def test_dmp_zz_wang_fail():
R, x, y, z = ring("x,y,z", ZZ)
UV, _x = ring("x", ZZ)
p = ZZ(nextprime(R.dmp_zz_mignotte_bound(w_1)))
assert p == 6291469
H_1 = [44 * x**2 + 42 * x + 1, 126 * x**2 - 9 * x + 28, 187 * x**2 - 23]
H_2 = [
-4 * x**2 * y - 12 * x**2 - 3 * x * y + 1,
-9 * x**2 * y - 9 * x - 2 * y, x**2 * y**2 - 9 * x**2 + y - 9
]
H_3 = [
-4 * x**2 * y - 12 * x**2 - 3 * x * y + 1,
-9 * x**2 * y - 9 * x - 2 * y, x**2 * y**2 - 9 * x**2 + y - 9
]
c_1 = -70686 * x**5 - 5863 * x**4 - 17826 * x**3 + 2009 * x**2 + 5031 * x + 74
c_2 = 9 * x**5 * y**4 + 12 * x**5 * y**3 - 45 * x**5 * y**2 - 108 * x**5 * y - 324 * x**5 + 18 * x**4 * y**3 - 216 * x**4 * y**2 - 810 * x**4 * y + 2 * x**3 * y**4 + 9 * x**3 * y**3 - 252 * x**3 * y**2 - 288 * x**3 * y - 945 * x**3 - 30 * x**2 * y**2 - 414 * x**2 * y + 2 * x * y**3 - 54 * x * y**2 - 3 * x * y + 81 * x + 12 * y
c_3 = -36 * x**4 * y**2 - 108 * x**4 * y - 27 * x**3 * y**2 - 36 * x**3 * y - 108 * x**3 - 8 * x**2 * y**2 - 42 * x**2 * y - 6 * x * y**2 + 9 * x + 2 * y
assert R.dmp_zz_diophantine(H_1, c_1, [], 5, p) == [-3 * x, -2, 1]
assert R.dmp_zz_diophantine(H_2, c_2, [ZZ(-14)], 5,
p) == [-x * y, -3 * x, -6]
assert R.dmp_zz_diophantine(H_3, c_3, [ZZ(-14)], 5, p) == [0, 0, -1]
def test_gf_ddf():
f = gf_from_dict({15: ZZ(1), 0: ZZ(-1)}, 11, ZZ)
g = [([1, 0, 0, 0, 0, 10], 1), ([1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1], 2)]
assert gf_ddf_zassenhaus(f, 11, ZZ) == g
assert gf_ddf_shoup(f, 11, ZZ) == g
f = gf_from_dict({63: ZZ(1), 0: ZZ(1)}, 2, ZZ)
g = [([1, 1], 1), ([1, 1, 1], 2), ([1, 1, 1, 1, 1, 1, 1], 3),
([
1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1,
1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1,
0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1
], 6)]
assert gf_ddf_zassenhaus(f, 2, ZZ) == g
assert gf_ddf_shoup(f, 2, ZZ) == g
f = gf_from_dict({
6: ZZ(1),
5: ZZ(-1),
4: ZZ(1),
3: ZZ(1),
1: ZZ(-1)
}, 3, ZZ)
g = [([1, 1, 0], 1), ([1, 1, 0, 1, 2], 2)]
assert gf_ddf_zassenhaus(f, 3, ZZ) == g
assert gf_ddf_shoup(f, 3, ZZ) == g
f = ZZ.map([1, 2, 5, 26, 677, 436, 791, 325, 456, 24, 577])
g = [([1, 701], 1), ([1, 110, 559, 532, 694, 151, 110, 70, 735, 122], 9)]
assert gf_ddf_zassenhaus(f, 809, ZZ) == g
assert gf_ddf_shoup(f, 809, ZZ) == g
p = ZZ(nextprime(int((2**15 * pi).evalf())))
f = gf_from_dict({15: 1, 1: 1, 0: 1}, p, ZZ)
g = [([1, 22730, 68144], 2),
([1, 64876, 83977, 10787, 12561, 68608, 52650, 88001, 84356], 4),
([1, 15347, 95022, 84569, 94508, 92335], 5)]
assert gf_ddf_zassenhaus(f, p, ZZ) == g
assert gf_ddf_shoup(f, p, ZZ) == g
def test_dmp_zz_wang():
R, x, y, z = ring("x,y,z", ZZ)
UV, _x = ring("x", ZZ)
p = ZZ(nextprime(R.dmp_zz_mignotte_bound(w_1)))
assert p == 6291469
t_1, k_1, e_1 = y, 1, ZZ(-14)
t_2, k_2, e_2 = z, 2, ZZ(3)
t_3, k_3, e_3 = y + z, 2, ZZ(-11)
t_4, k_4, e_4 = y - z, 1, ZZ(-17)
T = [t_1, t_2, t_3, t_4]
K = [k_1, k_2, k_3, k_4]
E = [e_1, e_2, e_3, e_4]
T = zip([t.drop(x) for t in T], K)
A = [ZZ(-14), ZZ(3)]
S = R.dmp_eval_tail(w_1, A)
cs, s = UV.dup_primitive(S)
assert cs == 1 and s == S == \
1036728*_x**6 + 915552*_x**5 + 55748*_x**4 + 105621*_x**3 - 17304*_x**2 - 26841*_x - 644
assert R.dmp_zz_wang_non_divisors(E, cs, ZZ(4)) == [7, 3, 11, 17]
assert UV.dup_sqf_p(s) and UV.dup_degree(s) == R.dmp_degree(w_1)
_, H = UV.dup_zz_factor_sqf(s)
h_1 = 44 * _x**2 + 42 * _x + 1
h_2 = 126 * _x**2 - 9 * _x + 28
h_3 = 187 * _x**2 - 23
assert H == [h_1, h_2, h_3]
LC = [lc.drop(x) for lc in [-4 * y - 4 * z, -y * z**2, y**2 - z**2]]
assert R.dmp_zz_wang_lead_coeffs(w_1, T, cs, E, H, A) == (w_1, H, LC)
factors = R.dmp_zz_wang_hensel_lifting(w_1, H, LC, A, p)
assert R.dmp_expand(factors) == w_1
def genprime(n, K):
return K(nextprime(int((2**n * pi).evalf())))
def next_prime(n: Union[int, float, complex]) -> int:
try:
from sympy.ntheory.generate import nextprime
except ModuleNotFoundError:
raise Exception("Install sympy to use number-theoretic functions!")
return nextprime(n.real) # exclusive of n
def generate_prime(self):
self.prime = generate.nextprime(self.sys_seed)
from Crypto.Util.number import *
from sage.all import *
from sympy.ntheory.generate import prevprime, nextprime
n = 22001778874542774315484392481115711539281104740723517828461360611903057304469869336789715900703500619163822273767393143914615001907123143200486464636351989898613180095341102875678204218769723325121832871221496816486100959384589443689594053640486953989205859492780929786509801664036223045197702752965199575588498118481259145703054094713019549136875163271600746675338534685099132138833920166786918380439074398183268612427028138632848870032333985485970488955991639327
c = 1067382668222320523824132555613324239857438151855225316282176402453660987952614935478188752664288189856467574123997124118639803436040589761488611318906877644244524931837804614243835412551576647161461088877884786181205274671088951504353502973964810690277238868854693198170257109413583371510824777614377906808757366142801309478368968340750993831416162099183649651151826983793949933939474873893278527484810417812120138131555544749220438456366110721231219155629863865
e = 65537
head = 0
tail = Integer(Integer(n).n().nth_root(2))
# p and q1 and q2 are chained together, so we can simply bruteforce p to get q1 and q1
# Binary search is faster!
while True:
mid = head + (tail - head) // 2
p = nextprime(mid)
q1 = nextprime(2 * p)
q2 = nextprime(3 * q1)
diff = (p * q1 * q2) - n
if diff > 0:
tail = mid
elif diff < 0:
head = mid
else:
break
phi = (p - 1) * (q1 - 1) * (q2 - 1)
tup = xgcd(e, phi)
d = Integer(mod(tup[1], phi))
def main():
lineLength, baseNum = getNumFromFile(PROTOTYPE_FILE)
prime = nextprime(baseNum)
printNumToFile(OUTPUT_FILE, prime, lineLength)
def test_gf_factor():
assert gf_factor([], 11, ZZ) == (0, [])
assert gf_factor([1], 11, ZZ) == (1, [])
assert gf_factor([1, 1], 11, ZZ) == (1, [([1, 1], 1)])
assert gf_factor_sqf([], 11, ZZ) == (0, [])
assert gf_factor_sqf([1], 11, ZZ) == (1, [])
assert gf_factor_sqf([1, 1], 11, ZZ) == (1, [[1, 1]])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor_sqf([], 11, ZZ) == (0, [])
assert gf_factor_sqf([1], 11, ZZ) == (1, [])
assert gf_factor_sqf([1, 1], 11, ZZ) == (1, [[1, 1]])
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor_sqf([], 11, ZZ) == (0, [])
assert gf_factor_sqf([1], 11, ZZ) == (1, [])
assert gf_factor_sqf([1, 1], 11, ZZ) == (1, [[1, 1]])
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor_sqf(ZZ.map([]), 11, ZZ) == (0, [])
assert gf_factor_sqf(ZZ.map([1]), 11, ZZ) == (1, [])
assert gf_factor_sqf(ZZ.map([1, 1]), 11, ZZ) == (1, [[1, 1]])
f, p = ZZ.map([1, 0, 0, 1, 0]), 2
g = (1, [([1, 0], 1), ([1, 1], 1), ([1, 1, 1], 1)])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
g = (1, [[1, 0], [1, 1], [1, 1, 1]])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor_sqf(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor_sqf(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor_sqf(f, p, ZZ) == g
f, p = gf_from_int_poly([1, -3, 1, -3, -1, -3, 1], 11), 11
g = (1, [([1, 1], 1), ([1, 5, 3], 1), ([1, 2, 3, 4], 1)])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
f, p = [1, 5, 8, 4], 11
g = (1, [([1, 1], 1), ([1, 2], 2)])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
f, p = [1, 1, 10, 1, 0, 10, 10, 10, 0, 0], 11
g = (1, [([1, 0], 2), ([1, 9, 5], 1), ([1, 3, 0, 8, 5, 2], 1)])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
f, p = gf_from_dict({32: 1, 0: 1}, 11, ZZ), 11
g = (1, [([1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 10], 1),
([1, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 10], 1)])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
f, p = gf_from_dict({32: ZZ(8), 0: ZZ(5)}, 11, ZZ), 11
g = (8, [([1, 3], 1), ([1, 8], 1), ([1, 0, 9], 1), ([1, 2, 2], 1),
([1, 9, 2], 1), ([1, 0, 5, 0, 7], 1), ([1, 0, 6, 0, 7], 1),
([1, 0, 0, 0, 1, 0, 0, 0, 6], 1), ([1, 0, 0, 0, 10, 0, 0, 0,
6], 1)])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
f, p = gf_from_dict({63: ZZ(8), 0: ZZ(5)}, 11, ZZ), 11
g = (8, [([1, 7], 1), ([1, 4, 5], 1), ([1, 6, 8, 2], 1), ([1, 9, 9, 2], 1),
([1, 0, 0, 9, 0, 0, 4], 1), ([1, 2, 0, 8, 4, 6, 4], 1),
([1, 2, 3, 8, 0, 6, 4], 1), ([1, 2, 6, 0, 8, 4, 4], 1),
([1, 3, 3, 1, 6, 8, 4], 1), ([1, 5, 6, 0, 8, 6, 4], 1),
([1, 6, 2, 7, 9, 8, 4], 1), ([1, 10, 4, 7, 10, 7, 4], 1),
([1, 10, 10, 1, 4, 9, 4], 1)])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
# Gathen polynomials: x**n + x + 1 (mod p > 2**n * pi)
p = ZZ(nextprime(int((2**15 * pi).evalf())))
f = gf_from_dict({15: 1, 1: 1, 0: 1}, p, ZZ)
assert gf_sqf_p(f, p, ZZ) is True
g = (1, [([1, 22730, 68144], 1), ([1, 81553, 77449, 86810, 4724], 1),
([1, 86276, 56779, 14859, 31575], 1),
([1, 15347, 95022, 84569, 94508, 92335], 1)])
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
g = (1, [[1, 22730, 68144], [1, 81553, 77449, 86810, 4724],
[1, 86276, 56779, 14859, 31575],
[1, 15347, 95022, 84569, 94508, 92335]])
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor_sqf(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor_sqf(f, p, ZZ) == g
# Shoup polynomials: f = a_0 x**n + a_1 x**(n-1) + ... + a_n
# (mod p > 2**(n-2) * pi), where a_n = a_{n-1}**2 + 1, a_0 = 1
p = ZZ(nextprime(int((2**4 * pi).evalf())))
f = ZZ.map([1, 2, 5, 26, 41, 39, 38])
assert gf_sqf_p(f, p, ZZ) is True
g = (1, [([1, 44, 26], 1), ([1, 11, 25, 18, 30], 1)])
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
g = (1, [[1, 44, 26], [1, 11, 25, 18, 30]])
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor_sqf(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor_sqf(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'other')
raises(KeyError, lambda: gf_factor([1, 1], 11, ZZ))
config.setup('GF_FACTOR_METHOD')