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 testFloorsqrt(self):
self.assertEqual(0, arith1.floorsqrt(0))
self.assertEqual(1, arith1.floorsqrt(1))
self.assertEqual(1, arith1.floorsqrt(3))
self.assertEqual(2, arith1.floorsqrt(4))
self.assertEqual(3, arith1.floorsqrt(10))
self.assertEqual(arith1.floorsqrt(400000000000000000000), 20000000000)
self.assertEqual(arith1.floorsqrt(400000000000000000000 - 1),
19999999999)
self.assertTrue(arith1.floorsqrt(2**60 - 1)**2 <= 2**60 - 1)
self.assertTrue(arith1.floorsqrt(2**59 - 1)**2 <= 2**59 - 1)
def _lucas_test_sequence(n, a, b):
"""
Return x_0, x_1, x_m, x_{m+1} of Lucas sequence of parameter a, b,
where m = (n - (a**2 - 4*b / n)) // 2.
"""
d = a**2 - 4*b
if (d >= 0 and arith1.floorsqrt(d) ** 2 == d) \
or not(gcd.coprime(n, 2*a*b*d)):
raise ValueError("Choose another parameters.")
x_0 = 2
inv_b = arith1.inverse(b, n)
x_1 = ((a ** 2) * inv_b - 2) % n
# Chain functions
def even_step(u):
"""
'double' u.
"""
return (u**2 - x_0) % n
def odd_step(u, v):
"""
'add' u and v.
"""
return (u*v - x_1) % n
m = (n - arith1.legendre(d, n)) // 2
x_m, x_mplus1 = Lucas_chain(m, even_step, odd_step, x_0, x_1)
return x_0, x_1, x_m, x_mplus1
def _lucas_test_sequence(n, a, b):
"""
Return x_0, x_1, x_m, x_{m+1} of Lucas sequence of parameter a, b,
where m = (n - (a**2 - 4*b / n)) >> 1.
"""
d = a**2 - 4 * b
if (d >= 0 and arith1.floorsqrt(d) ** 2 == d) \
or not(gcd.coprime(n, 2*a*b*d)):
raise ValueError("Choose another parameters.")
x_0 = 2
inv_b = arith1.inverse(b, n)
x_1 = ((a**2) * inv_b - 2) % n
# Chain functions
def even_step(u):
"""
'double' u.
"""
return (u**2 - x_0) % n
def odd_step(u, v):
"""
'add' u and v.
"""
return (u * v - x_1) % n
m = (n - arith1.legendre(d, n)) >> 1
x_m, x_mplus1 = Lucas_chain(m, even_step, odd_step, x_0, x_1)
return x_0, x_1, x_m, x_mplus1
def cornacchia(d, p):
"""
Return the solution of x^2 + d * y^2 = p .
p be a prime and d be an integer such that 0 < d < p.
"""
if (d <= 0) or (d >= p):
raise ValueError("invalid input")
k = arith1.legendre(-d, p)
if k == -1:
raise ValueError("no solution")
x0 = arith1.modsqrt(-d, p)
if x0 < (p / 2):
x0 = p - x0
a = p
b = x0
l = arith1.floorsqrt(p)
while b > l:
a, b = b, a % b
c, r = divmod(p - b * b, d)
if r:
raise ValueError("no solution")
t = arith1.issquare(c)
if t == 0:
raise ValueError("no solution")
else:
return (b, t)
def cornacchiamodify(d, p):
"""
Algorithm 26 (Modified cornacchia)
Input : p be a prime and d be an integer such that d < 0 and d > -4p with
d = 0, 1 (mod 4)
Output : the solution of u^2 -d * v^2 = 4p.
"""
q = 4 * p
if (d >= 0) or (d <= -q):
raise ValueError("invalid input")
if p == 2:
b = arith1.issquare(d + 8)
if b:
return (b, 1)
else:
raise ValueError("no solution")
if arith1.legendre(d, p) == -1:
raise ValueError("no solution")
x0 = arith1.modsqrt(d, p)
if (x0 - d) & 1:
x0 = p - x0
a = 2 * p
b = x0
l = arith1.floorsqrt(q)
while b > l:
a, b = b, a % b
c, r = divmod(q - b * b, -d)
if r:
raise ValueError("no solution")
t = arith1.issquare(c)
if t:
return (b, t)
else:
raise ValueError("no solution")
def cornacchia(d, p):
"""
Return the solution of x^2 + d * y^2 = p .
p be a prime and d be an integer such that 0 < d < p.
"""
if (d <= 0) or (d >= p):
raise ValueError("invalid input")
k = arith1.legendre(-d, p)
if k == -1:
raise ValueError("no solution")
x0 = arith1.modsqrt(-d, p)
if x0 < (p / 2):
x0 = p - x0
a = p
b = x0
l = arith1.floorsqrt(p)
while b > l:
a, b = b, a % b
c, r = divmod(p - b * b, d)
if r:
raise ValueError("no solution")
t = arith1.issquare(c)
if t == 0:
raise ValueError("no solution")
else:
return (b, t)
def cornacchiamodify(d, p):
"""
Algorithm 26 (Modified cornacchia)
Input : p be a prime and d be an integer such that d < 0 and d > -4p with
d = 0, 1 (mod 4)
Output : the solution of u^2 -d * v^2 = 4p.
"""
q = 4 * p
if (d >= 0) or (d <= -q):
raise ValueError("invalid input")
if p == 2:
b = arith1.issquare(d + 8)
if b:
return (b, 1)
else:
raise ValueError("no solution")
if arith1.legendre(d, p) == -1:
raise ValueError("no solution")
x0 = arith1.modsqrt(d, p)
if (x0 - d) % 2 != 0:
x0 = p - x0
a = 2 * p
b = x0
l = arith1.floorsqrt(q)
while b > l:
a, b = b, a % b
c, r = divmod(q - b * b, -d)
if r:
raise ValueError("no solution")
t = arith1.issquare(c)
if t:
return (b, t)
else:
raise ValueError("no solution")
def trialDivision(n, bound=0):
"""
Trial division primality test for an odd natural number.
Optional second argument is a search bound of primes.
If the bound is given and less than the sqaure root of n
and True is returned, it only means there is no prime factor
less than the bound.
"""
if bound:
m = min(bound, arith1.floorsqrt(n))
else:
m = arith1.floorsqrt(n)
for p in bigrange.range(3, m+1, 2):
if not (n % p):
return False
return True
def trialDivision(n, bound=0):
"""
Trial division primality test for an odd natural number.
Optional second argument is a search bound of primes.
If the bound is given and less than the sqaure root of n
and True is returned, it only means there is no prime factor
less than the bound.
"""
if bound:
m = min(bound, arith1.floorsqrt(n))
else:
m = arith1.floorsqrt(n)
for p in bigrange.range(3, m + 1, 2):
if not (n % p):
return False
return True
def generator_eratosthenes(n):
"""
Generate primes up to n (inclusive) using Eratosthenes sieve.
"""
if n < 2:
return
yield 2
if n <= 2:
return
yield 3
# make list for sieve
sieve_len_max = (n + 1) >> 1
sieve = [True, False, True]
sieve_len = 3
k = 5
i = 2
while sieve_len < sieve_len_max:
if sieve[i]:
yield k
sieve_len *= k
if sieve_len_max < sieve_len:
sieve_len //= k
# adjust sieve list length
sieve *= sieve_len_max // sieve_len
sieve += sieve[:(sieve_len_max - len(sieve))]
sieve_len = sieve_len_max
else:
sieve = sieve * k
for j in range(i, sieve_len, k):
sieve[j] = False
k += 2
i += 1
# sieve
limit = arith1.floorsqrt(n)
while k <= limit:
if sieve[i]:
yield k
j = (k**2 - 1) >> 1
while j < sieve_len_max:
sieve[j] = False
j += k
k += 2
i += 1
# output result
limit = (n - 1) >> 1
while i <= limit:
if sieve[i]:
yield 2 * i + 1
i += 1
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 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 apr(n):
"""
apr is the main function for Adleman-Pomerance-Rumery primality test.
Assuming n has no prime factors less than 32.
Assuming n is spsp for several bases.
"""
L = Status()
rb = arith1.floorsqrt(n) + 1
el = TestPrime()
while el.et <= rb:
el = el.next()
plist = el.t.factors.keys()
plist.remove(2)
L.yet(2)
for p in plist:
if pow(n, p-1, p*p) != 1:
L.done(p)
else:
L.yet(p)
qlist = el.et.factors.keys()
qlist.remove(2)
J = JacobiSum()
for q in qlist:
for p in plist:
if (q-1) % p != 0:
continue
if not L.subodd(p, q, n, J):
return False
k = arith1.vp(q-1, 2)[0]
if k == 1:
if not L.sub2(q, n):
return False
elif k == 2:
if not L.sub4(q, n, J):
return False
else:
if not L.sub8(q, k, n, J):
return False
for p in L.yet_keys():
if not L.subrest(p, n, el.et, J):
return False
r = int(n)
for _ in bigrange.range(1, el.t.integer):
r = (r*n) % el.et.integer
if n % r == 0 and r != 1 and r != n:
_log.info("%s divides %s.\n" %(r, n))
return False
return True
def apr(n):
"""
apr is the main function for Adleman-Pomerance-Rumery primality test.
Assuming n has no prime factors less than 32.
Assuming n is spsp for several bases.
"""
L = Status()
rb = arith1.floorsqrt(n) + 1
el = TestPrime()
while el.et <= rb:
el = next(el)
plist = list(el.t.factors.keys())
plist.remove(2)
L.yet(2)
for p in plist:
if pow(n, p - 1, p * p) != 1:
L.done(p)
else:
L.yet(p)
qlist = list(el.et.factors.keys())
qlist.remove(2)
J = JacobiSum()
for q in qlist:
for p in plist:
if (q - 1) % p != 0:
continue
if not L.subodd(p, q, n, J):
return False
k = arith1.vp(q - 1, 2)[0]
if k == 1:
if not L.sub2(q, n):
return False
elif k == 2:
if not L.sub4(q, n, J):
return False
else:
if not L.sub8(q, k, n, J):
return False
for p in L.yet_keys():
if not L.subrest(p, n, el.et, J):
return False
r = int(n)
for _ in bigrange.range(1, el.t.integer):
r = (r * n) % el.et.integer
if n % r == 0 and r != 1 and r != n:
_log.info("%s divides %s.\n" % (r, n))
return False
return True
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 _parse_seq(self, options):
"""
Parse 'options' to define trial sequaence.
"""
if 'start' in options and 'stop' in options:
if 'step' in options:
trials = bigrange.range(options['start'], options['stop'], options['step'])
else:
trials = bigrange.range(options['start'], options['stop'])
elif 'iterator' in options:
trials = options['iterator']
elif 'eratosthenes' in options:
trials = prime.generator_eratosthenes(options['eratosthenes'])
elif options['n'] < 1000000:
trials = prime.generator_eratosthenes(arith1.floorsqrt(options['n']))
else:
trials = prime.generator()
return trials
def cornacchia_smith(p, d):
'''
modified Cornacchia's Algorithm to solve a^2 + b^2 |D| = 4p for a and b
Args:
p:
d:
Returns:
a, b such that a^2 + b^2 |D| = 4p
'''
# check input
if not -4 * p < d < 0:
raise ValueError(" -4p < D < 0 not true.")
elif not (d % 4 in {0, 1}):
raise ValueError(" D = 0, 1 (mod 4) not true.")
# case where p=2
if p == 2:
r = sqrt(d + 8)
if r != -1:
return r, 1
else:
return None
# test for solvability
if jacobi(d % p, p) < 1:
return None
x = modsqrt(d, p)
if (x % 2) != (d % 2):
x = p - x
# euclid chain
a, b = (2 * p, x)
c = floorsqrt(4 * p)
while b > c:
a, b = b, a % b
t = 4 * p - b * b
if t % (-d) != 0:
return None
if not issquare(t / (-d)):
return None
return b, int(mpmath.sqrt(t / -d))
def _parse_seq(self, options):
"""
Parse 'options' to define trial sequaence.
"""
if 'start' in options and 'stop' in options:
if 'step' in options:
trials = bigrange.range(options['start'], options['stop'],
options['step'])
else:
trials = bigrange.range(options['start'], options['stop'])
elif 'iterator' in options:
trials = options['iterator']
elif 'eratosthenes' in options:
trials = prime.generator_eratosthenes(options['eratosthenes'])
elif options['n'] < 1000000:
trials = prime.generator_eratosthenes(
arith1.floorsqrt(options['n']))
else:
trials = prime.generator()
return trials
def mpqsfind(n, s=0, f=0, m=0, verbose=False):
"""
This is main function of MPQS.
Arguments are (composite_number, sieve_range, factorbase_size, multiplier)
You must input composite_number at least.
"""
# verbosity
if verbose:
_log.setLevel(logging.DEBUG)
_log.debug("verbose")
else:
_log.setLevel(logging.NOTSET)
starttime = time.time()
M = MPQS(n, s, f, m)
_log.info("Sieve range is [%d, %d]" % (M.move_range[0], M.move_range[-1]))
_log.info("Factorbase size = %d, Max Factorbase %d" % (len(M.FB), M.maxFB))
M.get_vector()
N = M.number // M.multiplier
V = Elimination(M.smooth)
A = V.gaussian()
_log.info("Found %d linearly dependent relations" % len(A))
differences = []
for i in A:
B = V.history[i].keys()
X = 1
Y = 1
for j in B:
X *= M.smooth[j][1][0]
Y *= M.smooth[j][1][1]
Y = Y % M.number
X = arith1.floorsqrt(X) % M.number
if X != Y:
differences.append(X-Y)
for diff in differences:
divisor = gcd.gcd(diff, N)
if 1 < divisor < N:
_log.info("Total time = %f sec" % (time.time() - starttime))
return divisor
def upper_bound_of_coefficient(f):
"""
upper_bound_of_coefficient(polynomial) -> int
Compute Landau-Mignotte bound of coefficients of factors, whose
degree is no greater than half of the given polynomial. The given
polynomial must have integer coefficients.
"""
weight = 0
for c in f.itercoefficients():
weight += abs(c)**2
weight = arith1.floorsqrt(weight) + 1
degree = f.degree()
lc = f[degree]
m = degree // 2 + 1
bound = 1
for i in range(1, m):
b = combinatorial.binomial(m - 1, i) * weight + \
combinatorial.binomial(m - 1, i - 1) * lc
if bound < b:
bound = b
return bound
def upper_bound_of_coefficient(f):
"""
upper_bound_of_coefficient(polynomial) -> int
Compute Landau-Mignotte bound of coefficients of factors, whose
degree is no greater than half of the given polynomial. The given
polynomial must have integer coefficients.
"""
weight = 0
for c in f.itercoefficients():
weight += abs(c)**2
weight = arith1.floorsqrt(weight) + 1
degree = f.degree()
lc = f[degree]
m = degree // 2 + 1
bound = 1
for i in range(1, m):
b = combinatorial.binomial(m - 1, i) * weight + \
combinatorial.binomial(m - 1, i - 1) * lc
if bound < b:
bound = b
return bound
def testFloorpowerroot(self):
self.assertEqual(0, arith1.floorpowerroot(0, 1))
self.assertEqual(0, arith1.floorpowerroot(0, 4))
self.assertEqual(0, arith1.floorpowerroot(0, 7))
self.assertEqual(1, arith1.floorpowerroot(1, 2))
self.assertEqual(1, arith1.floorpowerroot(1, 6))
self.assertEqual(1, arith1.floorpowerroot(1, 9))
self.assertEqual(1, arith1.floorpowerroot(2, 3))
self.assertEqual(1, arith1.floorpowerroot(2, 7))
self.assertEqual(2, arith1.floorpowerroot(8, 3))
self.assertEqual(2, arith1.floorpowerroot(128, 7))
self.assertEqual((5, 5), arith1.floorpowerroot(5, 1, True))
self.assertEqual((5, 25), arith1.floorpowerroot(27, 2, True))
self.assertEqual((0, 0), arith1.floorpowerroot(0, 7, True))
self.assertEqual((3, 243), arith1.floorpowerroot(245, 5, True))
self.assertEqual((-3, -243), arith1.floorpowerroot(-245, 5, True))
for j in range(3, 100, 10):
k = 5**j
self.assertEqual(4, arith1.floorpowerroot(k - 1, j))
self.assertEqual(5, arith1.floorpowerroot(k, j))
self.assertEqual(5, arith1.floorpowerroot(k + 1, j))
self.assertEqual(arith1.floorpowerroot(400000000000000000000, 4),
arith1.floorsqrt(20000000000))
def trialDivision(n, **options):
"""
Return a factor of given integer by trial division.
options can be either:
1) 'start' and 'stop' as range parameters.
2) 'iterator' as an iterator of primes.
If both options are not given, prime factor is searched from 2
to the square root of the given integer.
"""
# verbosity
verbose = options.get('verbose', False)
if not verbose:
_silence()
if 'start' in options and 'stop' in options:
if 'step' in options:
trials = range(options['start'], options['stop'], options['step'])
else:
trials = range(options['start'], options['stop'])
elif 'iterator' in options:
trials = options['iterator']
elif n < 1000000:
trials = prime.generator_eratosthenes(arith1.floorsqrt(n))
else:
trials = prime.generator()
for p in trials:
if not (n % p):
if not verbose:
_verbose()
return p
if p ** 2 > n:
break
if not verbose:
_verbose()
return 1
def _calc_bound(n, bound=0):
if bound:
m = min((bound, arith1.floorsqrt(n)))
else:
m = arith1.floorsqrt(n)
return m
def _calc_bound(n, bound=0):
if bound:
m = min((bound, arith1.floorsqrt(n)))
else:
m = arith1.floorsqrt(n)
return m