def __init__(self, begin, end, step=Integer(1), middle_point=Integer(1)):
r"""
TESTS::
sage: from sage.sets.integer_range import IntegerRangeFromMiddle
sage: I = IntegerRangeFromMiddle(-100,100,10,0)
sage: I.category()
Category of facade finite enumerated sets
sage: TestSuite(I).run()
sage: I = IntegerRangeFromMiddle(Infinity,-Infinity,-37,0)
sage: I.category()
Category of facade infinite enumerated sets
sage: TestSuite(I).run()
sage: IntegerRange(0, 5, 1, -3)
Traceback (most recent call last):
...
ValueError: middle_point is not in the interval
"""
self._begin = begin
self._end = end
self._step = step
self._middle_point = middle_point
if not middle_point in self:
raise ValueError("middle_point is not in the interval")
if (begin != Infinity and begin != -Infinity) and \
(end != Infinity and end != -Infinity):
Parent.__init__(self,
facade=IntegerRing(),
category=FiniteEnumeratedSets())
else:
Parent.__init__(self,
facade=IntegerRing(),
category=InfiniteEnumeratedSets())
def __dimension_Sp6Z(wt):
"""
Return the dimensions of subspaces of Siegel modular forms on $Sp(4,Z)$.
OUTPUT
("Total", "Miyawaki-Type-1", "Miyawaki-Type-2 (conjectured)", "Interesting")
Remember, Miywaki type 2 is ONLY CONJECTURED!!
"""
if not is_even(wt):
return (0, 0, 0, 0)
R = PowerSeriesRing(IntegerRing(), default_prec=wt + 1, names=('x', ))
(x, ) = R._first_ngens(1)
R = PowerSeriesRing(IntegerRing(), default_prec=2 * wt - 1, names=('y', ))
(y, ) = R._first_ngens(1)
H_all = 1 / (
(1 - x**4) * (1 - x**12)**2 * (1 - x**14) * (1 - x**18) * (1 - x**20) *
(1 - x**30)) * (1 + x**6 + x**10 + x**12 + 3 * x**16 + 2 * x**18 +
2 * x**20 + 5 * x**22 + 4 * x**24 + 5 * x**26 +
7 * x**28 + 6 * x**30 + 9 * x**32 + 10 * x**34 +
10 * x**36 + 12 * x**38 + 14 * x**40 + 15 * x**42 +
16 * x**44 + 18 * x**46 + 18 * x**48 + 19 * x**50 +
21 * x**52 + 19 * x**54 + 21 * x**56 + 21 * x**58 +
19 * x**60 + 21 * x**62 + 19 * x**64 + 18 * x**66 +
18 * x**68 + 16 * x**70 + 15 * x**72 + 14 * x**74 +
12 * x**76 + 10 * x**78 + 10 * x**80 + 9 * x**82 +
6 * x**84 + 7 * x**86 + 5 * x**88 + 4 * x**90 +
5 * x**92 + 2 * x**94 + 2 * x**96 + 3 * x**98 +
x**102 + x**104 + x**108 + x**114)
H_noncusp = 1 / (1 - x**4) / (1 - x**6) / (1 - x**10) / (1 - x**12)
H_E = y**12 / (1 - y**4) / (1 - y**6)
H_Miyawaki1 = H_E[wt] * H_E[2 * wt - 4]
H_Miyawaki2 = H_E[wt - 2] * H_E[2 * wt - 2]
a, b, c, d = H_all[wt], H_noncusp[wt], H_Miyawaki1, H_Miyawaki2
return (a, c, d, a - b - c - d)
def __getitem__(self, level):
"""
Return the modular polynomial of given level, or an error if
there is no such polynomial in the database.
EXAMPLES:
sage: DBMP = ClassicalModularPolynomialDatabase() # optional - database_kohel
sage: f = DBMP[29] # optional - database_kohel
sage: f.degree() # optional - database_kohel
58
sage: f.coefficient([28,28]) # optional - database_kohel
400152899204646997840260839128
sage: DBMP[50] # optional - database_kohel
Traceback (most recent call last):
...
RuntimeError: No database entry for modular polynomial of level 50
"""
if self.model in ("Atk", "Eta"):
level = Integer(level)
if not level.is_prime():
raise TypeError("Argument level (= %s) must be prime." % level)
elif self.model in ("AtkCrr", "EtaCrr"):
N = Integer(level[0])
if not N in (2, 3, 5, 7, 13):
raise TypeError("Argument level (= %s) must be prime." % N)
modpol = self._dbpath(level)
try:
coeff_list = _dbz_to_integer_list(modpol)
except RuntimeError as msg:
print(msg)
raise RuntimeError(
"No database entry for modular polynomial of level %s" % level)
if self.model == "Cls":
P = PolynomialRing(IntegerRing(), 2, "j")
else:
P = PolynomialRing(IntegerRing(), 2, "x,j")
poly = {}
if self.model == "Cls":
if level == 1:
return P({(1, 0): 1, (0, 1): -1})
for cff in coeff_list:
i = cff[0]
j = cff[1]
poly[(i, j)] = Integer(cff[2])
if i != j:
poly[(j, i)] = Integer(cff[2])
else:
for cff in coeff_list:
poly[(cff[0], cff[1])] = Integer(cff[2])
return P(poly)
def __getitem__(self, level):
"""
Return the modular polynomial of given level, or an error if
there is no such polynomial in the database.
EXAMPLES::
sage: DBMP = ClassicalModularPolynomialDatabase()
sage: f = DBMP[29] # optional - database_kohel
sage: f.degree() # optional - database_kohel
58
sage: f.coefficient([28,28]) # optional - database_kohel
400152899204646997840260839128
sage: DBMP[50] # optional - database_kohel
Traceback (most recent call last):
...
LookupError: filename .../kohel/PolMod/Cls/pol.050.dbz does not exist
"""
from sage.rings.integer import Integer
from sage.rings.integer_ring import IntegerRing
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
if self.model in ("Atk", "Eta"):
level = Integer(level)
if not level.is_prime():
raise TypeError("Argument level (= %s) must be prime." % level)
elif self.model in ("AtkCrr", "EtaCrr"):
N = Integer(level[0])
if not N in (2, 3, 5, 7, 13):
raise TypeError("Argument level (= %s) must be prime." % N)
modpol = self._dbpath(level)
coeff_list = _dbz_to_integer_list(modpol)
if self.model == "Cls":
P = PolynomialRing(IntegerRing(), 2, "j")
else:
P = PolynomialRing(IntegerRing(), 2, "x,j")
poly = {}
if self.model == "Cls":
if level == 1:
return P({(1, 0): 1, (0, 1): -1})
for cff in coeff_list:
i = cff[0]
j = cff[1]
poly[(i, j)] = Integer(cff[2])
if i != j:
poly[(j, i)] = Integer(cff[2])
else:
for cff in coeff_list:
poly[(cff[0], cff[1])] = Integer(cff[2])
return P(poly)
def global_genus_symbol(self):
"""
Returns the genus of a two times a quadratic form over ZZ. These
are defined by a collection of local genus symbols (a la Chapter
15 of Conway-Sloane), and a signature.
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,2,3,4])
sage: Q.global_genus_symbol()
Genus of [2 0 0 0]
[0 4 0 0]
[0 0 6 0]
[0 0 0 8]
::
sage: Q = QuadraticForm(ZZ, 4, range(10))
sage: Q.global_genus_symbol()
Genus of [ 0 1 2 3]
[ 1 8 5 6]
[ 2 5 14 8]
[ 3 6 8 18]
"""
## Check that the form is defined over ZZ
if not self.base_ring() == IntegerRing():
raise TypeError, "Oops! The quadratic form is not defined over the integers."
## Return the result
try:
return Genus(self.Hessian_matrix())
except Exception:
raise TypeError, "Oops! There is a problem computing the genus symbols for this form."
def __init__(self):
r"""
TESTS::
sage: TestSuite(SetsWithGrading().example()).run()
"""
Parent.__init__(self, category=SetsWithGrading(), facade=IntegerRing())
class ClassPolynomialDatabase:
def _dbpath(self, disc, level=1):
"""
TESTS:
sage: db = HilbertClassPolynomialDatabase()
sage: db.__getitem__(-23,level=2)
Traceback (most recent call last):
...
NotImplementedError: Level (= 2) > 1 not yet implemented.
"""
if level != 1:
raise NotImplementedError, "Level (= %s) > 1 not yet implemented." % level
n1 = 5000 * ((abs(disc) - 1) // 5000)
s1 = _pad_int_str(n1 + 1, disc_length)
s2 = _pad_int_str(n1 + 5000, disc_length)
subdir = "%s-%s" % (s1, s2)
discstr = _pad_int_str(abs(disc), disc_length)
return "PolHeeg/%s/%s/pol.%s.dbz" % (self.model, subdir, discstr)
def __getitem__(self, disc, level=1, var='x'):
classpol = self._dbpath(disc, level)
try:
coeff_list = _dbz_to_integers(classpol)
except RuntimeError, msg:
print msg
raise RuntimeError, \
"No database entry for class polynomial of discriminant %s"%disc
P = PolynomialRing(IntegerRing(), names=var)
return P(list(coeff_list))
def __init__(self):
"""
TESTS::
sage: P = Sets().example("inherits")
"""
Parent.__init__(self, facade=IntegerRing(), category=Sets())
def _dimension_Sp4Z(wt_range):
"""
Return the dimensions of subspaces of Siegel modular forms on $Sp(4,Z)$.
OUTPUT
("Total", "Eisenstein", "Klingen", "Maass", "Interesting")
"""
headers = ['Total', 'Eisenstein', 'Klingen', 'Maass', 'Interesting']
R = PowerSeriesRing(IntegerRing(),
default_prec=wt_range[-1] + 1,
names=('x', ))
(x, ) = R._first_ngens(1)
H_all = 1 / (1 - x**4) / (1 - x**6) / (1 - x**10) / (1 - x**12)
H_Kl = x**12 / (1 - x**4) / (1 - x**6)
H_MS = (x**10 + x**12) / (1 - x**4) / (1 - x**6)
dct = dict(
(k, {
'Total': H_all[k],
'Eisenstein': 1 if k >= 4 else 0,
'Klingen': H_Kl[k],
'Maass': H_MS[k],
'Interesting': H_all[k] - (1 if k >= 4 else 0) - H_Kl[k] - H_MS[k]
} if is_even(k) else {
'Total': H_all[k - 35],
'Eisenstein': 0,
'Klingen': 0,
'Maass': 0,
'Interesting': H_all[k - 35]
}) for k in wt_range)
return headers, dct
def __getitem__(self,disc,level=1,var='x'):
classpol = self._dbpath(disc,level)
try:
coeff_list = _dbz_to_integers(classpol)
except RuntimeError as msg:
print(msg)
raise RuntimeError("No database entry for class polynomial of discriminant %s"%disc)
P = PolynomialRing(IntegerRing(),names=var)
return P(list(coeff_list))
def __init__(self):
"""
TESTS::
sage: P = Sets().example("inherits")
sage: type(P(13)+P(17))
<type 'sage.rings.integer.Integer'>
sage: type(P(2)+P(3))
<type 'sage.rings.integer.Integer'>
"""
super(PrimeNumbers_Inherits, self).__init__()
self._populate_coercion_lists_(embedding=IntegerRing())
def __init__(self):
"""
TESTS::
sage: from sage.categories.examples.sets_cat import PrimeNumbers
sage: P = PrimeNumbers()
sage: P.category()
Category of facade sets
sage: P is Sets().example()
True
"""
Parent.__init__(self, facade=IntegerRing(), category=Sets())
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 __init__(self, begin, step=Integer(1)):
r"""
TESTS::
sage: I = IntegerRange(-57,Infinity,8)
sage: I.category()
Category of facade infinite enumerated sets
sage: TestSuite(I).run()
"""
assert isinstance(begin, Integer)
self._begin = begin
self._step = step
Parent.__init__(self, facade = IntegerRing(), category = InfiniteEnumeratedSets())
def __init__(self, begin, end, step=Integer(1)):
r"""
TESTS::
sage: I = IntegerRange(123,12,-4)
sage: I.category()
Category of facade finite enumerated sets
sage: TestSuite(I).run()
"""
self._begin = begin
self._end = end
self._step = step
Parent.__init__(self, facade = IntegerRing(), category = FiniteEnumeratedSets())
def __init__(self):
"""
TESTS::
sage: C = FiniteEnumeratedSets().example()
sage: C
An example of a finite enumerated set: {1,2,3}
sage: C.category()
Category of facade finite enumerated sets
sage: TestSuite(C).run()
"""
self._set = map(Integer, [1,2,3])
Parent.__init__(self, facade = IntegerRing(), category = FiniteEnumeratedSets())
def __init__(self, ambient = Example()):
"""
TESTS::
sage: C = FiniteEnumeratedSets().IsomorphicObjects().example()
sage: C
The image by some isomorphism of An example of a finite enumerated set: {1,2,3}
sage: C.category()
Category of facade isomorphic objects of finite enumerated sets
sage: TestSuite(C).run()
"""
self._ambient = ambient
Parent.__init__(self, facade = IntegerRing(), category = FiniteEnumeratedSets().IsomorphicObjects())
def _dimension_Gamma0_4(wt):
"""
Return the dimensions of subspaces of Siegel modular forms on $Gamma0(4)$.
OUTPUT
( "Total",)
REMARK
Not completely implemented
"""
R = PowerSeriesRing(IntegerRing(), default_prec=wt + 1, names=('x', ))
(x, ) = R._first_ngens(1)
H_all = (1 + x**4)(1 + x**11) / (1 - x**2)**3 / (1 - x**6)
return (H_all[wt], )
def __init__(self, begin, step=Integer(1)):
r"""
TESTS::
sage: I = IntegerRange(-57,Infinity,8)
sage: I.category()
Category of facade infinite enumerated sets
sage: TestSuite(I).run()
"""
if not isinstance(begin, Integer):
raise TypeError("begin should be Integer, not %r" % type(begin))
self._begin = begin
self._step = step
Parent.__init__(self, facade = IntegerRing(), category = InfiniteEnumeratedSets())
def tropical_coordinates(self):
r"""
Return the tropical coordinates of ``self`` in the braid group `B_n`.
OUTPUT:
- a list of `2n` tropical integers
EXAMPLES::
sage: B = BraidGroup(3)
sage: b = B([1])
sage: tc = b.tropical_coordinates(); tc
[1, 0, 0, 2, 0, 1]
sage: tc[0].parent()
Tropical semiring over Integer Ring
sage: b = B([-2, -2, -1, -1, 2, 2, 1, 1])
sage: b.tropical_coordinates()
[1, -19, -12, 9, 0, 13]
REFERENCES:
.. [Dynnikov07] I. Dynnikov and B. Wiest, On the complexity of braids,
J. Europ. Math. Soc. 9 (2007)
.. [Dehornoy] P. Dehornoy, Le probleme d'isotopie des tresses, in
lecons de mathematiques d'aujourd'hui vol. 4
"""
coord = [0, 1] * self.strands()
for s in self.Tietze():
k = 2 * (abs(s) - 1)
x1, y1, x2, y2 = coord[k:k + 4]
if s > 0:
sign = 1
z = x1 - min(y1, 0) - x2 + max(y2, 0)
coord[k + 1] = y2 - max(z, 0)
coord[k + 3] = y1 + max(z, 0)
else:
sign = -1
z = x1 + min(y1, 0) - x2 - max(y2, 0)
coord[k + 1] = y2 + min(z, 0)
coord[k + 3] = y1 - min(z, 0)
coord[k] = x1 + sign * (max(y1, 0) + max(max(y2, 0) - sign * z, 0))
coord[k +
2] = x2 + sign * (min(y2, 0) + min(min(y1, 0) + sign * z, 0))
from sage.rings.semirings.tropical_semiring import TropicalSemiring
T = TropicalSemiring(IntegerRing())
return map(T, coord)
def _dimension_Gamma0_3(wt):
"""
Return the dimensions of subspaces of Siegel modular forms on $Gamma0(3)$.
OUTPUT
( "Total")
REMARK
Only total dimension implemented.
"""
R = PowerSeriesRing(IntegerRing(), default_prec=wt + 1, names=('x', ))
(x, ) = R._first_ngens(1)
H_all = (1 + 2 * x**4 + x**6 + x**15 *
(1 + 2 * x**2 + x**6)) / (1 - x**2) / (1 - x**4) / (1 - x**6)**2
return (H_all[wt], )
def __init__(self, field):
"""
Initialize.
TESTS::
sage: K.<x> = FunctionField(GF(5)); _.<Y> = K[]
sage: F.<y> = K.extension(Y^2 - x^3 - 1)
sage: G = F.divisor_group()
sage: TestSuite(G).run()
"""
Parent.__init__(self,
base=IntegerRing(),
category=CommutativeAdditiveGroups())
self._field = field # function field
def generating_series(self, var='z'):
r"""
Return `1 / (1-z)`.
EXAMPLES::
sage: N = SetsWithGrading().example(); N
Non negative integers
sage: f = N.generating_series(); f
1/(-z + 1)
sage: LaurentSeriesRing(ZZ,'z')(f)
1 + z + z^2 + z^3 + z^4 + z^5 + z^6 + z^7 + z^8 + z^9 + z^10 + z^11 + z^12 + z^13 + z^14 + z^15 + z^16 + z^17 + z^18 + z^19 + O(z^20)
"""
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.rings.integer import Integer
R = PolynomialRing(IntegerRing(), var)
z = R.gen()
return Integer(1) / (Integer(1) - z)
def __init__( self, doc):
self.__collection = doc.get( 'collection')
self.__name = doc.get( 'name')
weight = doc.get( 'weight')
self.__weight = sage_eval( weight) if weight else weight
field = doc.get( 'field')
R = PolynomialRing( IntegerRing(), name = 'x')
self.__field = sage_eval( field, locals = R.gens_dict()) if field else field
self.__explicit_formula = doc.get( 'explicit_formula')
self.__type = doc.get( 'type')
self.__is_eigenform = doc.get( 'is_eigenform')
self.__is_integral = doc.get( 'is_integral')
self.__representation = doc.get( 'representation')
self.__id = doc.get( '_id')
def __init__(self, ainvs, verbose=False):
r"""
Create the mwrank elliptic curve with invariants
``ainvs``, which is a list of 5 or less *integers* `a_1`,
`a_2`, `a_3`, `a_4`, and `a_5`.
See the docstring of this class for full documentation.
EXAMPLES:
We create the elliptic curve `y^2 + y = x^3 + x^2 - 2x`::
sage: e = mwrank_EllipticCurve([0, 1, 1, -2, 0])
sage: e.ainvs()
[0, 1, 1, -2, 0]
"""
# import here to save time during startup (mwrank takes a while to init)
from sage.libs.mwrank.mwrank import _Curvedata
# if not isinstance(ainvs, list) and len(ainvs) <= 5:
if not isinstance(ainvs, (list,tuple)) or not len(ainvs) <= 5:
raise TypeError, "ainvs must be a list or tuple of length at most 5."
# Pad ainvs on the beginning by 0's, so e.g.
# [a4,a5] works.
ainvs = [0]*(5-len(ainvs)) + ainvs
# Convert each entry to an int
try:
a_int = [IntegerRing()(x) for x in ainvs]
except (TypeError, ValueError):
raise TypeError, "ainvs must be a list or tuple of integers."
self.__ainvs = a_int
self.__curve = _Curvedata(a_int[0], a_int[1], a_int[2],
a_int[3], a_int[4])
if verbose:
self.__verbose = True
else:
self.__verbose = False
# place holders
self.__saturate = -2 # not yet saturated
def fox_derivative(self, gen):
"""
Return the Fox derivative of self with respect to a given generator.
INPUT:
- ``gen`` : the generator with respect to which the derivative will be computed.
OUTPUT:
An element of the group algebra with integer coefficients.
EXAMPLES::
sage: G = FreeGroup(5)
sage: G.inject_variables()
Defining x0, x1, x2, x3, x4
sage: (~x0*x1*x0*x2*~x0).fox_derivative(x0)
-B[x0^-1] + B[x0^-1*x1] - B[x0^-1*x1*x0*x2*x0^-1]
sage: (~x0*x1*x0*x2*~x0).fox_derivative(x1)
B[x0^-1]
sage: (~x0*x1*x0*x2*~x0).fox_derivative(x2)
B[x0^-1*x1*x0]
sage: (~x0*x1*x0*x2*~x0).fox_derivative(x3)
0
"""
if not gen in self.parent().generators():
raise ValueError(
"Fox derivative can only be computed with respect to generators of the group"
)
l = list(self.Tietze())
R = self.parent().algebra(IntegerRing())
i = gen.Tietze()[0]
a = R.zero()
while len(l) > 0:
b = l.pop(-1)
if b == i:
p = R(self.parent()(l))
a += p
elif b == -i:
p = R(self.parent()(l + [b]))
a -= p
return a
def __init__(self):
"""
TESTS::
sage: P = Sets().example("wrapper")
sage: P.category()
Category of sets
sage: P(13) == 13
True
sage: ZZ(P(13)) == 13
True
sage: P(13) + 1 == 14
True
"""
Parent.__init__(self, category=Sets())
from sage.rings.integer_ring import IntegerRing
from sage.categories.homset import Hom
self.mor = Hom(self, IntegerRing())(lambda z: z.value)
self._populate_coercion_lists_(embedding=self.mor)
def __init__(self, field):
"""
Initialize.
INPUT:
- ``field`` -- function field
EXAMPLES::
sage: K.<x> = FunctionField(GF(5)); _.<t> = PolynomialRing(K)
sage: F.<y> = K.extension(t^2-x^3-1)
sage: F.divisor_group()
Divisor group of Function field in y defined by y^2 + 4*x^3 + 4
"""
Parent.__init__(self,
base=IntegerRing(),
category=CommutativeAdditiveGroups())
self._field = field # function field
def _dimension_Gamma0_3_psi_3(wt):
"""
Return the dimensions of the space of Siegel modular forms
on $Gamma_0(3)$ with character $\psi_3$.
OUTPUT
( "Total")
REMARK
Not completely implemented
"""
R = PowerSeriesRing(IntegerRing(), default_prec=wt + 1, names=('x', ))
(x, ) = R._first_ngens(1)
B = 1 / (1 - x**1) / (1 - x**3) / (1 - x**4) / (1 - x**3)
H_all_odd = B
H_all_even = B * x**14
if is_even(wt):
return (H_all_even[wt], )
else:
return (H_all_odd[wt], )
def __init__(self, ainvs, verbose=False):
r"""
Create the mwrank elliptic curve with invariants
``ainvs``, which is a list of 5 or less *integers* `a_1`,
`a_2`, `a_3`, `a_4`, and `a_5`.
See the docstring of this class for full documentation.
EXAMPLES:
We create the elliptic curve `y^2 + y = x^3 + x^2 - 2x`::
sage: e = mwrank_EllipticCurve([0, 1, 1, -2, 0])
sage: e.ainvs()
[0, 1, 1, -2, 0]
"""
ainvs = list(ainvs)
if len(ainvs) > 5:
raise TypeError("ainvs must have length at most 5")
# Pad ainvs on the beginning by 0's, so e.g. [a4, a6] works
ainvs = [0] * (5 - len(ainvs)) + ainvs
# Convert each entry to an int
try:
a_int = [IntegerRing()(x) for x in ainvs]
except (TypeError, ValueError):
raise TypeError("ainvs must be a list or tuple of integers.")
self.__ainvs = a_int
self.__curve = _Curvedata(a_int[0], a_int[1], a_int[2],
a_int[3], a_int[4])
if verbose:
self.__verbose = True
else:
self.__verbose = False
# place holders
self.__saturate = -2 # not yet saturated
self.__descent = None
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 gen_lattice(type='modular', n=4, m=8, q=11, seed=None,
quotient=None, dual=False, ntl=False, lattice=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`` -- 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 [R05]_.
- ``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]
[ 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]
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]
[ 4 1 4 -3 1 0 0 0]
[-3 4 1 4 0 1 0 0]
[ 4 -3 4 1 0 0 1 0]
[ 1 4 -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
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]
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 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
