, self.prg_size)
else:
self.H = H
self.Prog.append(f)
self.constant_terms.append(f.constant_coefficient())
for i in range(self.prg_size - 1):
self.Prog.append(Prg[1][i])
self.constant_terms.append(Prg[1][i].constant_coefficient())
tmp_deg = Prg[1][i].total_degree()
if self.total_degree < tmp_deg:
self.total_degree = tmp_deg
self.Settings = Settings
if 'AutoMatrix' in Settings:
self.AutoMatrix = Settings['AutoMatrix']
else:
self.AutoMatrix = False
if 'Order' in Settings:
self.Ord = max(Settings['Order'], self.total_degree)
else:
self.Ord = self.total_degree
if (self.Ord % 2) == 1:
self.Ord = self.Ord + 1
self.MarginalCoeffs = []
if 'precision' in Settings:
self.error_bound = 10**(-Settings['precision'])
else:
self.error_bound = 10**(-7)
self.Info = {'gp':0,'CPU':0,'Wall':0,'status':'Unknown','Message':''}
def invariant_bilinear_form(self):
"""
Return the symmetric bilinear form preserved by ``self``.
OUTPUT:
A matrix.
EXAMPLES::
sage: GO(2,3,+1).invariant_bilinear_form()
[0 1]
[1 0]
sage: GO(2,3,-1).invariant_bilinear_form()
[2 1]
[1 1]
sage: G = GO(4, QQ)
sage: G.invariant_bilinear_form()
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
sage: GO3m = GO(3,QQ, invariant_form=(1,0,0,0,2,0,0,0,3))
sage: GO3m.invariant_bilinear_form()
[1 0 0]
[0 2 0]
[0 0 3]
TESTS::
sage: GO3m.invariant_form()
[1 0 0]
[0 2 0]
[0 0 3]
"""
if self._invariant_form is not None:
return self._invariant_form
from sage.matrix.constructor import identity_matrix
m = identity_matrix(self.base_ring(), self.degree())
m.set_immutable()
return m
def invariant_bilinear_form(self):
"""
Return the symmetric bilinear form preserved by ``self``.
OUTPUT:
A matrix.
EXAMPLES::
sage: GO(2,3,+1).invariant_bilinear_form()
[0 1]
[1 0]
sage: GO(2,3,-1).invariant_bilinear_form()
[2 1]
[1 1]
sage: G = GO(4, QQ)
sage: G.invariant_bilinear_form()
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
sage: GO3m = GO(3,QQ, invariant_form=(1,0,0,0,2,0,0,0,3))
sage: GO3m.invariant_bilinear_form()
[1 0 0]
[0 2 0]
[0 0 3]
TESTS::
sage: GO3m.invariant_form()
[1 0 0]
[0 2 0]
[0 0 3]
"""
if self._invariant_form is not None:
return self._invariant_form
from sage.matrix.constructor import identity_matrix
m = identity_matrix(self.base_ring(), self.degree())
m.set_immutable()
return m
def cartan_matrix(self):
r"""
Returns the Cartan matrix for this Dynkin diagram
EXAMPLES::
sage: DynkinDiagram(['C',3]).cartan_matrix()
[ 2 -1 0]
[-1 2 -2]
[ 0 -1 2]
"""
from sage.matrix.constructor import identity_matrix
from sage.rings.all import ZZ
index_set = self.index_set()
reverse = dict((index_set[i], i) for i in range(len(index_set)))
m = 2*identity_matrix(ZZ, len(self.index_set()), sparse=True)
for (i,j,l) in self.edge_iterator():
m[reverse[j], reverse[i]] = -l
m.set_immutable()
return m
def cartan_matrix(self):
r"""
Returns the Cartan matrix for this Dynkin diagram
EXAMPLES::
sage: DynkinDiagram(['C',3]).cartan_matrix()
[ 2 -1 0]
[-1 2 -2]
[ 0 -1 2]
"""
from sage.matrix.constructor import identity_matrix
from sage.rings.all import ZZ
index_set = self.index_set()
reverse = dict((index_set[i], i) for i in range(len(index_set)))
m = 2*identity_matrix(ZZ, len(self.index_set()), sparse=True)
for (i,j,l) in self.edge_iterator():
m[reverse[j], reverse[i]] = -l
m.set_immutable()
return m
def invariant_quadratic_form(self):
"""
Return the quadratic form preserved by the orthogonal group.
OUTPUT:
A matrix.
EXAMPLES::
sage: GO(2,3,+1).invariant_quadratic_form()
[0 1]
[0 0]
sage: GO(2,3,-1).invariant_quadratic_form()
[1 1]
[0 2]
"""
from sage.matrix.constructor import identity_matrix
m = identity_matrix(self.base_ring(), self.degree())
m.set_immutable()
return m
def invariant_quadratic_form(self):
"""
Return the quadratic form preserved by the orthogonal group.
OUTPUT:
A matrix.
EXAMPLES::
sage: GO(2,3,+1).invariant_quadratic_form()
[0 1]
[0 0]
sage: GO(2,3,-1).invariant_quadratic_form()
[1 1]
[0 2]
"""
from sage.matrix.constructor import identity_matrix
m = identity_matrix(self.base_ring(), self.degree())
m.set_immutable()
return m
def _normalisation_factor_zz(self, tau=3):
r"""
This function returns an approximation of `∑_{x ∈ \ZZ^n}
\exp(-|x|_2^2/(2σ²))`, i.e. the normalisation factor such that the sum
over all probabilities is 1 for `\ZZⁿ`.
If this ``self.B`` is not an identity matrix over `\ZZ` a
``NotImplementedError`` is raised.
INPUT:
- ``tau`` -- all vectors `v` with `|v|_∞ ≤ τ·σ` are enumerated
(default: ``3``).
EXAMPLE::
sage: from sage.stats.distributions.discrete_gaussian_lattice import DiscreteGaussianDistributionLatticeSampler
sage: n = 3; sigma = 1.0; m = 1000
sage: D = DiscreteGaussianDistributionLatticeSampler(ZZ^n, sigma)
sage: f = D.f
sage: c = D._normalisation_factor_zz(); c
15.528...
sage: l = [D() for _ in xrange(m)]
sage: v = vector(ZZ, n, (0, 0, 0))
sage: l.count(v), ZZ(round(m*f(v)/c))
(57, 64)
"""
if self.B != identity_matrix(ZZ, self.B.nrows()):
raise NotImplementedError(
"This function is only implemented when B is an identity matrix."
)
f = self.f
n = self.B.ncols()
sigma = self._sigma
return sum(
f(x)
for x in _iter_vectors(n, -ceil(tau * sigma), ceil(tau * sigma)))
def hnf_with_transformation(A, proof=True):
"""
Compute the HNF H of A along with a transformation matrix U
such that U*A = H. Also return the pivots of H.
INPUT:
- A -- an n x m matrix A over the integers.
- proof -- whether or not to prove the result correct.
OUTPUT:
- matrix -- the Hermite normal form H of A
- U -- a unimodular matrix such that U * A = H
- pivots -- the pivot column positions of A
EXAMPLES::
sage: import sage.matrix.matrix_integer_dense_hnf as matrix_integer_dense_hnf
sage: A = matrix(ZZ, 2, [1, -5, -10, 1, 3, 197]); A
[ 1 -5 -10]
[ 1 3 197]
sage: H, U, pivots = matrix_integer_dense_hnf.hnf_with_transformation(A)
sage: H
[ 1 3 197]
[ 0 8 207]
sage: U
[ 0 1]
[-1 1]
sage: U*A
[ 1 3 197]
[ 0 8 207]
"""
# All we do is augment the input matrix with the identity matrix of the appropriate rank on the right.
C = A.augment(identity_matrix(ZZ, A.nrows()))
H, pivots = hnf(C, include_zero_rows=True, proof=proof)
U = H.matrix_from_columns(range(A.ncols(), H.ncols()))
H2 = H.matrix_from_columns(range(A.ncols()))
return H2, U, pivots
def fundamental_matrix_ordinary(dop, pt, eps, rows, maj, fail_fast):
eps_col = bounds.IR(eps) / bounds.IR(dop.order()).sqrt()
eps_col = accuracy.AbsoluteError(eps_col)
evpt = EvaluationPoint(pt, jet_order=rows)
bwrec = bw_shift_rec(dop)
inis = [
LogSeriesInitialValues(ZZ.zero(), ini, dop, check=False)
for ini in identity_matrix(dop.order())
]
stop = accuracy.StoppingCriterion(maj, eps_col.eps)
cols = [
interval_series_sum_wrapper(series_sum_ordinary,
dop,
ini,
evpt,
eps_col,
bwrec,
stop,
fail_fast,
max_prec=None) for ini in inis
]
return matrix(cols).transpose()
def hnf_with_transformation(A, proof=True):
"""
Compute the HNF H of A along with a transformation matrix U
such that U*A = H. Also return the pivots of H.
INPUT:
- A -- an n x m matrix A over the integers.
- proof -- whether or not to prove the result correct.
OUTPUT:
- matrix -- the Hermite normal form H of A
- U -- a unimodular matrix such that U * A = H
- pivots -- the pivot column positions of A
EXAMPLES::
sage: import sage.matrix.matrix_integer_dense_hnf as matrix_integer_dense_hnf
sage: A = matrix(ZZ, 2, [1, -5, -10, 1, 3, 197]); A
[ 1 -5 -10]
[ 1 3 197]
sage: H, U, pivots = matrix_integer_dense_hnf.hnf_with_transformation(A)
sage: H
[ 1 3 197]
[ 0 8 207]
sage: U
[ 0 1]
[-1 1]
sage: U*A
[ 1 3 197]
[ 0 8 207]
"""
# All we do is augment the input matrix with the identity matrix of the appropriate rank on the right.
C = A.augment(identity_matrix(ZZ, A.nrows()))
H, pivots = hnf(C, include_zero_rows=True, proof=proof)
U = H.matrix_from_columns(range(A.ncols(), H.ncols()))
H2 = H.matrix_from_columns(range(A.ncols()))
return H2, U, pivots
def identity(self):
r"""
Return identity morphism in an endomorphism ring.
EXAMPLES::
sage: V=FreeModule(ZZ,5)
sage: H=V.Hom(V)
sage: H.identity()
Free module morphism defined by the matrix
[1 0 0 0 0]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[0 0 0 0 1]
Domain: Ambient free module of rank 5 over the principal ideal domain ...
Codomain: Ambient free module of rank 5 over the principal ideal domain ...
"""
if self.is_endomorphism_set():
return self(identity_matrix(self.base_ring(),self.domain().rank()))
else:
raise TypeError("Identity map only defined for endomorphisms. Try natural_map() instead.")
def generator_matrix(self):
r"""
Return a generator matrix of ``self``.
EXAMPLES::
sage: C = codes.ParityCheckCode(GF(5),7)
sage: E = codes.encoders.ParityCheckCodeGeneratorMatrixEncoder(C)
sage: E.generator_matrix()
[1 0 0 0 0 0 0 4]
[0 1 0 0 0 0 0 4]
[0 0 1 0 0 0 0 4]
[0 0 0 1 0 0 0 4]
[0 0 0 0 1 0 0 4]
[0 0 0 0 0 1 0 4]
[0 0 0 0 0 0 1 4]
"""
k = self.code().dimension()
field = self.code().base_field()
G = identity_matrix(field, k)
G = G.augment(vector(field, [-field.one()] * k))
G.set_immutable()
return G
def _Hasse_Witt_cached(self):
r"""
INPUT:
- ``E`` : Hyperelliptic Curve of the form `y^2 = f(x)` over a finite field, `\mathbb{F}_q`
OUTPUT:
- ``N`` : The matrix `N = M M^p \dots M^{p^{g-1}}` where `M = c_{pi-j}, f(x)^{(p-1)/2} = \sum c_i x^i`
- ``E`` : The initial curve to check some caching conditions.
EXAMPLES::
sage: K.<x>=GF(9,'x')[]
sage: C=HyperellipticCurve(x^7-1,0)
sage: C._Hasse_Witt_cached()
(
[0 0 0]
[0 0 0]
[0 0 0], Hyperelliptic Curve over Finite Field in x of size 3^2 defined by y^2 = x^7 + 2
)
sage: K.<x>=GF(49,'x')[]
sage: C=HyperellipticCurve(x^5+1,0)
sage: C._Hasse_Witt_cached()
(
[0 0]
[0 0], Hyperelliptic Curve over Finite Field in x of size 7^2 defined by y^2 = x^5 + 1
)
sage: P.<x>=GF(9,'a')[]
sage: C=HyperellipticCurve(x^29+1,0)
sage: C._Hasse_Witt_cached()
(
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0], Hyperelliptic Curve over Finite Field in a of size 3^2 defined by y^2 = x^29 + 1
)
"""
# If Cartier Matrix is already cached for this curve, use that or evaluate it to get M,
#Coeffs, genus, Fq=base field of self, p=char(Fq). This is so we have one less matrix to
#compute.
#We use caching here since Cartier matrix is needed to compute Hasse Witt. So if the Cartier
#is already computed it is stored in list A. If it was not cached (i.e. A is empty), we simply
#compute it. If it is cached then we need to make sure that we have the correct one. So check
#which curve does the matrix correspond to. Since caching stores a lot of stuff, we only check
#the last entry in A. If it does not match, clear A and compute Cartier.
#
#Since Trac Ticket #11115, there is a different cache for methods
#that don't accept arguments. Anyway, the easiest is to call
#the cached method and simply see whether the data belong to self.
M, Coeffs, g, Fq, p, E = self._Cartier_matrix_cached()
if E != self:
self._Cartier_matrix_cached.clear_cache()
M, Coeffs, g, Fq, p, E = self._Cartier_matrix_cached()
#This compute the action of p^kth Frobenius on list of coefficients
def frob_mat(Coeffs, k):
a = p**k
mat = []
Coeffs_pow = [c**a for c in Coeffs]
for i in range(1, g + 1):
H = [(Coeffs[j])
for j in range((p * i - 1), (p * i - g - 1), -1)]
mat.append(H)
return matrix(Fq, mat)
#Computes all the different possible action of frobenius on matrix M and stores in list Mall
Mall = [M] + [frob_mat(Coeffs, k) for k in range(1, g)]
#initial N=I, so we can go through Mall and multiply all matrices with I and
#get the Hasse-Witt matrix.
N = identity_matrix(Fq, g)
for l in Mall:
N = N * l
return N, E
def gen_lattice(type='modular', n=4, m=8, q=11, seed=None, \
quotient=None, dual=False, ntl=False):
"""
This function generates different types of integral lattice bases
of row vectors relevant in cryptography.
Randomness can be set either with ``seed``, or by using
:func:`sage.misc.randstate.set_random_seed`.
INPUT:
* ``type`` - one of the following strings
* ``'modular'`` (default). A class of lattices for which
asymptotic worst-case to average-case connections hold. For
more refer to [A96]_.
* ``'random'`` - Special case of modular (n=1). A dense class
of lattice used for testing basis reduction algorithms
proposed by Goldstein and Mayer [GM02]_.
* ``'ideal'`` - Special case of modular. Allows for a more
compact representation proposed by [LM06]_.
* ``'cyclotomic'`` - Special case of ideal. Allows for
efficient processing proposed by [LM06]_.
* ``n`` - Determinant size, primal:`det(L) = q^n`, dual:`det(L) = q^{m-n}`.
For ideal lattices this is also the degree of the quotient polynomial.
* ``m`` - Lattice dimension, `L \subseteq Z^m`.
* ``q`` - Coefficent size, `q*Z^m \subseteq L`.
* ``seed`` - Randomness seed.
* ``quotient`` - For the type ideal, this determines the quotient
polynomial. Ignored for all other types.
* ``dual`` - Set this flag if you want a basis for `q*dual(L)`, for example
for Regev's LWE bases [R05]_.
* ``ntl`` - Set this flag if you want the lattice basis in NTL readable
format.
OUTPUT: ``B`` a unique size-reduced triangular (primal: lower_left,
dual: lower_right) basis of row vectors for the lattice in question.
EXAMPLES:
* Modular basis ::
sage: sage.crypto.gen_lattice(m=10, seed=42)
[11 0 0 0 0 0 0 0 0 0]
[ 0 11 0 0 0 0 0 0 0 0]
[ 0 0 11 0 0 0 0 0 0 0]
[ 0 0 0 11 0 0 0 0 0 0]
[ 2 4 3 5 1 0 0 0 0 0]
[ 1 -5 -4 2 0 1 0 0 0 0]
[-4 3 -1 1 0 0 1 0 0 0]
[-2 -3 -4 -1 0 0 0 1 0 0]
[-5 -5 3 3 0 0 0 0 1 0]
[-4 -3 2 -5 0 0 0 0 0 1]
* Random basis ::
sage: sage.crypto.gen_lattice(type='random', n=1, m=10, q=11^4, seed=42)
[14641 0 0 0 0 0 0 0 0 0]
[ 431 1 0 0 0 0 0 0 0 0]
[-4792 0 1 0 0 0 0 0 0 0]
[ 1015 0 0 1 0 0 0 0 0 0]
[-3086 0 0 0 1 0 0 0 0 0]
[-5378 0 0 0 0 1 0 0 0 0]
[ 4769 0 0 0 0 0 1 0 0 0]
[-1159 0 0 0 0 0 0 1 0 0]
[ 3082 0 0 0 0 0 0 0 1 0]
[-4580 0 0 0 0 0 0 0 0 1]
* Ideal bases with quotient x^n-1, m=2*n are NTRU bases ::
sage: sage.crypto.gen_lattice(type='ideal', seed=42, quotient=x^4-1)
[11 0 0 0 0 0 0 0]
[ 0 11 0 0 0 0 0 0]
[ 0 0 11 0 0 0 0 0]
[ 0 0 0 11 0 0 0 0]
[ 4 -2 -3 -3 1 0 0 0]
[-3 4 -2 -3 0 1 0 0]
[-3 -3 4 -2 0 0 1 0]
[-2 -3 -3 4 0 0 0 1]
* Cyclotomic bases with n=2^k are SWIFFT bases ::
sage: sage.crypto.gen_lattice(type='cyclotomic', seed=42)
[11 0 0 0 0 0 0 0]
[ 0 11 0 0 0 0 0 0]
[ 0 0 11 0 0 0 0 0]
[ 0 0 0 11 0 0 0 0]
[ 4 -2 -3 -3 1 0 0 0]
[ 3 4 -2 -3 0 1 0 0]
[ 3 3 4 -2 0 0 1 0]
[ 2 3 3 4 0 0 0 1]
* Dual modular bases are related to Regev's famous public-key
encryption [R05]_ ::
sage: sage.crypto.gen_lattice(type='modular', m=10, seed=42, dual=True)
[ 0 0 0 0 0 0 0 0 0 11]
[ 0 0 0 0 0 0 0 0 11 0]
[ 0 0 0 0 0 0 0 11 0 0]
[ 0 0 0 0 0 0 11 0 0 0]
[ 0 0 0 0 0 11 0 0 0 0]
[ 0 0 0 0 11 0 0 0 0 0]
[ 0 0 0 1 -5 -2 -1 1 -3 5]
[ 0 0 1 0 -3 4 1 4 -3 -2]
[ 0 1 0 0 -4 5 -3 3 5 3]
[ 1 0 0 0 -2 -1 4 2 5 4]
* Relation of primal and dual bases ::
sage: B_primal=sage.crypto.gen_lattice(m=10, q=11, seed=42)
sage: B_dual=sage.crypto.gen_lattice(m=10, q=11, seed=42, dual=True)
sage: B_dual_alt=transpose(11*B_primal.inverse()).change_ring(ZZ)
sage: B_dual_alt.hermite_form() == B_dual.hermite_form()
True
REFERENCES:
.. [A96] Miklos Ajtai.
Generating hard instances of lattice problems (extended abstract).
STOC, pp. 99--108, ACM, 1996.
.. [GM02] Daniel Goldstein and Andrew Mayer.
On the equidistribution of Hecke points.
Forum Mathematicum, 15:2, pp. 165--189, De Gruyter, 2003.
.. [LM06] Vadim Lyubashevsky and Daniele Micciancio.
Generalized compact knapsacks are collision resistant.
ICALP, pp. 144--155, Springer, 2006.
.. [R05] Oded Regev.
On lattices, learning with errors, random linear codes, and cryptography.
STOC, pp. 84--93, ACM, 2005.
"""
from sage.rings.finite_rings.integer_mod_ring \
import IntegerModRing
from sage.matrix.constructor import matrix, \
identity_matrix, block_matrix
from sage.matrix.matrix_space import MatrixSpace
from sage.rings.integer_ring import IntegerRing
if seed != None:
from sage.misc.randstate import set_random_seed
set_random_seed(seed)
if type == 'random':
if n != 1: raise ValueError('random bases require n = 1')
ZZ = IntegerRing()
ZZ_q = IntegerModRing(q)
A = identity_matrix(ZZ_q, n)
if type == 'random' or type == 'modular':
R = MatrixSpace(ZZ_q, m-n, n)
A = A.stack(R.random_element())
elif type == 'ideal':
if quotient == None: raise \
ValueError('ideal bases require a quotient polynomial')
x = quotient.default_variable()
if n != quotient.degree(x): raise \
ValueError('ideal bases require n = quotient.degree()')
R = ZZ_q[x].quotient(quotient, x)
for i in range(m//n):
A = A.stack(R.random_element().matrix())
elif type == 'cyclotomic':
from sage.rings.arith import euler_phi
from sage.misc.functional import cyclotomic_polynomial
# we assume that n+1 <= min( euler_phi^{-1}(n) ) <= 2*n
found = False
for k in range(2*n,n,-1):
if euler_phi(k) == n:
found = True
break
if not found: raise \
ValueError('cyclotomic bases require that n is an image of' + \
'Euler\'s totient function')
R = ZZ_q['x'].quotient(cyclotomic_polynomial(k, 'x'), 'x')
for i in range(m//n):
A = A.stack(R.random_element().matrix())
# switch from representatives 0,...,(q-1) to (1-q)/2,....,(q-1)/2
def minrep(a):
if abs(a-q) < abs(a): return a-q
else: return a
A_prime = A[n:m].lift().apply_map(minrep)
if not dual:
B = block_matrix([[ZZ(q), ZZ.zero()], [A_prime, ZZ.one()] ], \
subdivide=False)
else:
B = block_matrix([[ZZ.one(), -A_prime.transpose()], [ZZ.zero(), \
ZZ(q)]], subdivide=False)
for i in range(m//2): B.swap_rows(i,m-i-1)
if not ntl:
return B
else:
return B._ntl_()
def saturation(A, proof=True, p=0, max_dets=5):
"""
Compute a saturation matrix of A.
INPUT:
- A -- a matrix over ZZ
- proof -- bool (default: True)
- p -- int (default: 0); if not 0 only guarantees that output is
p-saturated
- max_dets -- int (default: 4) max number of dets of submatrices to
compute.
OUTPUT:
matrix -- saturation of the matrix A.
EXAMPLES::
sage: from sage.matrix.matrix_integer_dense_saturation import saturation
sage: A = matrix(ZZ, 2, 2, [3,2,3,4]); B = matrix(ZZ, 2,3,[1,2,3,4,5,6]); C = A*B
sage: C
[11 16 21]
[19 26 33]
sage: C.index_in_saturation()
18
sage: S = saturation(C); S
[11 16 21]
[-2 -3 -4]
sage: S.index_in_saturation()
1
sage: saturation(C, proof=False)
[11 16 21]
[-2 -3 -4]
sage: saturation(C, p=2)
[11 16 21]
[-2 -3 -4]
sage: saturation(C, p=2, max_dets=1)
[11 16 21]
[-2 -3 -4]
"""
# Find a submatrix of full rank and instead saturate that matrix.
r = A.rank()
if A.is_square() and r == A.nrows():
return identity_matrix(ZZ, r)
if A.nrows() > r:
P = []
while len(P) < r:
P = matrix_integer_dense_hnf.probable_pivot_rows(A)
A = A.matrix_from_rows(P)
# Factor out all common factors from all rows, just in case.
A = copy(A)
A._factor_out_common_factors_from_each_row()
if A.nrows() <= 1:
return A
A, zero_cols = A._delete_zero_columns()
if max_dets > 0:
# Take the GCD of at most num_dets randomly chosen determinants.
nr = A.nrows()
nc = A.ncols()
d = 0
trials = min(binomial(nc, nr), max_dets)
already_tried = []
while len(already_tried) < trials:
v = random_sublist_of_size(nc, nr)
tm = verbose('saturation -- checking det condition on submatrix')
d = gcd(d, A.matrix_from_columns(v).determinant(proof=proof))
verbose('saturation -- got det down to %s' % d, tm)
if gcd(d, p) == 1:
return A._insert_zero_columns(zero_cols)
already_tried.append(v)
if gcd(d, p) == 1:
# already p-saturated
return A._insert_zero_columns(zero_cols)
# Factor and p-saturate at each p.
# This is not a good algorithm, because all the HNF's in it are really slow!
#
#tm = verbose('factoring gcd %s of determinants'%d)
#limit = 2**31-1
#F = d.factor(limit = limit)
#D = [p for p, e in F if p <= limit]
#B = [n for n, e in F if n > limit] # all big factors -- there will only be at most one
#assert len(B) <= 1
#C = B[0]
#for p in D:
# A = p_saturation(A, p=p, proof=proof)
# This is a really simple but powerful algorithm.
# FACT: If A is a matrix of full rank, then hnf(transpose(A))^(-1)*A is a saturation of A.
# To make this practical we use solve_system_with_difficult_last_row, since the
# last column of HNF's are typically the only really big ones.
B = A.transpose().hermite_form(include_zero_rows=False, proof=proof)
B = B.transpose()
# Now compute B^(-1) * A
C = solve_system_with_difficult_last_row(B, A)
return C.change_ring(ZZ)._insert_zero_columns(zero_cols)
def simplicial_action_generators(self):
r"""
Return action of generators of the Veech group on homology
"""
from sage.matrix.constructor import matrix, identity_matrix
tree, reps, word_reps, gens = self._veech_group._spanning_tree_verrill(
on_right=False)
n = self[0].nb_squares()
l_mat = identity_matrix(2 * n)
s_mat = matrix(2 * n)
s_mat[:n, n:] = -identity_matrix(n)
reps = [None] * len(tree)
reps_o = [None] * len(tree)
reps[0] = identity_matrix(2 * n)
reps_o[0] = self.origami()
waiting = [0]
while waiting:
x = waiting.pop()
for (e0, e1, label) in tree.outgoing_edges(x):
waiting.append(e1)
o = reps_o[e0]
if label == 's':
m = copy(s_mat)
m[n:, :n] = o.u().matrix().transpose()
reps[e1] = m * reps[e0]
reps_o[e1] = o.S_action()
elif label == 'l':
o = o.L_action()
m = copy(l_mat)
m[:n, n:] = (~o.u()).matrix().transpose()
reps[e1] = m * reps[e0]
reps_o[e1] = o
elif label == 'r':
o = o.R_action()
m = copy(l_mat)
m[n:, :n] = (~o.r()).matrix().transpose()
reps[e1] = m * reps[e0]
reps_o[e1] = o
else:
raise ValueError("does know how to do only with slr")
m_gens = []
for (e0, e1, label) in gens:
o = reps_o[e0]
if label == 's':
oo = o.S_action()
m = copy(s_mat)
m[n:, :n] = o.u().matrix().transpose()
elif label == 'l':
oo = o.L_action()
m = copy(l_mat)
m[:n, n:] = (~oo.u()).matrix().transpose()
elif label == 'r':
oo = o.R_action()
m = copy(l_mat)
m[:n, n:] = (~oo.r()).matrix().transpose()
else:
raise ValueError("does know how to do only with slr")
ooo = reps_o[e1]
ot, oo_renum = oo.standard_form(True)
os, ooo_renum = ooo.standard_form(True)
assert (ot == os)
m_ren = (oo_renum * ~ooo_renum).matrix().transpose()
m_s = matrix(2 * n)
m_s[:n, :n] = m_ren
m_s[n:, n:] = m_ren
m_gens.append(~reps[e1] * m_s * m * reps[e0])
return tree, reps, reps_o, gens, m_gens, word_reps
def multiplier(self, P, n, check=True):
r"""
Returns the multiplier of the point ``P`` of period ``n`` by the map.
The map must be an endomorphism.
INPUT:
- ``P`` - a point on domain of the map.
- ``n`` - a positive integer, the period of ``P``.
- ``check`` -- verify that ``P`` has period ``n``, Default:True.
OUTPUT:
- a square matrix of size ``self.codomain().dimension_relative()`` in
the ``base_ring`` of the map.
EXAMPLES::
sage: P.<x,y> = AffineSpace(QQ, 2)
sage: H = End(P)
sage: f = H([x^2, y^2])
sage: f.multiplier(P([1, 1]), 1)
[2 0]
[0 2]
::
sage: P.<x,y,z> = AffineSpace(QQ, 3)
sage: H = End(P)
sage: f = H([x, y^2, z^2 - y])
sage: f.multiplier(P([1/2, 1, 0]), 2)
[1 0 0]
[0 4 0]
[0 0 0]
::
sage: P.<x> = AffineSpace(CC, 1)
sage: H = End(P)
sage: f = H([x^2 + 1/2])
sage: f.multiplier(P([0.5 + 0.5*I]), 1)
[1.00000000000000 + 1.00000000000000*I]
::
sage: R.<t> = PolynomialRing(CC, 1)
sage: P.<x> = AffineSpace(R, 1)
sage: H = End(P)
sage: f = H([x^2 - t^2 + t])
sage: f.multiplier(P([-t + 1]), 1)
[(-2.00000000000000)*t + 2.00000000000000]
::
sage: P.<x,y> = AffineSpace(QQ, 2)
sage: X = P.subscheme([x^2-y^2])
sage: H = End(X)
sage: f = H([x^2, y^2])
sage: f.multiplier(X([1, 1]), 1)
[2 0]
[0 2]
"""
if not self.is_endomorphism():
raise TypeError("must be an endomorphism")
if check:
if self.nth_iterate(P, n) != P:
raise ValueError("%s is not periodic of period %s" % (P, n))
if n < 1:
raise ValueError("period must be a positive integer")
N = self.domain().ambient_space().dimension_relative()
l = identity_matrix(FractionField(self.codomain().base_ring()), N, N)
Q = P
J = self.jacobian()
for i in range(0, n):
R = self(Q)
l = J(tuple(Q)) * l #chain rule matrix multiplication
Q = R
return l
def calculate_voronoi_cell(basis, radius=None, verbose=False):
"""
Calculate the Voronoi cell of the lattice defined by basis
INPUT:
- ``basis`` -- embedded basis matrix of the lattice
- ``radius`` -- radius of basis vectors to consider
- ``verbose`` -- whether to print debug information
OUTPUT:
A :class:`Polyhedron` instance.
EXAMPLES::
sage: from sage.modules.diamond_cutting import calculate_voronoi_cell
sage: V = calculate_voronoi_cell(matrix([[1, 0], [0, 1]]))
sage: V.volume()
1
"""
dim = basis.dimensions()
artificial_length = None
if dim[0] < dim[1]:
# introduce "artificial" basis points (representing infinity)
artificial_length = max(abs(v) for v in basis).ceil() * 2
additional_vectors = identity_matrix(dim[1]) * artificial_length
basis = basis.stack(additional_vectors)
# LLL-reduce to get quadratic matrix
basis = basis.LLL()
basis = matrix([v for v in basis if v])
dim = basis.dimensions()
if dim[0] != dim[1]:
raise ValueError("invalid matrix")
basis = basis / 2
ieqs = []
for v in basis:
ieqs.append(plane_inequality(v))
ieqs.append(plane_inequality(-v))
Q = Polyhedron(ieqs=ieqs)
# twice the length of longest vertex in Q is a safe choice
if radius is None:
radius = 2 * max(abs(v)**2 for v in basis)
V = diamond_cut(Q, basis, radius, verbose=verbose)
if artificial_length is not None:
# remove inequalities introduced by artificial basis points
H = V.Hrepresentation()
H = [
v for v in H if all(not V._is_zero(v.A() * w / 2 - v.b())
and not V._is_zero(v.A() * (-w) / 2 - v.b())
for w in additional_vectors)
]
V = Polyhedron(ieqs=H)
return V
def gen_lattice(type='modular', n=4, m=8, q=11, seed=None,
quotient=None, dual=False, ntl=False, lattice=False):
r"""
This function generates different types of integral lattice bases
of row vectors relevant in cryptography.
Randomness can be set either with ``seed``, or by using
:func:`sage.misc.randstate.set_random_seed`.
INPUT:
- ``type`` -- one of the following strings
- ``'modular'`` (default) -- A class of lattices for which
asymptotic worst-case to average-case connections hold. For
more refer to [Aj1996]_.
- ``'random'`` -- Special case of modular (n=1). A dense class
of lattice used for testing basis reduction algorithms
proposed by Goldstein and Mayer [GM2002]_.
- ``'ideal'`` -- Special case of modular. Allows for a more
compact representation proposed by [LM2006]_.
- ``'cyclotomic'`` -- Special case of ideal. Allows for
efficient processing proposed by [LM2006]_.
- ``n`` -- Determinant size, primal:`det(L) = q^n`, dual:`det(L) = q^{m-n}`.
For ideal lattices this is also the degree of the quotient polynomial.
- ``m`` -- Lattice dimension, `L \subseteq Z^m`.
- ``q`` -- Coefficient size, `q-Z^m \subseteq L`.
- ``seed`` -- Randomness seed.
- ``quotient`` -- For the type ideal, this determines the quotient
polynomial. Ignored for all other types.
- ``dual`` -- Set this flag if you want a basis for `q-dual(L)`, for example
for Regev's LWE bases [Reg2005]_.
- ``ntl`` -- Set this flag if you want the lattice basis in NTL readable
format.
- ``lattice`` -- Set this flag if you want a
:class:`FreeModule_submodule_with_basis_integer` object instead
of an integer matrix representing the basis.
OUTPUT: ``B`` a unique size-reduced triangular (primal: lower_left,
dual: lower_right) basis of row vectors for the lattice in question.
EXAMPLES:
Modular basis::
sage: sage.crypto.gen_lattice(m=10, seed=42)
[11 0 0 0 0 0 0 0 0 0]
[ 0 11 0 0 0 0 0 0 0 0]
[ 0 0 11 0 0 0 0 0 0 0]
[ 0 0 0 11 0 0 0 0 0 0]
[ 2 4 3 5 1 0 0 0 0 0]
[ 1 -5 -4 2 0 1 0 0 0 0]
[-4 3 -1 1 0 0 1 0 0 0]
[-2 -3 -4 -1 0 0 0 1 0 0]
[-5 -5 3 3 0 0 0 0 1 0]
[-4 -3 2 -5 0 0 0 0 0 1]
Random basis::
sage: sage.crypto.gen_lattice(type='random', n=1, m=10, q=11^4, seed=42)
[14641 0 0 0 0 0 0 0 0 0]
[ 431 1 0 0 0 0 0 0 0 0]
[-4792 0 1 0 0 0 0 0 0 0]
[ 1015 0 0 1 0 0 0 0 0 0]
[-3086 0 0 0 1 0 0 0 0 0]
[-5378 0 0 0 0 1 0 0 0 0]
[ 4769 0 0 0 0 0 1 0 0 0]
[-1159 0 0 0 0 0 0 1 0 0]
[ 3082 0 0 0 0 0 0 0 1 0]
[-4580 0 0 0 0 0 0 0 0 1]
Ideal bases with quotient x^n-1, m=2*n are NTRU bases::
sage: sage.crypto.gen_lattice(type='ideal', seed=42, quotient=x^4-1)
[11 0 0 0 0 0 0 0]
[ 0 11 0 0 0 0 0 0]
[ 0 0 11 0 0 0 0 0]
[ 0 0 0 11 0 0 0 0]
[-2 -3 -3 4 1 0 0 0]
[ 4 -2 -3 -3 0 1 0 0]
[-3 4 -2 -3 0 0 1 0]
[-3 -3 4 -2 0 0 0 1]
Ideal bases also work with polynomials::
sage: R.<t> = PolynomialRing(ZZ)
sage: sage.crypto.gen_lattice(type='ideal', seed=1234, quotient=t^4-1)
[11 0 0 0 0 0 0 0]
[ 0 11 0 0 0 0 0 0]
[ 0 0 11 0 0 0 0 0]
[ 0 0 0 11 0 0 0 0]
[ 1 4 -3 3 1 0 0 0]
[ 3 1 4 -3 0 1 0 0]
[-3 3 1 4 0 0 1 0]
[ 4 -3 3 1 0 0 0 1]
Cyclotomic bases with n=2^k are SWIFFT bases::
sage: sage.crypto.gen_lattice(type='cyclotomic', seed=42)
[11 0 0 0 0 0 0 0]
[ 0 11 0 0 0 0 0 0]
[ 0 0 11 0 0 0 0 0]
[ 0 0 0 11 0 0 0 0]
[-2 -3 -3 4 1 0 0 0]
[-4 -2 -3 -3 0 1 0 0]
[ 3 -4 -2 -3 0 0 1 0]
[ 3 3 -4 -2 0 0 0 1]
Dual modular bases are related to Regev's famous public-key
encryption [Reg2005]_::
sage: sage.crypto.gen_lattice(type='modular', m=10, seed=42, dual=True)
[ 0 0 0 0 0 0 0 0 0 11]
[ 0 0 0 0 0 0 0 0 11 0]
[ 0 0 0 0 0 0 0 11 0 0]
[ 0 0 0 0 0 0 11 0 0 0]
[ 0 0 0 0 0 11 0 0 0 0]
[ 0 0 0 0 11 0 0 0 0 0]
[ 0 0 0 1 -5 -2 -1 1 -3 5]
[ 0 0 1 0 -3 4 1 4 -3 -2]
[ 0 1 0 0 -4 5 -3 3 5 3]
[ 1 0 0 0 -2 -1 4 2 5 4]
Relation of primal and dual bases::
sage: B_primal=sage.crypto.gen_lattice(m=10, q=11, seed=42)
sage: B_dual=sage.crypto.gen_lattice(m=10, q=11, seed=42, dual=True)
sage: B_dual_alt=transpose(11*B_primal.inverse()).change_ring(ZZ)
sage: B_dual_alt.hermite_form() == B_dual.hermite_form()
True
TESTS:
Test some bad quotient polynomials::
sage: sage.crypto.gen_lattice(type='ideal', seed=1234, quotient=cos(x))
Traceback (most recent call last):
...
TypeError: unable to convert cos(x) to an integer
sage: sage.crypto.gen_lattice(type='ideal', seed=1234, quotient=x^23-1)
Traceback (most recent call last):
...
ValueError: ideal basis requires n = quotient.degree()
sage: R.<u,v> = ZZ[]
sage: sage.crypto.gen_lattice(type='ideal', seed=1234, quotient=u+v)
Traceback (most recent call last):
...
TypeError: quotient should be a univariate polynomial
We are testing output format choices::
sage: sage.crypto.gen_lattice(m=10, q=11, seed=42)
[11 0 0 0 0 0 0 0 0 0]
[ 0 11 0 0 0 0 0 0 0 0]
[ 0 0 11 0 0 0 0 0 0 0]
[ 0 0 0 11 0 0 0 0 0 0]
[ 2 4 3 5 1 0 0 0 0 0]
[ 1 -5 -4 2 0 1 0 0 0 0]
[-4 3 -1 1 0 0 1 0 0 0]
[-2 -3 -4 -1 0 0 0 1 0 0]
[-5 -5 3 3 0 0 0 0 1 0]
[-4 -3 2 -5 0 0 0 0 0 1]
sage: sage.crypto.gen_lattice(m=10, q=11, seed=42, ntl=True)
[
[11 0 0 0 0 0 0 0 0 0]
[0 11 0 0 0 0 0 0 0 0]
[0 0 11 0 0 0 0 0 0 0]
[0 0 0 11 0 0 0 0 0 0]
[2 4 3 5 1 0 0 0 0 0]
[1 -5 -4 2 0 1 0 0 0 0]
[-4 3 -1 1 0 0 1 0 0 0]
[-2 -3 -4 -1 0 0 0 1 0 0]
[-5 -5 3 3 0 0 0 0 1 0]
[-4 -3 2 -5 0 0 0 0 0 1]
]
sage: sage.crypto.gen_lattice(m=10, q=11, seed=42, lattice=True)
Free module of degree 10 and rank 10 over Integer Ring
User basis matrix:
[ 0 0 1 1 0 -1 -1 -1 1 0]
[-1 1 0 1 0 1 1 0 1 1]
[-1 0 0 0 -1 1 1 -2 0 0]
[-1 -1 0 1 1 0 0 1 1 -1]
[ 1 0 -1 0 0 0 -2 -2 0 0]
[ 2 -1 0 0 1 0 1 0 0 -1]
[-1 1 -1 0 1 -1 1 0 -1 -2]
[ 0 0 -1 3 0 0 0 -1 -1 -1]
[ 0 -1 0 -1 2 0 -1 0 0 2]
[ 0 1 1 0 1 1 -2 1 -1 -2]
"""
from sage.rings.finite_rings.integer_mod_ring import IntegerModRing
from sage.matrix.constructor import identity_matrix, block_matrix
from sage.matrix.matrix_space import MatrixSpace
from sage.rings.integer_ring import IntegerRing
if seed is not None:
from sage.misc.randstate import set_random_seed
set_random_seed(seed)
if type == 'random':
if n != 1: raise ValueError('random bases require n = 1')
ZZ = IntegerRing()
ZZ_q = IntegerModRing(q)
A = identity_matrix(ZZ_q, n)
if type == 'random' or type == 'modular':
R = MatrixSpace(ZZ_q, m-n, n)
A = A.stack(R.random_element())
elif type == 'ideal':
if quotient is None:
raise ValueError('ideal bases require a quotient polynomial')
try:
quotient = quotient.change_ring(ZZ_q)
except (AttributeError, TypeError):
quotient = quotient.polynomial(base_ring=ZZ_q)
P = quotient.parent()
# P should be a univariate polynomial ring over ZZ_q
if not is_PolynomialRing(P):
raise TypeError("quotient should be a univariate polynomial")
assert P.base_ring() is ZZ_q
if quotient.degree() != n:
raise ValueError('ideal basis requires n = quotient.degree()')
R = P.quotient(quotient)
for i in range(m//n):
A = A.stack(R.random_element().matrix())
elif type == 'cyclotomic':
from sage.arith.all import euler_phi
from sage.misc.functional import cyclotomic_polynomial
# we assume that n+1 <= min( euler_phi^{-1}(n) ) <= 2*n
found = False
for k in range(2*n,n,-1):
if euler_phi(k) == n:
found = True
break
if not found:
raise ValueError("cyclotomic bases require that n "
"is an image of Euler's totient function")
R = ZZ_q['x'].quotient(cyclotomic_polynomial(k, 'x'), 'x')
for i in range(m//n):
A = A.stack(R.random_element().matrix())
# switch from representatives 0,...,(q-1) to (1-q)/2,....,(q-1)/2
def minrep(a):
if abs(a-q) < abs(a): return a-q
else: return a
A_prime = A[n:m].lift().apply_map(minrep)
if not dual:
B = block_matrix([[ZZ(q), ZZ.zero()], [A_prime, ZZ.one()] ],
subdivide=False)
else:
B = block_matrix([[ZZ.one(), -A_prime.transpose()],
[ZZ.zero(), ZZ(q)]], subdivide=False)
for i in range(m//2):
B.swap_rows(i,m-i-1)
if ntl and lattice:
raise ValueError("Cannot specify ntl=True and lattice=True "
"at the same time")
if ntl:
return B._ntl_()
elif lattice:
from sage.modules.free_module_integer import IntegerLattice
return IntegerLattice(B)
else:
return B