def test_lfsr_sequence():
raises(TypeError, lambda: lfsr_sequence(1, [1], 1))
raises(TypeError, lambda: lfsr_sequence([1], 1, 1))
F = FF(2)
assert lfsr_sequence([F(1)], [F(1)], 2) == [F(1), F(1)]
assert lfsr_sequence([F(0)], [F(1)], 2) == [F(1), F(0)]
F = FF(3)
assert lfsr_sequence([F(1)], [F(1)], 2) == [F(1), F(1)]
assert lfsr_sequence([F(0)], [F(2)], 2) == [F(2), F(0)]
assert lfsr_sequence([F(1)], [F(2)], 2) == [F(2), F(2)]
def test_Domain_preprocess():
assert Domain.preprocess(ZZ) == ZZ
assert Domain.preprocess(QQ) == QQ
assert Domain.preprocess(EX) == EX
assert Domain.preprocess(FF(2)) == FF(2)
assert Domain.preprocess(ZZ[x, y]) == ZZ[x, y]
assert Domain.preprocess('Z') == ZZ
assert Domain.preprocess('Q') == QQ
assert Domain.preprocess('ZZ') == ZZ
assert Domain.preprocess('QQ') == QQ
assert Domain.preprocess('EX') == EX
assert Domain.preprocess('FF(23)') == FF(23)
assert Domain.preprocess('GF(23)') == GF(23)
raises(OptionError, lambda: Domain.preprocess('Z[]'))
assert Domain.preprocess('Z[x]') == ZZ[x]
assert Domain.preprocess('Q[x]') == QQ[x]
assert Domain.preprocess('ZZ[x]') == ZZ[x]
assert Domain.preprocess('QQ[x]') == QQ[x]
assert Domain.preprocess('Z[x,y]') == ZZ[x, y]
assert Domain.preprocess('Q[x,y]') == QQ[x, y]
assert Domain.preprocess('ZZ[x,y]') == ZZ[x, y]
assert Domain.preprocess('QQ[x,y]') == QQ[x, y]
raises(OptionError, lambda: Domain.preprocess('Z()'))
assert Domain.preprocess('Z(x)') == ZZ.frac_field(x)
assert Domain.preprocess('Q(x)') == QQ.frac_field(x)
assert Domain.preprocess('ZZ(x)') == ZZ.frac_field(x)
assert Domain.preprocess('QQ(x)') == QQ.frac_field(x)
assert Domain.preprocess('Z(x,y)') == ZZ.frac_field(x, y)
assert Domain.preprocess('Q(x,y)') == QQ.frac_field(x, y)
assert Domain.preprocess('ZZ(x,y)') == ZZ.frac_field(x, y)
assert Domain.preprocess('QQ(x,y)') == QQ.frac_field(x, y)
assert Domain.preprocess('Q<I>') == QQ.algebraic_field(I)
assert Domain.preprocess('QQ<I>') == QQ.algebraic_field(I)
assert Domain.preprocess('Q<sqrt(2), I>') == QQ.algebraic_field(sqrt(2), I)
assert Domain.preprocess('QQ<sqrt(2), I>') == QQ.algebraic_field(
sqrt(2), I)
raises(OptionError, lambda: Domain.preprocess('abc'))
def test_supplement_a_subspace_1():
M = DM([[1, 7, 0], [2, 3, 4]], QQ).transpose()
# First supplement over QQ:
B = supplement_a_subspace(M)
assert B[:, :2] == M
assert B[:, 2] == DomainMatrix.eye(3, QQ).to_dense()[:, 0]
# Now supplement over FF(7):
M = M.convert_to(FF(7))
B = supplement_a_subspace(M)
assert B[:, :2] == M
# When we work mod 7, first col of M goes to [1, 0, 0],
# so the supplementary vector cannot equal this, as it did
# when we worked over QQ. Instead, we get the second std basis vector:
assert B[:, 2] == DomainMatrix.eye(3, FF(7)).to_dense()[:, 1]
def test_lfsr_connection_polynomial():
F = FF(2)
x = symbols("x")
s = lfsr_sequence([F(1), F(0)], [F(0), F(1)], 5)
assert lfsr_connection_polynomial(s) == x**2 + 1
s = lfsr_sequence([F(1), F(1)], [F(0), F(1)], 5)
assert lfsr_connection_polynomial(s) == x**2 + x + 1
def test_dup_mul():
assert dup_mul([], [], ZZ) == []
assert dup_mul([], [ZZ(1)], ZZ) == []
assert dup_mul([ZZ(1)], [], ZZ) == []
assert dup_mul([ZZ(1)], [ZZ(1)], ZZ) == [ZZ(1)]
assert dup_mul([ZZ(5)], [ZZ(7)], ZZ) == [ZZ(35)]
assert dup_mul([], [], QQ) == []
assert dup_mul([], [QQ(1,2)], QQ) == []
assert dup_mul([QQ(1,2)], [], QQ) == []
assert dup_mul([QQ(1,2)], [QQ(4,7)], QQ) == [QQ(2,7)]
assert dup_mul([QQ(5,7)], [QQ(3,7)], QQ) == [QQ(15,49)]
f = dup_normal([3,0,0,6,1,2], ZZ)
g = dup_normal([4,0,1,0], ZZ)
h = dup_normal([12,0,3,24,4,14,1,2,0], ZZ)
assert dup_mul(f, g, ZZ) == h
assert dup_mul(g, f, ZZ) == h
f = dup_normal([2,0,0,1,7], ZZ)
h = dup_normal([4,0,0,4,28,0,1,14,49], ZZ)
assert dup_mul(f, f, ZZ) == h
K = FF(6)
assert dup_mul([K(2),K(1)], [K(3),K(4)], K) == [K(5),K(4)]
def __init__(self, a4, a6, a1=0, a2=0, a3=0, modulus=0):
if modulus == 0:
domain = QQ
else:
domain = FF(modulus)
a1, a2, a3, a4, a6 = map(domain.convert, (a1, a2, a3, a4, a6))
self._domain = domain
self.modulus = modulus
# Calculate discriminant
b2 = a1**2 + 4 * a2
b4 = 2 * a4 + a1 * a3
b6 = a3**2 + 4 * a6
b8 = a1**2 * a6 + 4 * a2 * a6 - a1 * a3 * a4 + a2 * a3**2 - a4**2
self._b2, self._b4, self._b6, self._b8 = b2, b4, b6, b8
self._discrim = -b2**2 * b8 - 8 * b4**3 - 27 * b6**2 + 9 * b2 * b4 * b6
self._a1 = a1
self._a2 = a2
self._a3 = a3
self._a4 = a4
self._a6 = a6
x, y, z = symbols('x y z')
self.x, self.y, self.z = x, y, z
self._eq = Eq(y**2 * z + a1 * x * y * z + a3 * y * z**2,
x**3 + a2 * x**2 * z + a4 * x * z**2 + a6 * z**3)
if isinstance(self._domain, FiniteField):
self._rank = 0
elif isinstance(self._domain, RationalField):
self._rank = None
def test_dmp_diff():
assert dmp_diff([], 1, 0, ZZ) == []
assert dmp_diff([[]], 1, 1, ZZ) == [[]]
assert dmp_diff([[[]]], 1, 2, ZZ) == [[[]]]
assert dmp_diff([[[1], [2]]], 1, 2, ZZ) == [[[]]]
assert dmp_diff([[[1]], [[]]], 1, 2, ZZ) == [[[1]]]
assert dmp_diff([[[3]], [[1]], [[]]], 1, 2, ZZ) == [[[6]], [[1]]]
assert dmp_diff([1, -1, 0, 0, 2], 1, 0, ZZ) == \
dup_diff([1, -1, 0, 0, 2], 1, ZZ)
assert dmp_diff(f_6, 0, 3, ZZ) == f_6
assert dmp_diff(f_6, 1, 3, ZZ) == [[[[8460]], [[]]],
[[[135, 0, 0], [], [], [-135, 0, 0]]],
[[[]]],
[[[-423]], [[-47]], [[]],
[[141], [], [94, 0], []], [[]]]]
assert dmp_diff(f_6, 2, 3, ZZ) == dmp_diff(dmp_diff(f_6, 1, 3, ZZ), 1, 3,
ZZ)
assert dmp_diff(f_6, 3, 3, ZZ) == dmp_diff(
dmp_diff(dmp_diff(f_6, 1, 3, ZZ), 1, 3, ZZ), 1, 3, ZZ)
K = FF(23)
F_6 = dmp_normal(f_6, 3, K)
assert dmp_diff(F_6, 0, 3, K) == F_6
assert dmp_diff(F_6, 1, 3, K) == dmp_diff(F_6, 1, 3, K)
assert dmp_diff(F_6, 2, 3, K) == dmp_diff(dmp_diff(F_6, 1, 3, K), 1, 3, K)
assert dmp_diff(F_6, 3, 3,
K) == dmp_diff(dmp_diff(dmp_diff(F_6, 1, 3, K), 1, 3, K),
1, 3, K)
def test_dup_diff():
assert dup_diff([], 1, ZZ) == []
assert dup_diff([7], 1, ZZ) == []
assert dup_diff([2, 7], 1, ZZ) == [2]
assert dup_diff([1, 2, 1], 1, ZZ) == [2, 2]
assert dup_diff([1, 2, 3, 4], 1, ZZ) == [3, 4, 3]
assert dup_diff([1, -1, 0, 0, 2], 1, ZZ) == [4, -3, 0, 0]
f = dup_normal([17, 34, 56, -345, 23, 76, 0, 0, 12, 3, 7], ZZ)
assert dup_diff(f, 0, ZZ) == f
assert dup_diff(f, 1, ZZ) == [170, 306, 448, -2415, 138, 380, 0, 0, 24, 3]
assert dup_diff(f, 2, ZZ) == dup_diff(dup_diff(f, 1, ZZ), 1, ZZ)
assert dup_diff(f, 3, ZZ) == dup_diff(dup_diff(dup_diff(f, 1, ZZ), 1, ZZ),
1, ZZ)
K = FF(3)
f = dup_normal([17, 34, 56, -345, 23, 76, 0, 0, 12, 3, 7], K)
assert dup_diff(f, 1, K) == dup_normal([2, 0, 1, 0, 0, 2, 0, 0, 0, 0], K)
assert dup_diff(f, 2, K) == dup_normal([1, 0, 0, 2, 0, 0, 0], K)
assert dup_diff(f, 3, K) == dup_normal([], K)
assert dup_diff(f, 0, K) == f
assert dup_diff(f, 2, K) == dup_diff(dup_diff(f, 1, K), 1, K)
assert dup_diff(f, 3, K) == dup_diff(dup_diff(dup_diff(f, 1, K), 1, K), 1,
K)
def test_dup_factor_list():
assert dup_factor_list([], ZZ) == (ZZ(0), [])
assert dup_factor_list([], QQ) == (QQ(0), [])
assert dup_factor_list([], ZZ['y']) == (DMP([],ZZ), [])
assert dup_factor_list([], QQ['y']) == (DMP([],QQ), [])
assert dup_factor_list_include([], ZZ) == [([], 1)]
assert dup_factor_list([ZZ(7)], ZZ) == (ZZ(7), [])
assert dup_factor_list([QQ(1,7)], QQ) == (QQ(1,7), [])
assert dup_factor_list([DMP([ZZ(7)],ZZ)], ZZ['y']) == (DMP([ZZ(7)],ZZ), [])
assert dup_factor_list([DMP([QQ(1,7)],QQ)], QQ['y']) == (DMP([QQ(1,7)],QQ), [])
assert dup_factor_list_include([ZZ(7)], ZZ) == [([ZZ(7)], 1)]
assert dup_factor_list([ZZ(1),ZZ(2),ZZ(1)], ZZ) == \
(ZZ(1), [([ZZ(1), ZZ(1)], 2)])
assert dup_factor_list([QQ(1,2),QQ(1),QQ(1,2)], QQ) == \
(QQ(1,2), [([QQ(1),QQ(1)], 2)])
assert dup_factor_list_include([ZZ(1),ZZ(2),ZZ(1)], ZZ) == \
[([ZZ(1), ZZ(1)], 2)]
K = FF(2)
assert dup_factor_list([K(1),K(0),K(1)], K) == \
(K(1), [([K(1), K(1)], 2)])
assert dup_factor_list([RR(1.0),RR(2.0),RR(1.0)], RR) == \
(RR(1.0), [([RR(1.0),RR(1.0)], 2)])
assert dup_factor_list([RR(2.0),RR(4.0),RR(2.0)], RR) == \
(RR(2.0), [([RR(1.0),RR(1.0)], 2)])
f = [DMP([ZZ(4),ZZ(0)],ZZ),DMP([ZZ(4),ZZ(0),ZZ(0)],ZZ),DMP([],ZZ)]
assert dup_factor_list(f, ZZ['y']) == \
(DMP([ZZ(4)],ZZ), [([DMP([ZZ(1),ZZ(0)],ZZ)], 1),
([DMP([ZZ(1)],ZZ),DMP([],ZZ)], 1),
([DMP([ZZ(1)],ZZ),DMP([ZZ(1),ZZ(0)],ZZ)], 1)])
f = [DMP([QQ(1,2),QQ(0)],ZZ),DMP([QQ(1,2),QQ(0),QQ(0)],ZZ),DMP([],ZZ)]
assert dup_factor_list(f, QQ['y']) == \
(DMP([QQ(1,2)],QQ), [([DMP([QQ(1),QQ(0)],QQ)], 1),
([DMP([QQ(1)],QQ),DMP([],QQ)], 1),
([DMP([QQ(1)],QQ),DMP([QQ(1),QQ(0)],QQ)], 1)])
K = QQ.algebraic_field(I)
h = [QQ(1,1), QQ(0,1), QQ(1,1)]
f = [ANP([QQ(1,1)], h, QQ), ANP([], h, QQ), ANP([QQ(2,1)], h, QQ), ANP([], h, QQ), ANP([], h, QQ)]
assert dup_factor_list(f, K) == \
(ANP([QQ(1,1)], h, QQ), [([ANP([QQ(1,1)], h, QQ), ANP([], h, QQ)], 2),
([ANP([QQ(1,1)], h, QQ), ANP([], h, QQ), ANP([QQ(2,1)], h, QQ)], 1)])
raises(DomainError, "dup_factor_list([EX(sin(1))], EX)")
def test_Modulus_postprocess():
opt = {'modulus': 5}
Modulus.postprocess(opt)
assert opt == {
'modulus': 5,
'domain': FF(5),
}
opt = {'modulus': 5, 'symmetric': False}
Modulus.postprocess(opt)
assert opt == {
'modulus': 5,
'domain': FF(5, False),
'symmetric': False,
}
def test_ModuleElement_column():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
e = A(0)
col1 = e.column()
assert col1 == e.col and col1 is not e.col
col2 = e.column(domain=FF(5))
assert col2.domain.is_FF
def test_dmp_sqf():
assert dmp_sqf_part([[]], 1, ZZ) == [[]]
assert dmp_sqf_p([[]], 1, ZZ) == True
assert dmp_sqf_part([[7]], 1, ZZ) == [[1]]
assert dmp_sqf_p([[7]], 1, ZZ) == True
assert dmp_sqf_p(f_0, 2, ZZ) == True
assert dmp_sqf_p(dmp_sqr(f_0, 2, ZZ), 2, ZZ) == False
assert dmp_sqf_p(f_1, 2, ZZ) == True
assert dmp_sqf_p(dmp_sqr(f_1, 2, ZZ), 2, ZZ) == False
assert dmp_sqf_p(f_2, 2, ZZ) == True
assert dmp_sqf_p(dmp_sqr(f_2, 2, ZZ), 2, ZZ) == False
assert dmp_sqf_p(f_3, 2, ZZ) == True
assert dmp_sqf_p(dmp_sqr(f_3, 2, ZZ), 2, ZZ) == False
assert dmp_sqf_p(f_5, 2, ZZ) == False
assert dmp_sqf_p(dmp_sqr(f_5, 2, ZZ), 2, ZZ) == False
assert dmp_sqf_p(f_4, 2, ZZ) == True
assert dmp_sqf_part(f_4, 2, ZZ) == dmp_neg(f_4, 2, ZZ)
assert dmp_sqf_p(f_6, 3, ZZ) == True
assert dmp_sqf_part(f_6, 3, ZZ) == f_6
assert dmp_sqf_part(f_5, 2, ZZ) == [[[1]], [[1], [-1, 0]]]
assert dup_sqf_list([], ZZ) == (ZZ(0), [])
assert dup_sqf_list_include([], ZZ) == [([], 1)]
assert dmp_sqf_list([[ZZ(3)]], 1, ZZ) == (ZZ(3), [])
assert dmp_sqf_list_include([[ZZ(3)]], 1, ZZ) == [([[ZZ(3)]], 1)]
f = [-1,1,0,0,1,-1]
assert dmp_sqf_list(f, 0, ZZ) == \
(-1, [([1,1,1,1], 1), ([1,-1], 2)])
assert dmp_sqf_list_include(f, 0, ZZ) == \
[([-1,-1,-1,-1], 1), ([1,-1], 2)]
f = [[-1],[1],[],[],[1],[-1]]
assert dmp_sqf_list(f, 1, ZZ) == \
(-1, [([[1],[1],[1],[1]], 1), ([[1],[-1]], 2)])
assert dmp_sqf_list_include(f, 1, ZZ) == \
[([[-1],[-1],[-1],[-1]], 1), ([[1],[-1]], 2)]
K = FF(2)
f = [[-1], [2], [-1]]
assert dmp_sqf_list_include(f, 1, ZZ) == \
[([[-1]], 1), ([[1], [-1]], 2)]
raises(DomainError, "dmp_sqf_list([[K(1), K(0), K(1)]], 1, K)")
def test_dmp_sqr():
assert dmp_sqr([ZZ(1),ZZ(2)], 0, ZZ) == \
dup_sqr([ZZ(1),ZZ(2)], ZZ)
assert dmp_sqr([[[]]], 2, ZZ) == [[[]]]
assert dmp_sqr([[[ZZ(2)]]], 2, ZZ) == [[[ZZ(4)]]]
assert dmp_sqr([[[]]], 2, QQ) == [[[]]]
assert dmp_sqr([[[QQ(2,3)]]], 2, QQ) == [[[QQ(4,9)]]]
K = FF(9)
assert dmp_sqr([[K(3)],[K(4)]], 1, K) == [[K(6)],[K(7)]]
def test_dmp_sqf():
R, x, y = ring("x,y", ZZ)
assert R.dmp_sqf_part(0) == 0
assert R.dmp_sqf_p(0) is True
assert R.dmp_sqf_part(7) == 1
assert R.dmp_sqf_p(7) is True
assert R.dmp_sqf_list(3) == (3, [])
assert R.dmp_sqf_list_include(3) == [(3, 1)]
R, x, y, z = ring("x,y,z", ZZ)
assert R.dmp_sqf_p(f_0) is True
assert R.dmp_sqf_p(f_0**2) is False
assert R.dmp_sqf_p(f_1) is True
assert R.dmp_sqf_p(f_1**2) is False
assert R.dmp_sqf_p(f_2) is True
assert R.dmp_sqf_p(f_2**2) is False
assert R.dmp_sqf_p(f_3) is True
assert R.dmp_sqf_p(f_3**2) is False
assert R.dmp_sqf_p(f_5) is False
assert R.dmp_sqf_p(f_5**2) is False
assert R.dmp_sqf_p(f_4) is True
assert R.dmp_sqf_part(f_4) == -f_4
assert R.dmp_sqf_part(f_5) == x + y - z
R, x, y, z, t = ring("x,y,z,t", ZZ)
assert R.dmp_sqf_p(f_6) is True
assert R.dmp_sqf_part(f_6) == f_6
R, x = ring("x", ZZ)
f = -(x**5) + x**4 + x - 1
assert R.dmp_sqf_list(f) == (-1, [(x**3 + x**2 + x + 1, 1), (x - 1, 2)])
assert R.dmp_sqf_list_include(f) == [(-(x**3) - x**2 - x - 1, 1),
(x - 1, 2)]
R, x, y = ring("x,y", ZZ)
f = -(x**5) + x**4 + x - 1
assert R.dmp_sqf_list(f) == (-1, [(x**3 + x**2 + x + 1, 1), (x - 1, 2)])
assert R.dmp_sqf_list_include(f) == [(-(x**3) - x**2 - x - 1, 1),
(x - 1, 2)]
f = -(x**2) + 2 * x - 1
assert R.dmp_sqf_list_include(f) == [(-1, 1), (x - 1, 2)]
R, x, y = ring("x,y", FF(2))
raises(NotImplementedError, lambda: R.dmp_sqf_list(y**2 + 1))
def test_dup_sqr():
assert dup_sqr([], ZZ) == []
assert dup_sqr([ZZ(2)], ZZ) == [ZZ(4)]
assert dup_sqr([ZZ(1),ZZ(2)], ZZ) == [ZZ(1),ZZ(4),ZZ(4)]
assert dup_sqr([], QQ) == []
assert dup_sqr([QQ(2,3)], QQ) == [QQ(4,9)]
assert dup_sqr([QQ(1,3),QQ(2,3)], QQ) == [QQ(1,9),QQ(4,9),QQ(4,9)]
f = dup_normal([2,0,0,1,7], ZZ)
assert dup_sqr(f, ZZ) == dup_normal([4,0,0,4,28,0,1,14,49], ZZ)
K = FF(9)
assert dup_sqr([K(3),K(4)], K) == [K(6),K(7)]
def test_DomainMatrix_from_list():
ddm = DDM([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
A = DomainMatrix.from_list([[1, 2], [3, 4]], ZZ)
assert A.rep == ddm
assert A.shape == (2, 2)
assert A.domain == ZZ
dom = FF(7)
ddm = DDM([[dom(1), dom(2)], [dom(3), dom(4)]], (2, 2), dom)
A = DomainMatrix.from_list([[1, 2], [3, 4]], dom)
assert A.rep == ddm
assert A.shape == (2, 2)
assert A.domain == dom
ddm = DDM([[QQ(1, 2), QQ(3, 1)], [QQ(1, 4), QQ(5, 1)]], (2, 2), QQ)
A = DomainMatrix.from_list([[(1, 2), (3, 1)], [(1, 4), (5, 1)]], QQ)
assert A.rep == ddm
assert A.shape == (2, 2)
assert A.domain == QQ
def test_dmp_mul():
assert dmp_mul([ZZ(5)], [ZZ(7)], 0, ZZ) == \
dup_mul([ZZ(5)], [ZZ(7)], ZZ)
assert dmp_mul([QQ(5,7)], [QQ(3,7)], 0, QQ) == \
dup_mul([QQ(5,7)], [QQ(3,7)], QQ)
assert dmp_mul([[[]]], [[[]]], 2, ZZ) == [[[]]]
assert dmp_mul([[[ZZ(1)]]], [[[]]], 2, ZZ) == [[[]]]
assert dmp_mul([[[]]], [[[ZZ(1)]]], 2, ZZ) == [[[]]]
assert dmp_mul([[[ZZ(2)]]], [[[ZZ(1)]]], 2, ZZ) == [[[ZZ(2)]]]
assert dmp_mul([[[ZZ(1)]]], [[[ZZ(2)]]], 2, ZZ) == [[[ZZ(2)]]]
assert dmp_mul([[[]]], [[[]]], 2, QQ) == [[[]]]
assert dmp_mul([[[QQ(1,2)]]], [[[]]], 2, QQ) == [[[]]]
assert dmp_mul([[[]]], [[[QQ(1,2)]]], 2, QQ) == [[[]]]
assert dmp_mul([[[QQ(2,7)]]], [[[QQ(1,3)]]], 2, QQ) == [[[QQ(2,21)]]]
assert dmp_mul([[[QQ(1,7)]]], [[[QQ(2,3)]]], 2, QQ) == [[[QQ(2,21)]]]
K = FF(6)
assert dmp_mul([[K(2)],[K(1)]], [[K(3)],[K(4)]], 1, K) == [[K(5)],[K(4)]]
def test_dmp_factor_list():
R, x, y = ring("x,y", ZZ)
assert R.dmp_factor_list(0) == (ZZ(0), [])
assert R.dmp_factor_list(7) == (7, [])
R, x, y = ring("x,y", QQ)
assert R.dmp_factor_list(0) == (QQ(0), [])
assert R.dmp_factor_list(QQ(1, 7)) == (QQ(1, 7), [])
R, x, y = ring("x,y", ZZ['y'])
assert R.dmp_factor_list(0) == (DMP([], ZZ), [])
assert R.dmp_factor_list(DMP([ZZ(7)], ZZ)) == (DMP([ZZ(7)], ZZ), [])
R, x, y = ring("x,y", QQ['y'])
assert R.dmp_factor_list(0) == (DMP([], QQ), [])
assert R.dmp_factor_list(DMP([QQ(1, 7)], QQ)) == (DMP([QQ(1, 7)], QQ), [])
R, x, y = ring("x,y", ZZ)
assert R.dmp_factor_list_include(0) == [(0, 1)]
assert R.dmp_factor_list_include(7) == [(7, 1)]
R, X = xring("x:200", ZZ)
f, g = X[0]**2 + 2 * X[0] + 1, X[0] + 1
assert R.dmp_factor_list(f) == (1, [(g, 2)])
f, g = X[-1]**2 + 2 * X[-1] + 1, X[-1] + 1
assert R.dmp_factor_list(f) == (1, [(g, 2)])
R, x = ring("x", ZZ)
assert R.dmp_factor_list(x**2 + 2 * x + 1) == (1, [(x + 1, 2)])
R, x = ring("x", QQ)
assert R.dmp_factor_list(QQ(1, 2) * x**2 + x + QQ(1, 2)) == (QQ(1, 2),
[(x + 1, 2)])
R, x, y = ring("x,y", ZZ)
assert R.dmp_factor_list(x**2 + 2 * x + 1) == (1, [(x + 1, 2)])
R, x, y = ring("x,y", QQ)
assert R.dmp_factor_list(QQ(1, 2) * x**2 + x + QQ(1, 2)) == (QQ(1, 2),
[(x + 1, 2)])
R, x, y = ring("x,y", ZZ)
f = 4 * x**2 * y + 4 * x * y**2
assert R.dmp_factor_list(f) == \
(4, [(y, 1),
(x, 1),
(x + y, 1)])
assert R.dmp_factor_list_include(f) == \
[(4*y, 1),
(x, 1),
(x + y, 1)]
R, x, y = ring("x,y", QQ)
f = QQ(1, 2) * x**2 * y + QQ(1, 2) * x * y**2
assert R.dmp_factor_list(f) == \
(QQ(1,2), [(y, 1),
(x, 1),
(x + y, 1)])
R, x, y = ring("x,y", RR)
f = 2.0 * x**2 - 8.0 * y**2
assert R.dmp_factor_list(f) == \
(RR(2.0), [(1.0*x - 2.0*y, 1),
(1.0*x + 2.0*y, 1)])
R, x, y = ring("x,y", ZZ['t'])
f = DMP([ZZ(4), ZZ(0)], ZZ) * x**2 + DMP([ZZ(4), ZZ(0), ZZ(0)], ZZ) * x
assert R.dmp_factor_list(f) == \
(DMP([ZZ(4)], ZZ), [(DMP([ZZ(1), ZZ(0)], ZZ), 1),
(DMP([ZZ(1)], ZZ)*x, 1),
(DMP([ZZ(1)], ZZ)*x + DMP([ZZ(1), ZZ(0)], ZZ), 1)])
R, x, y = ring("x,y", QQ['t'])
f = DMP([QQ(1, 2), QQ(0)], QQ) * x**2 + DMP(
[QQ(1, 2), QQ(0), QQ(0)], QQ) * x
assert R.dmp_factor_list(f) == \
(DMP([QQ(1, 2)], QQ), [(DMP([QQ(1), QQ(0)], QQ), 1),
(DMP([QQ(1)], QQ)*x, 1),
(DMP([QQ(1)], QQ)*x + DMP([QQ(1), QQ(0)], QQ), 1)])
R, x, y = ring("x,y", FF(2))
raises(NotImplementedError, lambda: R.dmp_factor_list(x**2 + y**2))
R, x, y = ring("x,y", EX)
raises(DomainError, lambda: R.dmp_factor_list(EX(sin(1))))
def test_dup_mul():
assert dup_mul([], [], ZZ) == []
assert dup_mul([], [ZZ(1)], ZZ) == []
assert dup_mul([ZZ(1)], [], ZZ) == []
assert dup_mul([ZZ(1)], [ZZ(1)], ZZ) == [ZZ(1)]
assert dup_mul([ZZ(5)], [ZZ(7)], ZZ) == [ZZ(35)]
assert dup_mul([], [], QQ) == []
assert dup_mul([], [QQ(1, 2)], QQ) == []
assert dup_mul([QQ(1, 2)], [], QQ) == []
assert dup_mul([QQ(1, 2)], [QQ(4, 7)], QQ) == [QQ(2, 7)]
assert dup_mul([QQ(5, 7)], [QQ(3, 7)], QQ) == [QQ(15, 49)]
f = dup_normal([3, 0, 0, 6, 1, 2], ZZ)
g = dup_normal([4, 0, 1, 0], ZZ)
h = dup_normal([12, 0, 3, 24, 4, 14, 1, 2, 0], ZZ)
assert dup_mul(f, g, ZZ) == h
assert dup_mul(g, f, ZZ) == h
f = dup_normal([2, 0, 0, 1, 7], ZZ)
h = dup_normal([4, 0, 0, 4, 28, 0, 1, 14, 49], ZZ)
assert dup_mul(f, f, ZZ) == h
K = FF(6)
assert dup_mul([K(2), K(1)], [K(3), K(4)], K) == [K(5), K(4)]
p1 = dup_normal([
79, -1, 78, -94, -10, 11, 32, -19, 78, 2, -89, 30, 73, 42, 85, 77, 83,
-30, -34, -2, 95, -81, 37, -49, -46, -58, -16, 37, 35, -11, -57, -15,
-31, 67, -20, 27, 76, 2, 70, 67, -65, 65, -26, -93, -44, -12, -92, 57,
-90, -57, -11, -67, -98, -69, 97, -41, 89, 33, 89, -50, 81, -31, 60,
-27, 43, 29, -77, 44, 21, -91, 32, -57, 33, 3, 53, -51, -38, -99, -84,
23, -50, 66, -100, 1, -75, -25, 27, -60, 98, -51, -87, 6, 8, 78, -28,
-95, -88, 12, -35, 26, -9, 16, -92, 55, -7, -86, 68, -39, -46, 84, 94,
45, 60, 92, 68, -75, -74, -19, 8, 75, 78, 91, 57, 34, 14, -3, -49, 65,
78, -18, 6, -29, -80, -98, 17, 13, 58, 21, 20, 9, 37, 7, -30, -53, -20,
34, 67, -42, 89, -22, 73, 43, -6, 5, 51, -8, -15, -52, -22, -58, -72,
-3, 43, -92, 82, 83, -2, -13, -23, -60, 16, -94, -8, -28, -95, -72, 63,
-90, 76, 6, -43, -100, -59, 76, 3, 3, 46, -85, 75, 62, -71, -76, 88,
97, -72, -1, 30, -64, 72, -48, 14, -78, 58, 63, -91, 24, -87, -27, -80,
-100, -44, 98, 70, 100, -29, -38, 11, 77, 100, 52, 86, 65, -5, -42,
-81, -38, -42, 43, -2, -70, -63, -52
], ZZ)
p2 = dup_normal([
65, -19, -47, 1, 90, 81, -15, -34, 25, -75, 9, -83, 50, -5, -44, 31, 1,
70, -7, 78, 74, 80, 85, 65, 21, 41, 66, 19, -40, 63, -21, -27, 32, 69,
83, 34, -35, 14, 81, 57, -75, 32, -67, -89, -100, -61, 46, 84, -78,
-29, -50, -94, -24, -32, -68, -16, 100, -7, -72, -89, 35, 82, 58, 81,
-92, 62, 5, -47, -39, -58, -72, -13, 84, 44, 55, -25, 48, -54, -31,
-56, -11, -50, -84, 10, 67, 17, 13, -14, 61, 76, -64, -44, -40, -96,
11, -11, -94, 2, 6, 27, -6, 68, -54, 66, -74, -14, -1, -24, -73, 96,
89, -11, -89, 56, -53, 72, -43, 96, 25, 63, -31, 29, 68, 83, 91, -93,
-19, -38, -40, 40, -12, -19, -79, 44, 100, -66, -29, -77, 62, 39, -8,
11, -97, 14, 87, 64, 21, -18, 13, 15, -59, -75, -99, -88, 57, 54, 56,
-67, 6, -63, -59, -14, 28, 87, -20, -39, 84, -91, -2, 49, -75, 11, -24,
-95, 36, 66, 5, 25, -72, -40, 86, 90, 37, -33, 57, -35, 29, -18, 4,
-79, 64, -17, -27, 21, 29, -5, -44, -87, -24, 52, 78, 11, -23, -53, 36,
42, 21, -68, 94, -91, -51, -21, 51, -76, 72, 31, 24, -48, -80, -9, 37,
-47, -6, -8, -63, -91, 79, -79, -100, 38, -20, 38, 100, 83, -90, 87,
63, -36, 82, -19, 18, -98, -38, 26, 98, -70, 79, 92, 12, 12, 70, 74,
36, 48, -13, 31, 31, -47, -71, -12, -64, 36, -42, 32, -86, 60, 83, 70,
55, 0, 1, 29, -35, 8, -82, 8, -73, -46, -50, 43, 48, -5, -86, -72, 44,
-90, 19, 19, 5, -20, 97, -13, -66, -5, 5, -69, 64, -30, 41, 51, 36, 13,
-99, -61, 94, -12, 74, 98, 68, 24, 46, -97, -87, -6, -27, 82, 62, -11,
-77, 86, 66, -47, -49, -50, 13, 18, 89, -89, 46, -80, 13, 98, -35, -36,
-25, 12, 20, 26, -52, 79, 27, 79, 100, 8, 62, -58, -28, 37
], ZZ)
res = dup_normal([
5135, -1566, 1376, -7466, 4579, 11710, 8001, -7183, -3737, -7439, 345,
-10084, 24522, -1201, 1070, -10245, 9582, 9264, 1903, 23312, 18953,
10037, -15268, -5450, 6442, -6243, -3777, 5110, 10936, -16649, -6022,
16255, 31300, 24818, 31922, 32760, 7854, 27080, 15766, 29596, 7139,
31945, -19810, 465, -38026, -3971, 9641, 465, -19375, 5524, -30112,
-11960, -12813, 13535, 30670, 5925, -43725, -14089, 11503, -22782,
6371, 43881, 37465, -33529, -33590, -39798, -37854, -18466, -7908,
-35825, -26020, -36923, -11332, -5699, 25166, -3147, 19885, 12962,
-20659, -1642, 27723, -56331, -24580, -11010, -20206, 20087, -23772,
-16038, 38580, 20901, -50731, 32037, -4299, 26508, 18038, -28357,
31846, -7405, -20172, -15894, 2096, 25110, -45786, 45918, -55333,
-31928, -49428, -29824, -58796, -24609, -15408, 69, -35415, -18439,
10123, -20360, -65949, 33356, -20333, 26476, -32073, 33621, 930, 28803,
-42791, 44716, 38164, 12302, -1739, 11421, 73385, -7613, 14297, 38155,
-414, 77587, 24338, -21415, 29367, 42639, 13901, -288, 51027, -11827,
91260, 43407, 88521, -15186, 70572, -12049, 5090, -12208, -56374,
15520, -623, -7742, 50825, 11199, -14894, 40892, 59591, -31356, -28696,
-57842, -87751, -33744, -28436, -28945, -40287, 37957, -35638, 33401,
-61534, 14870, 40292, 70366, -10803, 102290, -71719, -85251, 7902,
-22409, 75009, 99927, 35298, -1175, -762, -34744, -10587, -47574,
-62629, -19581, -43659, -54369, -32250, -39545, 15225, -24454, 11241,
-67308, -30148, 39929, 37639, 14383, -73475, -77636, -81048, -35992,
41601, -90143, 76937, -8112, 56588, 9124, -40094, -32340, 13253, 10898,
-51639, 36390, 12086, -1885, 100714, -28561, -23784, -18735, 18916,
16286, 10742, -87360, -13697, 10689, -19477, -29770, 5060, 20189,
-8297, 112407, 47071, 47743, 45519, -4109, 17468, -68831, 78325, -6481,
-21641, -19459, 30919, 96115, 8607, 53341, 32105, -16211, 23538, 57259,
-76272, -40583, 62093, 38511, -34255, -40665, -40604, -37606, -15274,
33156, -13885, 103636, 118678, -14101, -92682, -100791, 2634, 63791,
98266, 19286, -34590, -21067, -71130, 25380, -40839, -27614, -26060,
52358, -15537, 27138, -6749, 36269, -33306, 13207, -91084, -5540,
-57116, 69548, 44169, -57742, -41234, -103327, -62904, -8566, 41149,
-12866, 71188, 23980, 1838, 58230, 73950, 5594, 43113, -8159, -15925,
6911, 85598, -75016, -16214, -62726, -39016, 8618, -63882, -4299,
23182, 49959, 49342, -3238, -24913, -37138, 78361, 32451, 6337, -11438,
-36241, -37737, 8169, -3077, -24829, 57953, 53016, -31511, -91168,
12599, -41849, 41576, 55275, -62539, 47814, -62319, 12300, -32076,
-55137, -84881, -27546, 4312, -3433, -54382, 113288, -30157, 74469,
18219, 79880, -2124, 98911, 17655, -33499, -32861, 47242, -37393,
99765, 14831, -44483, 10800, -31617, -52710, 37406, 22105, 29704,
-20050, 13778, 43683, 36628, 8494, 60964, -22644, 31550, -17693, 33805,
-124879, -12302, 19343, 20400, -30937, -21574, -34037, -33380, 56539,
-24993, -75513, -1527, 53563, 65407, -101, 53577, 37991, 18717, -23795,
-8090, -47987, -94717, 41967, 5170, -14815, -94311, 17896, -17734,
-57718, -774, -38410, 24830, 29682, 76480, 58802, -46416, -20348,
-61353, -68225, -68306, 23822, -31598, 42972, 36327, 28968, -65638,
-21638, 24354, -8356, 26777, 52982, -11783, -44051, -26467, -44721,
-28435, -53265, -25574, -2669, 44155, 22946, -18454, -30718, -11252,
58420, 8711, 67447, 4425, 41749, 67543, 43162, 11793, -41907, 20477,
-13080, 6559, -6104, -13244, 42853, 42935, 29793, 36730, -28087, 28657,
17946, 7503, 7204, 21491, -27450, -24241, -98156, -18082, -42613,
-24928, 10775, -14842, -44127, 55910, 14777, 31151, -2194, 39206,
-2100, -4211, 11827, -8918, -19471, 72567, 36447, -65590, -34861,
-17147, -45303, 9025, -7333, -35473, 11101, 11638, 3441, 6626, -41800,
9416, 13679, 33508, 40502, -60542, 16358, 8392, -43242, -35864, -34127,
-48721, 35878, 30598, 28630, 20279, -19983, -14638, -24455, -1851,
-11344, 45150, 42051, 26034, -28889, -32382, -3527, -14532, 22564,
-22346, 477, 11706, 28338, -25972, -9185, -22867, -12522, 32120, -4424,
11339, -33913, -7184, 5101, -23552, -17115, -31401, -6104, 21906,
25708, 8406, 6317, -7525, 5014, 20750, 20179, 22724, 11692, 13297,
2493, -253, -16841, -17339, -6753, -4808, 2976, -10881, -10228, -13816,
-12686, 1385, 2316, 2190, -875, -1924
], ZZ)
assert dup_mul(p1, p2, ZZ) == res
p1 = dup_normal([
83, -61, -86, -24, 12, 43, -88, -9, 42, 55, -66, 74, 95, -25, -12, 68,
-99, 4, 45, 6, -15, -19, 78, 65, -55, 47, -13, 17, 86, 81, -58, -27,
50, -40, -24, 39, -41, -92, 75, 90, -1, 40, -15, -27, -35, 68, 70, -64,
-40, 78, -88, -58, -39, 69, 46, 12, 28, -94, -37, -50, -80, -96, -61,
25, 1, 71, 4, 12, 48, 4, 34, -47, -75, 5, 48, 82, 88, 23, 98, 35, 17,
-10, 48, -61, -95, 47, 65, -19, -66, -57, -6, -51, -42, -89, 66, -13,
18, 37, 90, -23, 72, 96, -53, 0, 40, -73, -52, -68, 32, -25, -53, 79,
-52, 18, 44, 73, -81, 31, -90, 70, 3, 36, 48, 76, -24, -44, 23, 98, -4,
73, 69, 88, -70, 14, -68, 94, -78, -15, -64, -97, -70, -35, 65, 88, 49,
-53, -7, 12, -45, -7, 59, -94, 99, -2, 67, -60, -71, 29, -62, -77, 1,
51, 17, 80, -20, -47, -19, 24, -9, 39, -23, 21, -84, 10, 84, 56, -17,
-21, -66, 85, 70, 46, -51, -22, -95, 78, -60, -96, -97, -45, 72, 35,
30, -61, -92, -93, -60, -61, 4, -4, -81, -73, 46, 53, -11, 26, 94, 45,
14, -78, 55, 84, -68, 98, 60, 23, 100, -63, 68, 96, -16, 3, 56, 21,
-58, 62, -67, 66, 85, 41, -79, -22, 97, -67, 82, 82, -96, -20, -7, 48,
-67, 48, -9, -39, 78
], ZZ)
p2 = dup_normal([
52, 88, 76, 66, 9, -64, 46, -20, -28, 69, 60, 96, -36, -92, -30, -11,
-35, 35, 55, 63, -92, -7, 25, -58, 74, 55, -6, 4, 47, -92, -65, 67,
-45, 74, -76, 59, -6, 69, 39, 24, -71, -7, 39, -45, 60, -68, 98, 97,
-79, 17, 4, 94, -64, 68, -100, -96, -2, 3, 22, 96, 54, -77, -86, 67, 6,
57, 37, 40, 89, -78, 64, -94, -45, -92, 57, 87, -26, 36, 19, 97, 25,
77, -87, 24, 43, -5, 35, 57, 83, 71, 35, 63, 61, 96, -22, 8, -1, 96,
43, 45, 94, -93, 36, 71, -41, -99, 85, -48, 59, 52, -17, 5, 87, -16,
-68, -54, 76, -18, 100, 91, -42, -70, -66, -88, -12, 1, 95, -82, 52,
43, -29, 3, 12, 72, -99, -43, -32, -93, -51, 16, -20, -12, -11, 5, 33,
-38, 93, -5, -74, 25, 74, -58, 93, 59, -63, -86, 63, -20, -4, -74, -73,
-95, 29, -28, 93, -91, -2, -38, -62, 77, -58, -85, -28, 95, 38, 19,
-69, 86, 94, 25, -2, -4, 47, 34, -59, 35, -48, 29, -63, -53, 34, 29,
66, 73, 6, 92, -84, 89, 15, 81, 93, 97, 51, -72, -78, 25, 60, 90, -45,
39, 67, -84, -62, 57, 26, -32, -56, -14, -83, 76, 5, -2, 99, -100, 28,
46, 94, -7, 53, -25, 16, -23, -36, 89, -78, -63, 31, 1, 84, -99, -52,
76, 48, 90, -76, 44, -19, 54, -36, -9, -73, -100, -69, 31, 42, 25, -39,
76, -26, -8, -14, 51, 3, 37, 45, 2, -54, 13, -34, -92, 17, -25, -65,
53, -63, 30, 4, -70, -67, 90, 52, 51, 18, -3, 31, -45, -9, 59, 63, -87,
22, -32, 29, -38, 21, 36, -82, 27, -11
], ZZ)
res = dup_normal([
4316, 4132, -3532, -7974, -11303, -10069, 5484, -3330, -5874, 7734,
4673, 11327, -9884, -8031, 17343, 21035, -10570, -9285, 15893, 3780,
-14083, 8819, 17592, 10159, 7174, -11587, 8598, -16479, 3602, 25596,
9781, 12163, 150, 18749, -21782, -12307, 27578, -2757, -12573, 12565,
6345, -18956, 19503, -15617, 1443, -16778, 36851, 23588, -28474, 5749,
40695, -7521, -53669, -2497, -18530, 6770, 57038, 3926, -6927, -15399,
1848, -64649, -27728, 3644, 49608, 15187, -8902, -9480, -7398, -40425,
4824, 23767, -7594, -6905, 33089, 18786, 12192, 24670, 31114, 35334,
-4501, -14676, 7107, -59018, -21352, 20777, 19661, 20653, 33754, -885,
-43758, 6269, 51897, -28719, -97488, -9527, 13746, 11644, 17644,
-21720, 23782, -10481, 47867, 20752, 33810, -1875, 39918, -7710,
-40840, 19808, -47075, 23066, 46616, 25201, 9287, 35436, -1602, 9645,
-11978, 13273, 15544, 33465, 20063, 44539, 11687, 27314, -6538, -37467,
14031, 32970, -27086, 41323, 29551, 65910, -39027, -37800, -22232,
8212, 46316, -28981, -55282, 50417, -44929, -44062, 73879, 37573,
-2596, -10877, -21893, -133218, -33707, -25753, -9531, 17530, 61126,
2748, -56235, 43874, -10872, -90459, -30387, 115267, -7264, -44452,
122626, 14839, -599, 10337, 57166, -67467, -54957, 63669, 1202, 18488,
52594, 7205, -97822, 612, 78069, -5403, -63562, 47236, 36873, -154827,
-26188, 82427, -39521, 5628, 7416, 5276, -53095, 47050, 26121, -42207,
79021, -13035, 2499, -66943, 29040, -72355, -23480, 23416, -12885,
-44225, -42688, -4224, 19858, 55299, 15735, 11465, 101876, -39169,
51786, 14723, 43280, -68697, 16410, 92295, 56767, 7183, 111850, 4550,
115451, -38443, -19642, -35058, 10230, 93829, 8925, 63047, 3146, 29250,
8530, 5255, -98117, -115517, -76817, -8724, 41044, 1312, -35974, 79333,
-28567, 7547, -10580, -24559, -16238, 10794, -3867, 24848, 57770,
-51536, -35040, 71033, 29853, 62029, -7125, -125585, -32169, -47907,
156811, -65176, -58006, -15757, -57861, 11963, 30225, -41901, -41681,
31310, 27982, 18613, 61760, 60746, -59096, 33499, 30097, -17997, 24032,
56442, -83042, 23747, -20931, -21978, -158752, -9883, -73598, -7987,
-7333, -125403, -116329, 30585, 53281, 51018, -29193, 88575, 8264,
-40147, -16289, 113088, 12810, -6508, 101552, -13037, 34440, -41840,
101643, 24263, 80532, 61748, 65574, 6423, -20672, 6591, -10834, -71716,
86919, -92626, 39161, 28490, 81319, 46676, 106720, 43530, 26998, 57456,
-8862, 60989, 13982, 3119, -2224, 14743, 55415, -49093, -29303, 28999,
1789, 55953, -84043, -7780, -65013, 57129, -47251, 61484, 61994,
-78361, -82778, 22487, -26894, 9756, -74637, -15519, -4360, 30115,
42433, 35475, 15286, 69768, 21509, -20214, 78675, -21163, 13596, 11443,
-10698, -53621, -53867, -24155, 64500, -42784, -33077, -16500, 873,
-52788, 14546, -38011, 36974, -39849, -34029, -94311, 83068, -50437,
-26169, -46746, 59185, 42259, -101379, -12943, 30089, -59086, 36271,
22723, -30253, -52472, -70826, -23289, 3331, -31687, 14183, -857,
-28627, 35246, -51284, 5636, -6933, 66539, 36654, 50927, 24783, 3457,
33276, 45281, 45650, -4938, -9968, -22590, 47995, 69229, 5214, -58365,
-17907, -14651, 18668, 18009, 12649, -11851, -13387, 20339, 52472,
-1087, -21458, -68647, 52295, 15849, 40608, 15323, 25164, -29368,
10352, -7055, 7159, 21695, -5373, -54849, 101103, -24963, -10511,
33227, 7659, 41042, -69588, 26718, -20515, 6441, 38135, -63, 24088,
-35364, -12785, -18709, 47843, 48533, -48575, 17251, -19394, 32878,
-9010, -9050, 504, -12407, 28076, -3429, 25324, -4210, -26119, 752,
-29203, 28251, -11324, -32140, -3366, -25135, 18702, -31588, -7047,
-24267, 49987, -14975, -33169, 37744, -7720, -9035, 16964, -2807, -421,
14114, -17097, -13662, 40628, -12139, -9427, 5369, 17551, -13232,
-16211, 9804, -7422, 2677, 28635, -8280, -4906, 2908, -22558, 5604,
12459, 8756, -3980, -4745, -18525, 7913, 5970, -16457, 20230, -6247,
-13812, 2505, 11899, 1409, -15094, 22540, -18863, 137, 11123, -4516,
2290, -8594, 12150, -10380, 3005, 5235, -7350, 2535, -858
], ZZ)
assert dup_mul(p1, p2, ZZ) == res
def test_lfsr_autocorrelation():
raises(TypeError, lambda: lfsr_autocorrelation(1, 2, 3))
F = FF(2)
s = lfsr_sequence([F(1), F(0)], [F(0), F(1)], 5)
assert lfsr_autocorrelation(s, 2, 0) == 1
assert lfsr_autocorrelation(s, 2, 1) == -1
def test_issue_21410():
R, x = ring('x', FF(2))
p = x**6 + x**5 + x**4 + x**3 + 1
assert p._pow_multinomial(4) == x**24 + x**20 + x**16 + x**12 + 1
def test_dmp_factor_list():
assert dmp_factor_list([[]], 1, ZZ) == (ZZ(0), [])
assert dmp_factor_list([[]], 1, QQ) == (QQ(0), [])
assert dmp_factor_list([[]], 1, ZZ['y']) == (DMP([],ZZ), [])
assert dmp_factor_list([[]], 1, QQ['y']) == (DMP([],QQ), [])
assert dmp_factor_list_include([[]], 1, ZZ) == [([[]], 1)]
assert dmp_factor_list([[ZZ(7)]], 1, ZZ) == (ZZ(7), [])
assert dmp_factor_list([[QQ(1,7)]], 1, QQ) == (QQ(1,7), [])
assert dmp_factor_list([[DMP([ZZ(7)],ZZ)]], 1, ZZ['y']) == (DMP([ZZ(7)],ZZ), [])
assert dmp_factor_list([[DMP([QQ(1,7)],QQ)]], 1, QQ['y']) == (DMP([QQ(1,7)],QQ), [])
assert dmp_factor_list_include([[ZZ(7)]], 1, ZZ) == [([[ZZ(7)]], 1)]
f, g = [ZZ(1),ZZ(2),ZZ(1)], [ZZ(1),ZZ(1)]
assert dmp_factor_list(dmp_nest(f, 200, ZZ), 200, ZZ) == \
(ZZ(1), [(dmp_nest(g, 200, ZZ), 2)])
assert dmp_factor_list(dmp_raise(f, 200, 0, ZZ), 200, ZZ) == \
(ZZ(1), [(dmp_raise(g, 200, 0, ZZ), 2)])
assert dmp_factor_list([ZZ(1),ZZ(2),ZZ(1)], 0, ZZ) == \
(ZZ(1), [([ZZ(1), ZZ(1)], 2)])
assert dmp_factor_list([QQ(1,2),QQ(1),QQ(1,2)], 0, QQ) == \
(QQ(1,2), [([QQ(1),QQ(1)], 2)])
assert dmp_factor_list([[ZZ(1)],[ZZ(2)],[ZZ(1)]], 1, ZZ) == \
(ZZ(1), [([[ZZ(1)], [ZZ(1)]], 2)])
assert dmp_factor_list([[QQ(1,2)],[QQ(1)],[QQ(1,2)]], 1, QQ) == \
(QQ(1,2), [([[QQ(1)],[QQ(1)]], 2)])
f = [[ZZ(4),ZZ(0)],[ZZ(4),ZZ(0),ZZ(0)],[]]
assert dmp_factor_list(f, 1, ZZ) == \
(ZZ(4), [([[ZZ(1),ZZ(0)]], 1),
([[ZZ(1)],[]], 1),
([[ZZ(1)],[ZZ(1),ZZ(0)]], 1)])
assert dmp_factor_list_include(f, 1, ZZ) == \
[([[ZZ(4),ZZ(0)]], 1),
([[ZZ(1)],[]], 1),
([[ZZ(1)],[ZZ(1),ZZ(0)]], 1)]
f = [[QQ(1,2),QQ(0)],[QQ(1,2),QQ(0),QQ(0)],[]]
assert dmp_factor_list(f, 1, QQ) == \
(QQ(1,2), [([[QQ(1),QQ(0)]], 1),
([[QQ(1)],[]], 1),
([[QQ(1)],[QQ(1),QQ(0)]], 1)])
f = [[RR(2.0)],[],[-RR(8.0),RR(0.0),RR(0.0)]]
assert dmp_factor_list(f, 1, RR) == \
(RR(2.0), [([[RR(1.0)],[-RR(2.0),RR(0.0)]], 1),
([[RR(1.0)],[ RR(2.0),RR(0.0)]], 1)])
f = [[DMP([ZZ(4),ZZ(0)],ZZ)],[DMP([ZZ(4),ZZ(0),ZZ(0)],ZZ)],[DMP([],ZZ)]]
assert dmp_factor_list(f, 1, ZZ['y']) == \
(DMP([ZZ(4)],ZZ), [([[DMP([ZZ(1),ZZ(0)],ZZ)]], 1),
([[DMP([ZZ(1)],ZZ)],[]], 1),
([[DMP([ZZ(1)],ZZ)],[DMP([ZZ(1),ZZ(0)],ZZ)]], 1)])
f = [[DMP([QQ(1,2),QQ(0)],ZZ)],[DMP([QQ(1,2),QQ(0),QQ(0)],ZZ)],[DMP([],ZZ)]]
assert dmp_factor_list(f, 1, QQ['y']) == \
(DMP([QQ(1,2)],QQ), [([[DMP([QQ(1),QQ(0)],QQ)]], 1),
([[DMP([QQ(1)],QQ)],[]], 1),
([[DMP([QQ(1)],QQ)],[DMP([QQ(1),QQ(0)],QQ)]], 1)])
K = FF(2)
raises(DomainError, "dmp_factor_list([[K(1)],[],[K(1),K(0),K(0)]], 1, K)")
raises(DomainError, "dmp_factor_list([[EX(sin(1))]], 1, EX)")
def lfsr_connection_polynomial(s):
"""
This function computes the lsfr connection polynomial.
INPUT:
``s``: a sequence of elements of even length, with entries in a finite field
OUTPUT:
``C(x)``: the connection polynomial of a minimal LFSR yielding ``s``.
This implements the algorithm in section 3 of J. L. Massey's article [M]_.
References
==========
.. [M] James L. Massey, "Shift-Register Synthesis and BCH Decoding."
IEEE Trans. on Information Theory, vol. 15(1), pp. 122-127, Jan 1969.
Examples
========
>>> from sympy.crypto.crypto import lfsr_sequence, lfsr_connection_polynomial
>>> from sympy.polys.domains import FF
>>> F = FF(2)
>>> fill = [F(1), F(1), F(0), F(1)]
>>> key = [F(1), F(0), F(0), F(1)]
>>> s = lfsr_sequence(key, fill, 20)
>>> lfsr_connection_polynomial(s)
x**4 + x + 1
>>> fill = [F(1), F(0), F(0), F(1)]
>>> key = [F(1), F(1), F(0), F(1)]
>>> s = lfsr_sequence(key, fill, 20)
>>> lfsr_connection_polynomial(s)
x**3 + 1
>>> fill = [F(1), F(0), F(1)]
>>> key = [F(1), F(1), F(0)]
>>> s = lfsr_sequence(key, fill, 20)
>>> lfsr_connection_polynomial(s)
x**3 + x**2 + 1
>>> fill = [F(1), F(0), F(1)]
>>> key = [F(1), F(0), F(1)]
>>> s = lfsr_sequence(key, fill, 20)
>>> lfsr_connection_polynomial(s)
x**3 + x + 1
"""
# Initialization:
p = s[0].mod
F = FF(p)
x = Symbol("x")
C = 1 * x**0
B = 1 * x**0
m = 1
b = 1 * x**0
L = 0
N = 0
while N < len(s):
if L > 0:
dC = Poly(C).degree()
r = min(L + 1, dC + 1)
coeffsC = [C.subs(x, 0)
] + [C.coeff(x**i) for i in range(1, dC + 1)]
d = (s[N].to_int() +
sum([coeffsC[i] * s[N - i].to_int()
for i in range(1, r)])) % p
if L == 0:
d = s[N].to_int() * x**0
if d == 0:
m += 1
N += 1
if d > 0:
if 2 * L > N:
C = (C - d * ((b**(p - 2)) % p) * x**m * B).expand()
m += 1
N += 1
else:
T = C
C = (C - d * ((b**(p - 2)) % p) * x**m * B).expand()
L = N + 1 - L
m = 1
b = d
B = T
N += 1
dC = Poly(C).degree()
coeffsC = [C.subs(x, 0)] + [C.coeff(x**i) for i in range(1, dC + 1)]
return sum([
coeffsC[i] % p * x**i for i in range(dC + 1) if coeffsC[i] is not None
])
def lfsr_sequence(key, fill, n):
r"""
This function creates an lfsr sequence.
INPUT:
``key``: a list of finite field elements,
`[c_0, c_1, \ldots, c_k].`
``fill``: the list of the initial terms of the lfsr
sequence, `[x_0, x_1, \ldots, x_k].`
``n``: number of terms of the sequence that the
function returns.
OUTPUT:
The lfsr sequence defined by `x_{n+1} = c_k x_n + \ldots + c_0 x_{n-k}`, for
`n \leq k`.
Notes
=====
S. Golomb [G]_ gives a list of three statistical properties a
sequence of numbers `a = \{a_n\}_{n=1}^\infty`,
`a_n \in \{0,1\}`, should display to be considered
"random". Define the autocorrelation of `a` to be
.. math::
C(k) = C(k,a) = \lim_{N\rightarrow \infty} {1\over N}\sum_{n=1}^N (-1)^{a_n + a_{n+k}}.
In the case where `a` is periodic with period
`P` then this reduces to
.. math::
C(k) = {1\over P}\sum_{n=1}^P (-1)^{a_n + a_{n+k}}.
Assume `a` is periodic with period `P`.
- balance:
.. math::
\left|\sum_{n=1}^P(-1)^{a_n}\right| \leq 1.
- low autocorrelation:
.. math::
C(k) = \left\{ \begin{array}{cc} 1,& k = 0,\\ \epsilon, & k \ne 0. \end{array} \right.
(For sequences satisfying these first two properties, it is known
that `\epsilon = -1/P` must hold.)
- proportional runs property: In each period, half the runs have
length `1`, one-fourth have length `2`, etc.
Moreover, there are as many runs of `1`'s as there are of
`0`'s.
References
==========
.. [G] Solomon Golomb, Shift register sequences, Aegean Park Press, Laguna Hills, Ca, 1967
Examples
========
>>> from sympy.crypto.crypto import lfsr_sequence
>>> from sympy.polys.domains import FF
>>> F = FF(2)
>>> fill = [F(1), F(1), F(0), F(1)]
>>> key = [F(1), F(0), F(0), F(1)]
>>> lfsr_sequence(key, fill, 10)
[1 mod 2, 1 mod 2, 0 mod 2, 1 mod 2, 0 mod 2, 1 mod 2, 1 mod 2, 0 mod 2, 0 mod 2, 1 mod 2]
"""
if not isinstance(key, list):
raise TypeError("key must be a list")
if not isinstance(fill, list):
raise TypeError("fill must be a list")
p = key[0].mod
F = FF(p)
s = fill
k = len(fill)
L = []
for i in range(n):
s0 = s[:]
L.append(s[0])
s = s[1:k]
x = sum([int(key[i] * s0[i]) for i in range(k)])
s.append(F(x))
return L # use [x.to_int() for x in L] for int version
def test_dup_sqf():
assert dup_sqf_part([], ZZ) == []
assert dup_sqf_p([], ZZ) == True
assert dup_sqf_part([7], ZZ) == [1]
assert dup_sqf_p([7], ZZ) == True
assert dup_sqf_part([2,2], ZZ) == [1,1]
assert dup_sqf_p([2,2], ZZ) == True
assert dup_sqf_part([1,0,1,1], ZZ) == [1,0,1,1]
assert dup_sqf_p([1,0,1,1], ZZ) == True
assert dup_sqf_part([-1,0,1,1], ZZ) == [1,0,-1,-1]
assert dup_sqf_p([-1,0,1,1], ZZ) == True
assert dup_sqf_part([2,3,0,0], ZZ) == [2,3,0]
assert dup_sqf_p([2,3,0,0], ZZ) == False
assert dup_sqf_part([-2,3,0,0], ZZ) == [2,-3,0]
assert dup_sqf_p([-2,3,0,0], ZZ) == False
assert dup_sqf_list([], ZZ) == (0, [])
assert dup_sqf_list([1], ZZ) == (1, [])
assert dup_sqf_list([1,0], ZZ) == (1, [([1,0], 1)])
assert dup_sqf_list([2,0,0], ZZ) == (2, [([1,0], 2)])
assert dup_sqf_list([3,0,0,0], ZZ) == (3, [([1,0], 3)])
assert dup_sqf_list([ZZ(2),ZZ(4),ZZ(2)], ZZ) == \
(ZZ(2), [([ZZ(1),ZZ(1)], 2)])
assert dup_sqf_list([QQ(2),QQ(4),QQ(2)], QQ) == \
(QQ(2), [([QQ(1),QQ(1)], 2)])
assert dup_sqf_list([-1,1,0,0,1,-1], ZZ) == \
(-1, [([1,1,1,1], 1), ([1,-1], 2)])
assert dup_sqf_list([1,0,6,0,12,0,8,0,0], ZZ) == \
(1, [([1,0], 2), ([1,0,2], 3)])
K = FF(2)
f = map(K, [1,0,1])
assert dup_sqf_list(f, K) == \
(K(1), [([K(1),K(1)], 2)])
K = FF(3)
f = map(K, [1,0,0,2,0,0,2,0,0,1,0])
assert dup_sqf_list(f, K) == \
(K(1), [([K(1), K(0)], 1),
([K(1), K(1)], 3),
([K(1), K(2)], 6)])
f = [1,0,0,1]
g = map(K, f)
assert dup_sqf_part(f, ZZ) == f
assert dup_sqf_part(g, K) == [K(1), K(1)]
assert dup_sqf_p(f, ZZ) == True
assert dup_sqf_p(g, K) == False
A = [[1],[],[-3],[],[6]]
D = [[1],[],[-5],[],[5],[],[4]]
f, g = D, dmp_sub(A, dmp_mul(dmp_diff(D, 1, 1, ZZ), [[1,0]], 1, ZZ), 1, ZZ)
res = dmp_resultant(f, g, 1, ZZ)
assert dup_sqf_list(res, ZZ) == (45796, [([4,0,1], 3)])
assert dup_sqf_list_include([DMP([1, 0, 0, 0], ZZ), DMP([], ZZ), DMP([], ZZ)], ZZ[x]) == \
[([DMP([1, 0, 0, 0], ZZ)], 1), ([DMP([1], ZZ), DMP([], ZZ)], 2)]
def test_FF_of_type():
assert FF(3).of_type(FF(3)(1)) is True
assert FF(5).of_type(FF(5)(3)) is True
assert FF(5).of_type(FF(7)(3)) is False
def test_ModularInteger():
F3 = FF(3)
a = F3(0)
assert isinstance(a, F3.dtype) and a == 0
a = F3(1)
assert isinstance(a, F3.dtype) and a == 1
a = F3(2)
assert isinstance(a, F3.dtype) and a == 2
a = F3(3)
assert isinstance(a, F3.dtype) and a == 0
a = F3(4)
assert isinstance(a, F3.dtype) and a == 1
a = F3(F3(0))
assert isinstance(a, F3.dtype) and a == 0
a = F3(F3(1))
assert isinstance(a, F3.dtype) and a == 1
a = F3(F3(2))
assert isinstance(a, F3.dtype) and a == 2
a = F3(F3(3))
assert isinstance(a, F3.dtype) and a == 0
a = F3(F3(4))
assert isinstance(a, F3.dtype) and a == 1
a = -F3(1)
assert isinstance(a, F3.dtype) and a == 2
a = -F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = 2 + F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = F3(2) + 2
assert isinstance(a, F3.dtype) and a == 1
a = F3(2) + F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = F3(2) + F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = 3 - F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = F3(3) - 2
assert isinstance(a, F3.dtype) and a == 1
a = F3(3) - F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = F3(3) - F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = 2*F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = F3(2)*2
assert isinstance(a, F3.dtype) and a == 1
a = F3(2)*F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = F3(2)*F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = 2/F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = F3(2)/2
assert isinstance(a, F3.dtype) and a == 1
a = F3(2)/F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = F3(2)/F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = 1 % F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = F3(1) % 2
assert isinstance(a, F3.dtype) and a == 1
a = F3(1) % F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = F3(1) % F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = F3(2)**0
assert isinstance(a, F3.dtype) and a == 1
a = F3(2)**1
assert isinstance(a, F3.dtype) and a == 2
a = F3(2)**2
assert isinstance(a, F3.dtype) and a == 1
F7 = FF(7)
a = F7(3)**100000000000
assert isinstance(a, F7.dtype) and a == 4
a = F7(3)**-100000000000
assert isinstance(a, F7.dtype) and a == 2
a = F7(3)**S(2)
assert isinstance(a, F7.dtype) and a == 2
assert bool(F3(3)) is False
assert bool(F3(4)) is True
F5 = FF(5)
a = F5(1)**(-1)
assert isinstance(a, F5.dtype) and a == 1
a = F5(2)**(-1)
assert isinstance(a, F5.dtype) and a == 3
a = F5(3)**(-1)
assert isinstance(a, F5.dtype) and a == 2
a = F5(4)**(-1)
assert isinstance(a, F5.dtype) and a == 4
assert (F5(1) < F5(2)) is True
assert (F5(1) <= F5(2)) is True
assert (F5(1) > F5(2)) is False
assert (F5(1) >= F5(2)) is False
assert (F5(3) < F5(2)) is False
assert (F5(3) <= F5(2)) is False
assert (F5(3) > F5(2)) is True
assert (F5(3) >= F5(2)) is True
assert (F5(1) < F5(7)) is True
assert (F5(1) <= F5(7)) is True
assert (F5(1) > F5(7)) is False
assert (F5(1) >= F5(7)) is False
assert (F5(3) < F5(7)) is False
assert (F5(3) <= F5(7)) is False
assert (F5(3) > F5(7)) is True
assert (F5(3) >= F5(7)) is True
assert (F5(1) < 2) is True
assert (F5(1) <= 2) is True
assert (F5(1) > 2) is False
assert (F5(1) >= 2) is False
assert (F5(3) < 2) is False
assert (F5(3) <= 2) is False
assert (F5(3) > 2) is True
assert (F5(3) >= 2) is True
assert (F5(1) < 7) is True
assert (F5(1) <= 7) is True
assert (F5(1) > 7) is False
assert (F5(1) >= 7) is False
assert (F5(3) < 7) is False
assert (F5(3) <= 7) is False
assert (F5(3) > 7) is True
assert (F5(3) >= 7) is True
raises(NotInvertible, lambda: F5(0)**(-1))
raises(NotInvertible, lambda: F5(5)**(-1))
raises(ValueError, lambda: FF(0))
raises(ValueError, lambda: FF(2.1))
def test_dup_factor_list():
R, x = ring("x", ZZ)
assert R.dup_factor_list(0) == (0, [])
assert R.dup_factor_list(7) == (7, [])
R, x = ring("x", QQ)
assert R.dup_factor_list(0) == (0, [])
assert R.dup_factor_list(QQ(1, 7)) == (QQ(1, 7), [])
R, x = ring("x", ZZ['t'])
assert R.dup_factor_list(0) == (0, [])
assert R.dup_factor_list(7) == (7, [])
R, x = ring("x", QQ['t'])
assert R.dup_factor_list(0) == (0, [])
assert R.dup_factor_list(QQ(1, 7)) == (QQ(1, 7), [])
R, x = ring("x", ZZ)
assert R.dup_factor_list_include(0) == [(0, 1)]
assert R.dup_factor_list_include(7) == [(7, 1)]
assert R.dup_factor_list(x**2 + 2*x + 1) == (1, [(x + 1, 2)])
assert R.dup_factor_list_include(x**2 + 2*x + 1) == [(x + 1, 2)]
R, x = ring("x", QQ)
assert R.dup_factor_list(QQ(1,2)*x**2 + x + QQ(1,2)) == (QQ(1, 2), [(x + 1, 2)])
R, x = ring("x", FF(2))
assert R.dup_factor_list(x**2 + 1) == (1, [(x + 1, 2)])
R, x = ring("x", RR)
assert R.dup_factor_list(1.0*x**2 + 2.0*x + 1.0) == (1.0, [(1.0*x + 1.0, 2)])
assert R.dup_factor_list(2.0*x**2 + 4.0*x + 2.0) == (2.0, [(1.0*x + 1.0, 2)])
f = 6.7225336055071*x**2 - 10.6463972754741*x - 0.33469524022264
coeff, factors = R.dup_factor_list(f)
assert coeff == RR(1.0) and len(factors) == 1 and factors[0][0].almosteq(f, 1e-10) and factors[0][1] == 1
Rt, t = ring("t", ZZ)
R, x = ring("x", Rt)
f = 4*t*x**2 + 4*t**2*x
assert R.dup_factor_list(f) == \
(4*t, [(x, 1),
(x + t, 1)])
Rt, t = ring("t", QQ)
R, x = ring("x", Rt)
f = QQ(1, 2)*t*x**2 + QQ(1, 2)*t**2*x
assert R.dup_factor_list(f) == \
(QQ(1, 2)*t, [(x, 1),
(x + t, 1)])
R, x = ring("x", QQ.algebraic_field(I))
def anp(element):
return ANP(element, [QQ(1), QQ(0), QQ(1)], QQ)
f = anp([QQ(1, 1)])*x**4 + anp([QQ(2, 1)])*x**2
assert R.dup_factor_list(f) == \
(anp([QQ(1, 1)]), [(anp([QQ(1, 1)])*x, 2),
(anp([QQ(1, 1)])*x**2 + anp([])*x + anp([QQ(2, 1)]), 1)])
R, x = ring("x", EX)
raises(DomainError, lambda: R.dup_factor_list(EX(sin(1))))
def test_PolyElement_is_():
R, x,y,z = ring("x,y,z", QQ)
assert (x - x).is_generator == False
assert (x - x).is_ground == True
assert (x - x).is_monomial == True
assert (x - x).is_term == True
assert (x - x + 1).is_generator == False
assert (x - x + 1).is_ground == True
assert (x - x + 1).is_monomial == True
assert (x - x + 1).is_term == True
assert x.is_generator == True
assert x.is_ground == False
assert x.is_monomial == True
assert x.is_term == True
assert (x*y).is_generator == False
assert (x*y).is_ground == False
assert (x*y).is_monomial == True
assert (x*y).is_term == True
assert (3*x).is_generator == False
assert (3*x).is_ground == False
assert (3*x).is_monomial == False
assert (3*x).is_term == True
assert (3*x + 1).is_generator == False
assert (3*x + 1).is_ground == False
assert (3*x + 1).is_monomial == False
assert (3*x + 1).is_term == False
assert R(0).is_zero is True
assert R(1).is_zero is False
assert R(0).is_one is False
assert R(1).is_one is True
assert (x - 1).is_monic is True
assert (2*x - 1).is_monic is False
assert (3*x + 2).is_primitive is True
assert (4*x + 2).is_primitive is False
assert (x + y + z + 1).is_linear is True
assert (x*y*z + 1).is_linear is False
assert (x*y + z + 1).is_quadratic is True
assert (x*y*z + 1).is_quadratic is False
assert (x - 1).is_squarefree is True
assert ((x - 1)**2).is_squarefree is False
assert (x**2 + x + 1).is_irreducible is True
assert (x**2 + 2*x + 1).is_irreducible is False
_, t = ring("t", FF(11))
assert (7*t + 3).is_irreducible is True
assert (7*t**2 + 3*t + 1).is_irreducible is False
_, u = ring("u", ZZ)
f = u**16 + u**14 - u**10 - u**8 - u**6 + u**2
assert f.is_cyclotomic is False
assert (f + 1).is_cyclotomic is True
raises(MultivariatePolynomialError, lambda: x.is_cyclotomic)
def test_dmp_factor_list():
R, x, y = ring("x,y", ZZ)
assert R.dmp_factor_list(0) == (ZZ(0), [])
assert R.dmp_factor_list(7) == (7, [])
R, x, y = ring("x,y", QQ)
assert R.dmp_factor_list(0) == (QQ(0), [])
assert R.dmp_factor_list(QQ(1, 7)) == (QQ(1, 7), [])
Rt, t = ring("t", ZZ)
R, x, y = ring("x,y", Rt)
assert R.dmp_factor_list(0) == (0, [])
assert R.dmp_factor_list(7) == (ZZ(7), [])
Rt, t = ring("t", QQ)
R, x, y = ring("x,y", Rt)
assert R.dmp_factor_list(0) == (0, [])
assert R.dmp_factor_list(QQ(1, 7)) == (QQ(1, 7), [])
R, x, y = ring("x,y", ZZ)
assert R.dmp_factor_list_include(0) == [(0, 1)]
assert R.dmp_factor_list_include(7) == [(7, 1)]
R, X = xring("x:200", ZZ)
f, g = X[0]**2 + 2*X[0] + 1, X[0] + 1
assert R.dmp_factor_list(f) == (1, [(g, 2)])
f, g = X[-1]**2 + 2*X[-1] + 1, X[-1] + 1
assert R.dmp_factor_list(f) == (1, [(g, 2)])
R, x = ring("x", ZZ)
assert R.dmp_factor_list(x**2 + 2*x + 1) == (1, [(x + 1, 2)])
R, x = ring("x", QQ)
assert R.dmp_factor_list(QQ(1,2)*x**2 + x + QQ(1,2)) == (QQ(1,2), [(x + 1, 2)])
R, x, y = ring("x,y", ZZ)
assert R.dmp_factor_list(x**2 + 2*x + 1) == (1, [(x + 1, 2)])
R, x, y = ring("x,y", QQ)
assert R.dmp_factor_list(QQ(1,2)*x**2 + x + QQ(1,2)) == (QQ(1,2), [(x + 1, 2)])
R, x, y = ring("x,y", ZZ)
f = 4*x**2*y + 4*x*y**2
assert R.dmp_factor_list(f) == \
(4, [(y, 1),
(x, 1),
(x + y, 1)])
assert R.dmp_factor_list_include(f) == \
[(4*y, 1),
(x, 1),
(x + y, 1)]
R, x, y = ring("x,y", QQ)
f = QQ(1,2)*x**2*y + QQ(1,2)*x*y**2
assert R.dmp_factor_list(f) == \
(QQ(1,2), [(y, 1),
(x, 1),
(x + y, 1)])
R, x, y = ring("x,y", RR)
f = 2.0*x**2 - 8.0*y**2
assert R.dmp_factor_list(f) == \
(RR(2.0), [(1.0*x - 2.0*y, 1),
(1.0*x + 2.0*y, 1)])
f = 6.7225336055071*x**2*y**2 - 10.6463972754741*x*y - 0.33469524022264
coeff, factors = R.dmp_factor_list(f)
assert coeff == RR(1.0) and len(factors) == 1 and factors[0][0].almosteq(f, 1e-10) and factors[0][1] == 1
Rt, t = ring("t", ZZ)
R, x, y = ring("x,y", Rt)
f = 4*t*x**2 + 4*t**2*x
assert R.dmp_factor_list(f) == \
(4*t, [(x, 1),
(x + t, 1)])
Rt, t = ring("t", QQ)
R, x, y = ring("x,y", Rt)
f = QQ(1, 2)*t*x**2 + QQ(1, 2)*t**2*x
assert R.dmp_factor_list(f) == \
(QQ(1, 2)*t, [(x, 1),
(x + t, 1)])
R, x, y = ring("x,y", FF(2))
raises(NotImplementedError, lambda: R.dmp_factor_list(x**2 + y**2))
R, x, y = ring("x,y", EX)
raises(DomainError, lambda: R.dmp_factor_list(EX(sin(1))))