def isfinished_bsgscv(n, sossp, sogsp, nt, lpt, qpt, disc, classnum, indofg):
"""
Determine whether the bsgs algorithm is finished or not yet.
This is a submodule called by the bsgs module.
"""
lpt.append(n)
sumn = 1
for nid in lpt:
sumn = sumn * nid
if sumn == qpt:
return True, sossp, sogsp
elif sumn > qpt:
raise ValueError
if n == 1:
return False, sossp, sogsp
else:
tpsq = misc.primePowerTest(n)
if (tpsq[1] != 0) and ((tpsq[1] % 2) == 0):
q = arith1.floorsqrt(n)
else:
q = arith1.floorsqrt(n) + 1
ss = sossp.retel()
new_sossp = ClassGroup(disc, classnum, [])
tnt = copy.deepcopy(nt)
for i in range(q):
base = tnt ** i
for ssi in ss:
newel = base * ssi
if new_sossp.search(newel) is False:
newel.alpha = ssi.alpha[:]
lenal = len(newel.alpha)
sfind = indofg - lenal
for sit in range(sfind):
newel.alpha.append([lenal + sit, 0, 0])
newel.alpha.append([indofg, tnt, i])
new_sossp.insttree(newel) # multiple of two elements of G
y = nt ** q
ltl = sogsp.retel()
new_sogsp = ClassGroup(disc, classnum, [])
for i in range(q + 1):
base = y ** (-i)
for eol in ltl:
newel2 = base * eol
if new_sogsp.search(newel2) is False:
newel2.beta = eol.beta[:]
lenbt = len(newel2.beta)
gfind = indofg - lenbt
for git in range(gfind):
newel2.beta.append([lenbt + git, 0, 0])
newel2.beta.append([indofg, tnt, q * (-i)])
new_sogsp.insttree(newel2) # multiple of two elements of G
return False, new_sossp, new_sogsp
def isfinished_bsgscv(n, sossp, sogsp, nt, lpt, qpt, disc, classnum, indofg):
"""
Determine whether the bsgs algorithm is finished or not yet.
This is a submodule called by the bsgs module.
"""
lpt.append(n)
sumn = 1
for nid in lpt:
sumn = sumn * nid
if sumn == qpt:
return True, sossp, sogsp
elif sumn > qpt:
raise ValueError
if n == 1:
return False, sossp, sogsp
else:
tpsq = misc.primePowerTest(n)
if (tpsq[1] != 0) and ((tpsq[1] & 1) == 0):
q = arith1.floorsqrt(n)
else:
q = arith1.floorsqrt(n) + 1
ss = sossp.retel()
new_sossp = ClassGroup(disc, classnum, [])
tnt = copy.deepcopy(nt)
for i in range(q):
base = tnt**i
for ssi in ss:
newel = base * ssi
if new_sossp.search(newel) is False:
newel.alpha = ssi.alpha[:]
lenal = len(newel.alpha)
sfind = indofg - lenal
for sit in range(sfind):
newel.alpha.append([lenal + sit, 0, 0])
newel.alpha.append([indofg, tnt, i])
new_sossp.insttree(newel) # multiple of two elements of G
y = nt**q
ltl = sogsp.retel()
new_sogsp = ClassGroup(disc, classnum, [])
for i in range(q + 1):
base = y**(-i)
for eol in ltl:
newel2 = base * eol
if new_sogsp.search(newel2) is False:
newel2.beta = eol.beta[:]
lenbt = len(newel2.beta)
gfind = indofg - lenbt
for git in range(gfind):
newel2.beta.append([lenbt + git, 0, 0])
newel2.beta.append([indofg, tnt, q * (-i)])
new_sogsp.insttree(newel2) # multiple of two elements of G
return False, new_sossp, new_sogsp
def class_number_bsgs(disc):
"""
Return the class number with the given discriminant.
"""
if disc & 3 not in (0, 1):
raise ValueError("a discriminant must be 0 or 1 mod 4")
if disc >= 0:
raise ValueError("a discriminant must be negative")
lx = max(arith1.floorpowerroot(abs(disc), 5),
500 * (math.log(abs(disc)))**2)
uprbd = int(class_formula(disc, int(lx)) * 3 / 2)
lwrbd = (uprbd >> 1) + 1
bounds = [lwrbd, uprbd]
# get the unit
element = RetNext(disc)
ut = element.unit()
# append the unit to subset of G
sossp = ClassGroup(disc, 0, [])
sogsp = ClassGroup(disc, 0, [])
sossp.insttree(ut)
sogsp.insttree(ut)
h = 1 # order
finished = False
while not finished:
mstp1 = bounds[1] - bounds[0]
if mstp1 <= 1:
q = 1
else:
q = arith1.floorsqrt(mstp1)
if misc.primePowerTest(mstp1)[1] != 2:
q += 1
# get next element
nt = element.retnext()
# x is the set of elements of G
x = [ut, nt**h]
if q > 2:
x.extend([0] * (q - 2))
# compute small steps
if x[1] == ut:
# compute the order of nt
n = trorder(h, sossp, sogsp, nt, disc)
else:
n = trbabysp(q, x, bounds, sossp, sogsp, ut, h, nt, disc)
# finished?
finished, h, sossp, sogsp = isfinished_trbsgs(lwrbd, bounds, h, n,
sossp, sogsp, nt, disc)
return h
def class_number_bsgs(disc):
"""
Return the class number with the given discriminant.
"""
if disc % 4 not in (0, 1):
raise ValueError("a discriminant must be 0 or 1 mod 4")
if disc >= 0:
raise ValueError("a discriminant must be negative")
lx = max(arith1.floorpowerroot(abs(disc), 5), 500 * (math.log(abs(disc)))**2)
uprbd = int(class_formula(disc, int(lx)) * 3 / 2)
lwrbd = uprbd // 2 + 1
bounds = [lwrbd, uprbd]
# get the unit
element = RetNext(disc)
ut = element.unit()
# append the unit to subset of G
sossp = ClassGroup(disc, 0, [])
sogsp = ClassGroup(disc, 0, [])
sossp.insttree(ut)
sogsp.insttree(ut)
h = 1 # order
finished = False
while not finished:
mstp1 = bounds[1] - bounds[0]
if mstp1 <= 1:
q = 1
else:
q = arith1.floorsqrt(mstp1)
if misc.primePowerTest(mstp1)[1] != 2:
q += 1
# get next element
nt = element.retnext()
# x is the set of elements of G
x = [ut, nt ** h]
if q > 2:
x.extend([0] * (q - 2))
# compute small steps
if x[1] == ut:
# compute the order of nt
n = trorder(h, sossp, sogsp, nt, disc)
else:
n = trbabysp(q, x, bounds, sossp, sogsp, ut, h, nt, disc)
# finished?
finished, h, sossp, sogsp = isfinished_trbsgs(lwrbd, bounds, h, n, sossp, sogsp, nt, disc)
return h
def testPrimePowerTest(self):
# not a power
self.assertEqual((12, 0), misc.primePowerTest(12))
self.assertEqual((53, 1), misc.primePowerTest(53))
# not a prime power
self.assertEqual((36, 0), misc.primePowerTest(36))
# powers
self.assertEqual((7, 2), misc.primePowerTest(49))
self.assertEqual((3, 4), misc.primePowerTest(81))
self.assertEqual((5, 3), misc.primePowerTest(125))
self.assertEqual((2, 7), misc.primePowerTest(128))
def babyspcv(bounds, sossp, sogsp, utwi, nt, disc, classnum):
"""
Compute small steps
"""
mstp1 = bounds[1] - bounds[0]
if (mstp1 == 0) or (mstp1 == 1):
q = 1
else:
tppm = misc.primePowerTest(mstp1)
q = arith1.floorsqrt(mstp1)
if (tppm[1] == 0) or (tppm[1] % 2):
q += 1
n_should_be_set = True
# initialize
c_s1 = ClassGroup(disc, classnum, []) # a subset of G
# extracting i = 0 case of main loop
for ttr in sossp.retel():
tmpx = ttr
tmpx.s_parent = ttr # tmpx belongs ttr in the set of smallstep
# index of the element
tmpx.ind = 0
c_s1.insttree(tmpx)
# main loop
x_i = nt
for i in range(1, q):
for ttr in sossp.retel():
tmpx = x_i * ttr
tmpx.s_parent = ttr # tmpx belongs ttr in the set of smallstep
if n_should_be_set and tmpx == utwi:
n = i
tmp_ss = tmpx.s_parent
tmp_gs = utwi
n_should_be_set = False
# index of the element
tmpx.ind = i
c_s1.insttree(tmpx)
x_i = nt * x_i
assert x_i == nt ** q
if n_should_be_set:
sz = nt ** bounds[0]
n, tmp_ss, tmp_gs = giantspcv(q, sz, x_i, c_s1, bounds, sogsp)
return ordercv(n, sossp, sogsp, nt, disc, classnum, tmp_ss, tmp_gs)
def generate(self, target, **options):
"""
Generate squarefree factors of the target number with their
valuations. The method may terminate with yielding (1, 1)
to indicate the factorization is incomplete.
If a keyword option 'strict' is False (default to True),
factorization will stop after the first square factor no
matter whether it is squarefree or not.
"""
strict = options.get('strict', True)
options['n'] = target
primeseq = self._parse_seq(options)
for p in primeseq:
if not (target % p):
e, target = arith1.vp(target, p)
yield p, e
if target == 1:
break
elif e > 1 and not strict:
yield 1, 1
break
elif trivial_test_ternary(target):
# the factor remained is squarefree.
yield target, 1
break
q, e = factor_misc.primePowerTest(target)
if e:
yield q, e
break
sqrt = arith1.issquare(target)
if sqrt:
if strict:
for q, e in self.factor(sqrt, iterator=primeseq):
yield q, 2 * e
else:
yield sqrt, 2
break
if p ** 3 > target:
# there are no more square factors of target,
# thus target is squarefree
yield target, 1
break
else:
# primeseq is exhausted but target has not been proven prime
yield 1, 1
def babyspcv(bounds, sossp, sogsp, utwi, nt, disc, classnum):
"""
Compute small steps
"""
mstp1 = bounds[1] - bounds[0]
if (mstp1 == 0) or (mstp1 == 1):
q = 1
else:
tppm = misc.primePowerTest(mstp1)
q = arith1.floorsqrt(mstp1)
if (tppm[1] == 0) or (tppm[1] & 1):
q += 1
n_should_be_set = True
# initialize
c_s1 = ClassGroup(disc, classnum, []) # a subset of G
# extracting i = 0 case of main loop
for ttr in sossp.retel():
tmpx = ttr
tmpx.s_parent = ttr # tmpx belongs ttr in the set of smallstep
# index of the element
tmpx.ind = 0
c_s1.insttree(tmpx)
# main loop
x_i = nt
for i in range(1, q):
for ttr in sossp.retel():
tmpx = x_i * ttr
tmpx.s_parent = ttr # tmpx belongs ttr in the set of smallstep
if n_should_be_set and tmpx == utwi:
n = i
tmp_ss = tmpx.s_parent
tmp_gs = utwi
n_should_be_set = False
# index of the element
tmpx.ind = i
c_s1.insttree(tmpx)
x_i = nt * x_i
assert x_i == nt**q
if n_should_be_set:
sz = nt**bounds[0]
n, tmp_ss, tmp_gs = giantspcv(q, sz, x_i, c_s1, bounds, sogsp)
return ordercv(n, sossp, sogsp, nt, disc, classnum, tmp_ss, tmp_gs)
def generate(self, target, **options):
"""
Generate squarefree factors of the target number with their
valuations. The method may terminate with yielding (1, 1)
to indicate the factorization is incomplete.
If a keyword option 'strict' is False (default to True),
factorization will stop after the first square factor no
matter whether it is squarefree or not.
"""
strict = options.get('strict', True)
options['n'] = target
primeseq = self._parse_seq(options)
for p in primeseq:
if not (target % p):
e, target = arith1.vp(target, p)
yield p, e
if target == 1:
break
elif e > 1 and not strict:
yield 1, 1
break
elif trivial_test_ternary(target):
# the factor remained is squarefree.
yield target, 1
break
q, e = factor_misc.primePowerTest(target)
if e:
yield q, e
break
sqrt = arith1.issquare(target)
if sqrt:
if strict:
for q, e in self.factor(sqrt, iterator=primeseq):
yield q, 2 * e
else:
yield sqrt, 2
break
if p ** 3 > target:
# there are no more square factors of target,
# thus target is squarefree
yield target, 1
break
else:
# primeseq is exhausted but target has not been proven prime
yield 1, 1
def isfinished_trbsgs(lwrbd, bounds, h, n, sossp, sogsp, nt, disc):
"""
Determine whether bsgs is finished or not yet.
This is a submodule called by the bsgs module.
lwrbd, h, n: int
nt: element
"""
h *= n
if h >= lwrbd:
result = True
elif n == 1:
result = False
else:
bounds[0] = (bounds[0] + n - 1) // n # ceil of lower bound // n
bounds[1] = bounds[1] // n # floor of upper bound // n
q = arith1.floorsqrt(n)
if misc.primePowerTest(n)[1] != 2:
q = arith1.floorsqrt(n) + 1 # ceil of sqrt
sossp, sogsp = _update_subgrps(q, nt, sossp, sogsp, disc)
result = False
return result, h, sossp, sogsp
def isfinished_trbsgs(lwrbd, bounds, h, n, sossp, sogsp, nt, disc):
"""
Determine whether bsgs is finished or not yet.
This is a submodule called by the bsgs module.
lwrbd, h, n: int
nt: element
"""
h *= n
if h >= lwrbd:
result = True
elif n == 1:
result = False
else:
bounds[0] = (bounds[0] + n - 1) // n # ceil of lower bound // n
bounds[1] = bounds[1] // n # floor of upper bound // n
q = arith1.floorsqrt(n)
if misc.primePowerTest(n)[1] != 2:
q = arith1.floorsqrt(n) + 1 # ceil of sqrt
sossp, sogsp = _update_subgrps(q, nt, sossp, sogsp, disc)
result = False
return result, h, sossp, sogsp