def pohlig_hellman(g, h, M, factors):
M1 = M - 1
xs = []
for f in factors:
pf = f
pe = factors[f]
subgroup_exponent = gmpy2.div(M1, gmpy2.powmod(pf, pe, M))
gi = gmpy2.powmod(g, subgroup_exponent, M)
hi = gmpy2.powmod(h, subgroup_exponent, M)
xi = log_prime_power(gi, hi, pf, pe, M)
xs.append(xi)
crt_coeffs = []
for f in factors:
pf = f
pe = factors[f]
mi = pf**pe
bi = gmpy2.div(M, mi)
bi_inv = gmpy2.invert(bi, mi)
crt_coeffs.append(gmpy2.mul(bi, bi_inv))
x = 0
for i in range(len(crt_coeffs)):
x = gmpy2.t_mod(x + gmpy2.t_mod(xs[i] * crt_coeffs[i], M1), M1)
return x
def ch3_factor(N):
""" Valid when |3p - 2q| < N^(1/4)
"""
A = ceil_sqrt(6 * N)
# let M = (3p+2q)/2
# M is not an integer since 3p + 2q is odd
# So there is some integer A = M + 0.5 and some integer i such that
# 3p = M + i - 0.5 = A + i - 1
# and
# 2q = M - i + 0.5 = A - i
#
# N = pq = (A-i)(A+i-1)/6 = (A^2 - i^2 - A + i)/6
# So 6N = A^2 - i^2 - A + i
# i^2 - i = A^2 - A - 6N
# Solve using the quadratic equation!
a = mpz(1)
b = mpz(-1)
c = -(A**2 - A - 6 * N)
det = b**2 - 4 * a * c
roots = (div(-b + isqrt(b**2 - 4 * a * c),
2 * a), div(-b - isqrt(b**2 - 4 * a * c), 2 * a))
for i in roots:
if i >= 0:
f = check_ch3(i, A, N)
if f:
return f
# We should have found the root
assert (False)
def calc_near6(self):
""" Solves the Extra Credit question Q3
See:
Uses only integer arithmetic to avoid issues with rounding errors
Solution credit to Francois Degros:
https://class.coursera.org/crypto-011/forum/thread?thread_id=517#post-2279
:return: the prime factors of ```self.n```, p and q
:rtype: tuple
"""
# A = ceil(sqrt(24 N)) - the use of isqrt() won't matter, as we seek the ceil
A = add(isqrt(mul(self.n, mpz(24))), mpz(1))
# D = A^2 - 24 N
D = sub(mul(A, A), mul(24, self.n))
# E = sqrt(D) - note D is a perfect square and we can use integer arithmetic
E = isqrt(D)
assert sub(mul(E, E), D) == mpz(0)
p = div(sub(A, E), 6)
q = div(add(A, E), 4)
if self._check_sol(p, q):
return p, q
# The above is the right solution, however, there is another possible solution:
p = div(add(A, E), 6)
q = div(sub(A, E), 4)
if self._check_sol(p, q):
return p, q
print 'Could not find a solution'
return 0, 0
def main(argv):
lines = files.read_lines(argv[0])
N_1 = mpz(lines[0])
N_2 = mpz(lines[1])
N_3 = mpz(lines[2])
lines = files.read_lines(argv[1])
C = mpz(lines[0])
E = int(lines[1])
p1, q1, i1 = number.factorize(N_1, 1)
p2, q2, i2 = number.factorize(N_2, 1048576)
p3, q3, i3 = number.factorize(mul(N_3, 24), 1)
print
print 'Factorization Demo'
print
print ' 1st:'
print 'Iterations: ', i1
print ' p: ', p1
print ' q: ', q1
print
print ' 2nd:'
print 'Iterations: ', i2
print ' p: ', p2
print ' q: ', q2
print
print ' 3rd:'
print 'Iterations: ', i3
print ' p: ', div(p3, 6)
print ' q: ', div(q3, 4)
print
print
print ' Plaintext: ', encdec.rsa_decrypt(C, E, p1, q1, N_1)
print
def runPart3(N):
A2 = add(isqrt(mul(24, N)), 1)
x2 = isqrt(sub(mul(A2, A2), mul(24, N)))
p = div(sub(A2, x2), 6)
q = div(add(A2, x2), 4)
return p, q
def solve_quadratic(a, b, c):
""" Solves the quadratic equation ```a x2 + b x + c = 0```
:return: the GMP result of solving the quadratic equation usign multi-precision numbers
"""
bb = sqrt(sub(mul(b, b), mul(mpz(4), mul(a, c))))
x1 = gmpy2.div(sub(-b, bb), mul(mpz(2), a))
x2 = gmpy2.div(add(-b, bb), mul(mpz(2), a))
return x1, x2
def check_ch3(i, A, N):
p, q = (div(A + i - 1, 3), div(A - i, 2))
if check_factors(p, q, N):
return p, q
p, q = (div(A - i, 3), div(A + i - 1, 2))
if check_factors(p, q, N):
return p, q
return None
def check_ch3(i,A,N):
p,q = (div(A + i - 1,3), div(A - i,2))
if check_factors(p,q,N):
return p,q
p,q = (div(A - i,3), div(A + i - 1,2))
if check_factors(p,q,N):
return p,q
return None
def challenge3():
"""
Valid when |3p - 2q| < N^(1/4)
Solving
>>> 6N = A^2 - i^2 - A + i
>>> [ax^2 + bx + c = 0]
>>> 1i^2 - 1i - (A^2 - A - 6N) = 0
Therefore:
a = 1
b = -1
c = (A^2 - A - 6N)
"""
print("> Challenge 3")
N = 720062263747350425279564435525583738338084451473999841826653057981916355690188337790423408664187663938485175264994017897083524079135686877441155132015188279331812309091996246361896836573643119174094961348524639707885238799396839230364676670221627018353299443241192173812729276147530748597302192751375739387929
A = gmpy2.ceil(gmpy2.sqrt(6*N))
a = gmpy2.mpz(1)
b = gmpy2.mpz(-1)
c = gmpy2.mpz(-(A**2 - A - 6*N))
# Solving using quadratic formula
#
# x = (-b (+/-) b^2 - 4ac) / 2a
#
# b^2 - 4ac
det = gmpy2.isqrt(b**2 - 4*a*c)
plus = gmpy2.div(-b + det, 2*a)
minus = gmpy2.div(-b - det, 2*a)
answers = [plus, minus]
for i in answers:
# check for correct answer
p = gmpy2.mpz(gmpy2.div(A + i - 1, 3))
q = gmpy2.mpz(gmpy2.div(A - i, 2))
# values found
if p*q == N:
printSmallest(p, q)
return
raise(Exception("[Challenge 3] > Could not find p or q"))
def isqrt(x):
'''
Writing your own square root function
'''
if x < 0: raise ValueError('square root not defined for negative numbers')
n = mpz(x)
if n == 0: return 0
a, b = t_divmod(n.bit_length(), mpz(2))
x = 2**(a + b)
while True:
y = div(add(x, div(n, x)), 2)
if y >= x: return x
x = y
def isqrt(x):
'''
Writing your own square root function
'''
if x < 0: raise ValueError('square root not defined for negative numbers')
n = mpz(x)
if n == 0: return 0
a, b = t_divmod(n.bit_length(), mpz(2))
x = 2**(a + b)
while True:
y = div(add(x, div(n, x)), 2 )
if y >= x: return x
x = y
def computeN3Factors():
A = gmpy2.ceil(gmpy2.mul(2, gmpy2.sqrt(gmpy2.mul(6, N3))))
X = gmpy2.ceil(
gmpy2.sqrt(
gmpy2.sub(
pow(A, 2),
gmpy2.mul(24, N3)
)
)
)
p = gmpy2.ceil(gmpy2.div(gmpy2.sub(A, X), 6)) # Only round one.
q = gmpy2.div(N3, p)
confirmed(N3, p, q)
def computepi():
# N: number of decimals
f = StringIO()
w = xmpz(0)
k = 1
n1 = xmpz(4)
n2 = xmpz(3)
d = xmpz(1)
f10 = xmpz(10)
n10 = xmpz(-10)
i = 0
URL = request.args.get('URL', type=str)
N = request.args.get('N', default=10, type=int)
lenght = request.args.get('lenght', default=1, type=int)
while True:
# digit
u = int(div(n1, d))
v = int(div(n2, d))
if u == v:
f.write(chr(48 + u))
i += 1
if i % 10 == 0:
f.write("\t:%d\n" % i)
if i == N:
break
# extract
u = mul(d, mul(n10, u))
n1 = mul(n1, f10)
n1 = add(n1, u)
n2 = mul(n2, f10)
n2 = add(n2, u)
else:
# produce
k2 = k << 1
u = mul(n1, k2 - 1)
v = add(n2, n2)
w = mul(n1, k - 1)
n1 = add(u, v)
u = mul(n2, k + 2)
n2 = add(w, u)
d = mul(d, k2 + 1)
k += 1
if lenght != 0:
contents = urllib.request.urlopen(URL + "?N=" + str(N) + "&lenght=" +
str(lenght - 1)).read()
return (f.getvalue())
def Dec(self, PP, sk, c):
p = self.PP['p']
a = c['a']
b = c['b']
m = 0
if b < 0:
b_mod = -b
m_mod = gmpy2.div(a, b_mod**sk) % p
m = -m_mod
else:
b_mod = b
m_mod = gmpy2.div(a, b_mod**sk) % p
m = m_mod
return m
def bruteSmall(N):
zero = g.mpz(0)
two = g.mpz(2)
if g.f_mod(N, two) == zero:
return two, g.mpz(g.div(N, two))
i = g.mpz(3)
while True:
if g.f_mod(N, i) == zero:
p = g.mpz(i)
q = g.mpz( g.div(N,i) )
if checkFactors(p,q,N) :
return p,q
i = g.add(i, two)
def find_factors_6n(n):
n24 = gmp.mul(n, 24)
a = gmp.isqrt(n24)
p = 0
q = 0
while not check(n, p, q):
a = gmp.add(a, 1)
a2 = gmp.mul(a, a)
x = gmp.isqrt(gmp.sub(a2, n24))
p = gmp.mpz(gmp.div(gmp.sub(a, x), 6))
q = gmp.mpz(gmp.div(gmp.add(a, x), 4))
return (p, q)
def attack3(N):
"""
|3p-2q|<N^(1/4)
"""
N *= 4
A, r = gmpy2.isqrt_rem(6 * N)
if (r > 0):
A += 1
x, r = gmpy2.isqrt_rem(A * A - 6 * N)
if (r > 0):
x += 1
if (N % div(A - x, 6) == 0 and N % div(A + x, 4) == 0):
print("Factors are ", div(A - x, 6), div(A + x, 4))
def regenerate_key(bits,x,x_i,L,leave):
""" Regenerate the key r when some peer join or leave the team. All the variables' name are the same as those used in the paper.
Args:
bits: int, the number of bits of p, which can be set to be larger than the length of r.
x: list, x_i which has been distributed members.
x_i: int, the x_i of a member who wants to join or leave the team.
L: int, the value of product of x_is before the member join or leave.
leave: bool. True if the x_i is leaving the team. False if the x_i is join the team.
"""
p=generate_large_prime(bits)
(m,q)=generate_m(p)
delta= generate_large_prime(len(bin(min(x)))-2)
k=delta-p
a=generate_large_prime(bits)
g=gmpy2.powmod(a,q,m)
r=gmpy2.powmod(g,k,m)
if leave :
L=gmpy2.div(L,x_i)
else:
L=gmpy2.mul(L,x_i)
u=gmpy2.invert(delta,L)
while get_key(g,m,u,x[0])!=r:
p=generate_large_prime(bits)
(m,q)=generate_m(p)
delta= generate_large_prime(len(bin(min(x)))-2)
k=delta-p
a=generate_large_prime(bits)
g=gmpy2.powmod(a,q,m)
r=gmpy2.powmod(g,k,m)
if leave :
L=gmpy2.div(L,x_i)
else:
L=gmpy2.mul(L,x_i)
u=gmpy2.invert(delta,L)
return (g,m,u,L,r)
def compute_middle_point_given_interval(low, upper):
with gmpy2.local_context(gmpy2.context(),
round=gmpy2.RoundDown,
precision=mpfr_proxy_precision) as ctx:
res_left = gmpy2.add(gmpy2.div(mpfr(upper), mpfr("2.0")),
gmpy2.div(mpfr(low), mpfr("2.0")))
with gmpy2.local_context(gmpy2.context(),
round=gmpy2.RoundUp,
precision=mpfr_proxy_precision) as ctx:
res_right = gmpy2.add(gmpy2.div(mpfr(upper), mpfr("2.0")),
gmpy2.div(mpfr(low), mpfr("2.0")))
return Interval(
round_number_down_to_digits(res_left, digits_for_range),
round_number_up_to_digits(res_right, digits_for_range), True, True,
digits_for_range)
def prime_factor(n):
assert n % 2 != 0
a = isqrt(24 * n) + 1 # làm tròn đến phần nguyên của A = căn(6N)
# print(a)
x2 = add(square(a), -24 * n) # x^2 = a^2 - n
# print(x2)
'''
while not is_square(x2):
a += 1
x2 = square(a) - 24*n
'''
p = div(a - isqrt(x2), 6)
q = div(a + isqrt(x2), 4)
return int(p), int(q)
def chinrest(aas, ns):
count = len(aas)
m = 1
ms = [1] * count
ees = [mpz(0)] * count
# product of all ns
for i in range(0, count):
m = gmpy2.mul(m, ns[i])
# products of all but one ns
for i in range(0, count):
ms[i] = gmpy2.div(m, ns[i])
# extended euclid to get the factors
for i in range(0, count):
ggtn, r, s = gmpy2.gcdext(mpz(ns[i]), mpz(ms[i]))
ees[i] = gmpy2.mul(s, ms[i])
# calculating x
x = 0
for i in range(0, count):
x = gmpy2.add(x, gmpy2.mul(aas[i], ees[i]))
# making x positive. just in case
x = gmpy2.t_mod(mpz(x), mpz(m))
while x < 0:
x = gmpy2.t_mod(mpz(x + m), mpz(m))
return m, x
def legendre_symbol():
p = 101524035174539890485408575671085261788758965189060164484385690801466167356667036677932998889725476582421738788500738738503134356158197247473850273565349249573867251280253564698939768700489401960767007716413932851838937641880157263936985954881657889497583485535527613578457628399173971810541670838543309159139
ints = [
25081841204695904475894082974192007718642931811040324543182130088804239047149283334700530600468528298920930150221871666297194395061462592781551275161695411167049544771049769000895119729307495913024360169904315078028798025169985966732789207320203861858234048872508633514498384390497048416012928086480326832803,
45471765180330439060504647480621449634904192839383897212809808339619841633826534856109999027962620381874878086991125854247108359699799913776917227058286090426484548349388138935504299609200377899052716663351188664096302672712078508601311725863678223874157861163196340391008634419348573975841578359355931590555,
17364140182001694956465593533200623738590196990236340894554145562517924989208719245429557645254953527658049246737589538280332010533027062477684237933221198639948938784244510469138826808187365678322547992099715229218615475923754896960363138890331502811292427146595752813297603265829581292183917027983351121325,
14388109104985808487337749876058284426747816961971581447380608277949200244660381570568531129775053684256071819837294436069133592772543582735985855506250660938574234958754211349215293281645205354069970790155237033436065434572020652955666855773232074749487007626050323967496732359278657193580493324467258802863,
4379499308310772821004090447650785095356643590411706358119239166662089428685562719233435615196994728767593223519226235062647670077854687031681041462632566890129595506430188602238753450337691441293042716909901692570971955078924699306873191983953501093343423248482960643055943413031768521782634679536276233318,
85256449776780591202928235662805033201684571648990042997557084658000067050672130152734911919581661523957075992761662315262685030115255938352540032297113615687815976039390537716707854569980516690246592112936796917504034711418465442893323439490171095447109457355598873230115172636184525449905022174536414781771,
50576597458517451578431293746926099486388286246142012476814190030935689430726042810458344828563913001012415702876199708216875020997112089693759638454900092580746638631062117961876611545851157613835724635005253792316142379239047654392970415343694657580353333217547079551304961116837545648785312490665576832987,
96868738830341112368094632337476840272563704408573054404213766500407517251810212494515862176356916912627172280446141202661640191237336568731069327906100896178776245311689857997012187599140875912026589672629935267844696976980890380730867520071059572350667913710344648377601017758188404474812654737363275994871,
4881261656846638800623549662943393234361061827128610120046315649707078244180313661063004390750821317096754282796876479695558644108492317407662131441224257537276274962372021273583478509416358764706098471849536036184924640593888902859441388472856822541452041181244337124767666161645827145408781917658423571721,
18237936726367556664171427575475596460727369368246286138804284742124256700367133250078608537129877968287885457417957868580553371999414227484737603688992620953200143688061024092623556471053006464123205133894607923801371986027458274343737860395496260538663183193877539815179246700525865152165600985105257601565
]
x = -1
for i in ints:
if euler_criterion(
i, p) == 1: # 1 means that the number is a quadratic residue
x = i # this is the quadratic residue
# a^2 = x mod p
# b^p cong b mod p
gmpy2.get_context().precision = 10000
# temp = gmpy2.div(p+1, 4)
# print(temp)
a = pow(x, int(gmpy2.div(p + 1, 4)), p)
# a = (pow(x, p/2, p))
# print(int(a) == a)
print(a)
def diophantine(a, b, c):
q, r = gmpy2.f_divmod(a, b)
if (r == 0):
print "remainder = ", gmpy2.div(c, b)
return ([0, gmpy2.div(c, b)])
else:
sol = diophantine(b, r, c)
x = sol[1]
y = sol[0]
print "x = ", x
print "y = ", y
return ([x, gmpy2.sub(y, gmpy2.mul(x, q))])
def diophantine(a, b, c):
q,r = gmpy2.f_divmod(a, b)
if (r == 0):
print "remainder = ", gmpy2.div(c, b)
return([0, gmpy2.div(c, b)])
else:
sol = diophantine(b, r, c)
x = sol[1]
y = sol[0]
print "x = ", x
print "y = ", y
return ([x, gmpy2.sub(y, gmpy2.mul(x, q))])
def evaluate_error_at_sample(self, tree):
""" Sample from the leaf then evaluate tree in the tree's working precision"""
if tree.left is not None or tree.right is not None:
if tree.left is not None:
sample_l, lp_sample_l = self.evaluate_error_at_sample(
tree.left)
if tree.right is not None:
sample_r, lp_sample_r = self.evaluate_error_at_sample(
tree.right)
if tree.root_name == "+":
return (sample_l + sample_r), gmpy2.add(
mpfr(str(lp_sample_l)), mpfr(str(lp_sample_r)))
elif tree.root_name == "-":
return (sample_l - sample_r), gmpy2.sub(
mpfr(str(lp_sample_l)), mpfr(str(lp_sample_r)))
elif tree.root_name == "*":
return (sample_l * sample_r), gmpy2.mul(
mpfr(str(lp_sample_l)), mpfr(str(lp_sample_r)))
elif tree.root_name == "/":
return (sample_l / sample_r), gmpy2.div(
mpfr(str(lp_sample_l)), mpfr(str(lp_sample_r)))
elif tree.root_name == "exp":
return np.exp(sample_l), gmpy2.exp(mpfr(str(sample_l)))
elif tree.root_name == "sin":
return np.sin(sample_l), gmpy2.sin(mpfr(str(sample_l)))
elif tree.root_name == "cos":
return np.cos(sample_l), gmpy2.cos(mpfr(str(sample_l)))
elif tree.root_name == "abs":
return np.abs(sample_l), abs(mpfr(str(sample_l)))
else:
print("Operation not supported!")
exit(-1)
else:
sample = tree.root_value[0].getSampleSet(n=1)[0]
return sample, mpfr(str(sample))
def compute_uncertainty_given_interval(low, upper):
coefficients = {}
with gmpy2.local_context(gmpy2.context(),
round=gmpy2.RoundDown,
precision=mpfr_proxy_precision) as ctx:
res_left = gmpy2.sub(gmpy2.div(mpfr(upper), mpfr("2.0")),
gmpy2.div(mpfr(low), mpfr("2.0")))
with gmpy2.local_context(gmpy2.context(),
round=gmpy2.RoundUp,
precision=mpfr_proxy_precision) as ctx:
res_right = gmpy2.sub(gmpy2.div(mpfr(upper), mpfr("2.0")),
gmpy2.div(mpfr(low), mpfr("2.0")))
coefficients[AffineManager.get_new_error_index()]=\
Interval(round_number_down_to_digits(res_left, digits_for_range),
round_number_up_to_digits(res_right, digits_for_range), True, True, digits_for_range)
return coefficients
def Q3():
N = mpz('72006226374735042527956443552558373833808445147399984182665305798191\
63556901883377904234086641876639384851752649940178970835240791356868\
77441155132015188279331812309091996246361896836573643119174094961348\
52463970788523879939683923036467667022162701835329944324119217381272\
9276147530748597302192751375739387929')
A = isqrt(mul(6, N)) + 1
# (3 * p + 2 * q) / 2 is not an integer. or, A - 0.5 = (3p + 2q) / 2
# so, x = sqrt((A-0.5)^2 - 6N) = sqrt(A^2 - A + 0.25 - 6N). the 0.25 term can be thrown away.
x = isqrt(sub(sub(mul(A, A), A), mul(6, N)))
p = div(sub(A, x), 3)
q = div(add(A, x), 2)
if mul(p, q) == N:
print("Q3:")
print(p)
def chinese_remainder(n, a):
sum = 0
prod = reduce(gmpy2.mul, n)
for n_i, a_i in zip(n, a):
p = gmpy2.div(prod, n_i)
sum += gmpy2.mul(gmpy2.mul(a_i, mul_inv(p, n_i)), p)
return sum % prod
def computepi():
# N: number of decimals
f = StringIO()
w = xmpz(0)
k = 1
n1 = xmpz(4)
n2 = xmpz(3)
d = xmpz(1)
f10 = xmpz(10)
n10 = xmpz(-10)
i = 0
N = request.args.get('N', default=10, type=int)
while True:
# digit
u = int(div(n1, d))
v = int(div(n2, d))
if u == v:
f.write(chr(48 + u))
i += 1
if i % 10 == 0:
f.write("\t:%d\n" % i)
if i == N:
break
# extract
u = mul(d, mul(n10, u))
n1 = mul(n1, f10)
n1 = add(n1, u)
n2 = mul(n2, f10)
n2 = add(n2, u)
else:
# produce
k2 = k << 1
u = mul(n1, k2 - 1)
v = add(n2, n2)
w = mul(n1, k - 1)
n1 = add(u, v)
u = mul(n2, k + 2)
n2 = add(w, u)
d = mul(d, k2 + 1)
k += 1
return (f.getvalue())
def get_smoothness(_start, delta, list_primes, pos, output):
print("start process " + str(pos))
segment = math.ceil(delta / 10)
V = []
for k in range(0, 10):
start_p = pos * delta + k * segment
end_p = pos * delta + (k + 1) * segment
if end_p > offset:
end_p = offset
listNN = []
for j in range(start_p, end_p):
listNN.append(gmpy2.sub(gmpy2.mul(gmpy2.add(_start, j), gmpy2.add(_start, j)), N))
for i in range(0, n_smooth + 1):
p = list_primes[i]
index = 0
for index in range(0, len(listNN)):
if gmpy2.f_mod(listNN[index], p) == 0:
break
index += start_p
start_index1 = gmpy2.f_mod(index, p)
for n in range(index - start_p, len(listNN), p):
while gmpy2.f_mod(listNN[n], p) == 0:
listNN[n] = gmpy2.div(listNN[n], p)
start_index2 = gmpy2.f_mod(gmpy2.sub(gmpy2.sub(p, gmpy2.f_mod(gmpy2.add(_start, start_index1), p)), _start),p)
index2 = gmpy2.sub(p, gmpy2.f_mod(gmpy2.sub(start_p, start_index2), p))
if start_index2 != start_index1:
for n in range(index2, len(listNN), p):
while gmpy2.f_mod(listNN[n], p) == 0:
listNN[n] = gmpy2.div(listNN[n], p)
for i in range(0, len(listNN)):
if listNN[i] == 1:
start1 = gmpy2.add(_start, start_p)
num = gmpy2.sub(gmpy2.mul(gmpy2.add(start1, i), gmpy2.add(start1, i)), N)
V.append(num)
output.put((pos, V))
print("end process " + str(pos))
def H_mpc(a,z):
'''Returns generating function value. Takes in the a vector, the current position
on the parametrized variable t and the gamma function. Uses MPC and is
not vectorized.'''
temp=1
n=len(a)
for k in range(0,n):
temp = gmpy2.div(temp, mpc((1-((z))**a[k])))
return temp
def main():
big = int(input("Big: "))
small = int(input("Small: "))
print("Sml Stat:", gmpy2.is_prime(small))
result = gmpy2.div(big, small)
print("Res Stat:", gmpy2.is_prime(result))
print(result)
def main():
start = time.clock()
try:
N_prime = mpz(mul(24,N))
A_prime = isqrt(N_prime)+1
x_prime = mpz(isqrt(pow(A_prime,2) - N_prime))
p_prime = mpz(A_prime - x_prime)
q_prime = mpz(A_prime + x_prime)
assert mpz(mul(p_prime,q_prime)) == N_prime
p = mpz(div(p_prime,6))
q = mpz(div(q_prime,4))
print 'p = ' + str(p)
print 'q = ' + str(q)
print 'Time: ' + str(time.clock() - start)
assert mpz(mul(p,q)) == N
except KeyboardInterrupt:
print 'Keyboard interrupt!'
print 'Time: ' + str(time.clock() - start)
sys.exit()
def factorial_div (one, two):
'''
computes a!/b!
'''
if one < two:
return div(1., reduce (mul, xrange(one, two)))
elif one > two:
return reduce (mul, xrange(two, one))
else:
return mpfr (1.)
def factor(N):
"""
:param N: mpz(N)
:return: mpz(p), mpz(q)
"""
N = gmpy2.mpz(N)
A = ceil_sqrt(6 * N)
a = 1
b = -1
c = -(A ** 2 - A - 6 * N)
det = b ** 2 - 4 * a * c
roots = (gmpy2.div(-b + gmpy2.isqrt(det), 2 * a), # 这里还非得用isqrt不能用sqrt了
gmpy2.div(-b - gmpy2.isqrt(det), 2 * a))
for x in roots:
p = (A - x) // 3
q = (A + x - 1) // 2
if p * q == N:
return p, q
def main():
start = time.clock()
try:
N_prime = mpz(mul(24, N))
A_prime = isqrt(N_prime) + 1
x_prime = mpz(isqrt(pow(A_prime, 2) - N_prime))
p_prime = mpz(A_prime - x_prime)
q_prime = mpz(A_prime + x_prime)
assert mpz(mul(p_prime, q_prime)) == N_prime
p = mpz(div(p_prime, 6))
q = mpz(div(q_prime, 4))
print 'p = ' + str(p)
print 'q = ' + str(q)
print 'Time: ' + str(time.clock() - start)
assert mpz(mul(p, q)) == N
except KeyboardInterrupt:
print 'Keyboard interrupt!'
print 'Time: ' + str(time.clock() - start)
sys.exit()
def factor2(n):
avg_guess = gmpy2.isqrt(6 * n) + (0 if gmpy2.is_square(n) else 1)
while True:
a = gmpy2.mpz(1)
b = gmpy2.mpz(1)
c = 6 * n - avg_guess ** 2 + avg_guess
x_roots = quadform(a, b, c)
for x in x_roots:
# Check for 3p < 2q
p = gmpy2.div((avg_guess - x - 1), 3)
q = gmpy2.div((avg_guess + x), 2)
if (p * q == n):
return (p, q)
# Check for 3p > 2q
p = gmpy2.div((avg_guess + x), 3)
q = gmpy2.div((avg_guess - x - 1), 2)
if (p * q == n):
return (p, q)
avg_guess += 1
def eea(a, b):
r, r1 = a, b
s, s1 = 1, 0
t, t1 = 0, 1
while r1 != 0:
q = gmpy2.div(r, r1)
r2 = gmpy2.f_mod(r, r1)
r, s, t, r1, s1, t1 = r1, s1, t1, r2, gmpy2.sub(s, gmpy2.mul(s1, q)), gmpy2.sub(t, gmpy2.mul(t1, q))
d = r
return d, s, t
def mul_inv(a, b):
b0 = b
x0, x1 = 0, 1
if b == 1: return 1
while a > 1:
q = gmpy2.div(a, b)
a, b = b, a % b
x0, x1 = x1 - gmpy2.mul(q, x0), x0
if x1 < 0:
x1 = gmpy2.add(x1, b0)
return x1
def swing(m, primes):
if m < 33: return small_swing[m]
s = bisect.bisect_left(primes, 1 + isqrt(m))
d = bisect.bisect_left(primes, 1 + m // 3)
e = bisect.bisect_left(primes, 1 + m // 2)
g = bisect.bisect_left(primes, 1 + m)
factors = primes[e:g]
factors += filter(lambda x: div(m, x) & 1 == 1, primes[s:d])
for prime in primes[1:s]:
p, q = mpz(1), m
while True:
q = div(q, prime)
if q == 0: break
if q & 1 == 1:
p = mul(p, prime)
if p > 1: factors.append(p)
return product(factors, 0, len(factors) - 1)
def swing(m, primes):
if m < 33: return small_swing[m]
s = bisect.bisect_left(primes, 1 + isqrt(m))
d = bisect.bisect_left(primes, 1 + m // 3)
e = bisect.bisect_left(primes, 1 + m // 2)
g = bisect.bisect_left(primes, 1 + m)
factors = primes[e:g]
factors += filter(lambda x: div(m, x) & 1 == 1, primes[s:d])
for prime in primes[1:s]:
p, q = mpz(1), m
while True:
q = div(q, prime )
if q == 0: break
if q & 1 == 1:
p = mul(p, prime )
if p > 1: factors.append(p)
return product(factors, 0, len(factors) - 1)
def verify_A3(A, N):
'''
serves factor_3
'''
c = -A**2 + A + 6 * N
b = mpz(1)
a = mpz(1)
if (b**2 - 4 * a * c) < 0:
return False
x_candidates = (gmpy2.div((-b + gmpy2.isqrt(b**2 - 4 * a * c)), 2 * a),
gmpy2.div((-b - gmpy2.isqrt(b**2 - 4 * a * c)), 2 * a))
for x in x_candidates:
if x < 0:
continue
# 2q < 3p
p = gmpy2.div((A + x), 3)
q = gmpy2.div((A - x - 1), 2)
if p * q == N:
return p, q
# 3p < 2q
p = gmpy2.div((A - x - 1), 3)
q = gmpy2.div((A + x), 2)
if p * q == N:
return p, q
return False
def verify_A3(A, N):
'''
serves factor_3
'''
c = -A**2 + A + 6*N
b = mpz(1)
a = mpz(1)
if (b**2 - 4*a*c) < 0:
return False
x_candidates = (
gmpy2.div((-b + gmpy2.isqrt(b**2 - 4*a*c)), 2*a),
gmpy2.div((-b - gmpy2.isqrt(b**2 - 4*a*c)), 2*a))
for x in x_candidates:
if x < 0:
continue
# 2q < 3p
p = gmpy2.div((A + x), 3)
q = gmpy2.div((A - x - 1), 2)
if p*q == N:
return p, q
# 3p < 2q
p = gmpy2.div((A - x - 1), 3)
q = gmpy2.div((A + x), 2)
if p*q == N:
return p, q
return False
def element_order_general(element, modulus, order, order_decomposition):
if element == g.mpz(1):
return 1 # by definition
if g.powmod(element, order, modulus) != g.mpz(1):
return None # not an element of the group
for factor, power in order_decomposition.items():
for p in range(1, power + 1):
next_order = g.div(order, factor)
if g.powmod(element, next_order, modulus) == g.mpz(1):
order = next_order
else:
break
return order
def get_min_prime_root(euler_value,prime_factors,m):
"""求最小的原根"""
a = 1
i = 0
while(a<m):
for prime_factor in prime_factors:
if gmpy2.powmod(a, gmpy2.div(euler_value, prime_factor), m) == 1:
a = gmpy2.add(a,1)
else:
i = gmpy2.add(i,1)
break
if i == len(prime_factors):
break
return a
def ch3_factor(N):
""" Valid when |3p - 2q| < N^(1/4)
"""
A = ceil_sqrt(6*N)
# let M = (3p+2q)/2
# M is not an integer since 3p + 2q is odd
# So there is some integer A = M + 0.5 and some integer i such that
# 3p = M + i - 0.5 = A + i - 1
# and
# 2q = M - i + 0.5 = A - i
#
# N = pq = (A-i)(A+i-1)/6 = (A^2 - i^2 - A + i)/6
# So 6N = A^2 - i^2 - A + i
# i^2 - i = A^2 - A - 6N
# Solve using the quadratic equation!
a = mpz(1)
b = mpz(-1)
c = -(A**2 - A - 6*N)
det = b**2 - 4*a*c
roots = (div(-b + isqrt(b**2 - 4*a*c), 2*a),
div(-b - isqrt(b**2 - 4*a*c), 2*a))
for i in roots:
if i >= 0:
f = check_ch3(i,A,N)
if f:
return f
# We should have found the root
assert(False)
def create_gauss_matrix(V, list_primes):
A = [[]]
for i in range(0, len(V)):
num = V[i]
vector = []
for p in range(0, n_smooth + 1):
value = 0
while gmpy2.c_mod(num, list_primes[p]) == 0:
num = gmpy2.div(num, list_primes[p])
value ^= 1
vector.append(value)
A.append(vector)
print("cac so co the tao tich binh phuong: " + str(len(V)))
print(V[:20])
return A[1:]
def generate():
r = 257
while True:
p = getPrime(1024)
if (p - 1) % r == 0 and gcd(r, div(p - 1, r)) == 1:
break
while True:
q = getPrime(1024)
if gcd(r, q - 1) == 1:
break
n = p * q
phi = (p - 1) * (q - 1)
while True:
y = getRandomRange(0, n)
x = powmod(y, phi * invert(r, n) % n, n)
if x != 1:
break
return (n, y), (n, phi, x)
def find_sqrt_ceil(N):
lower_bound = mpz(2)
upper_bound = N
while True:
times_two = mul(lower_bound, 2)
if square(times_two) > N:
upper_bound = times_two
break
else:
lower_bound = times_two
while True:
middle = div(lower_bound + upper_bound, 2)
if square(middle) == N:
return middle
elif square(middle) < N and square(middle + 1) > N:
return middle + 1
elif square(middle) < N:
lower_bound = middle
else:
upper_bound = middle
def find_factors3(self, N):
A = gmpy2.isqrt(6 * N) * 2
print("Origi A: ", A)
x = gmpy2.div(A, 5)
p = gmpy2.mul(x, 2)
q = gmpy2.mul(x, 3)
print("x: ", x)
print("p: ", p)
print("q: ", q)
product = gmpy2.mul(p, q)
difference = gmpy2.sub(product, N)
if difference == 0:
print("Found A: ", A)
print("p: ", p)
print("q: ", q)
print("product: ", product)
print("difference: ", difference)
def rsa_mitm(n, e, c, b):
S = random.randint(pow(2,b)/2 , pow(2,b))
T = random.randint(pow(2,b)/2 , pow(2,b))
n=gmpy2.mpz(n)
cs=[gmpy2.mpz()]*(S)
print S
print T
print n
for s in range(1,S):
cs[s] = gmpy2.mpz( gmpy2.c_mod(gmpy2.div(c,( gmpy2.powmod(s,e,n))) , n) )
#print (s,cs[s])
S_ = (T)/100
for t in range(1,T):
for s in range(1,S):
if cs[s] == (gmpy2.powmod(t,e,n)):
print (s,s*t)
return s*t
print cs
return "Plaintext nicht gefunden"
def factors2(n):
n = gmpy2.mpz(n)
init_guess = gmpy2.isqrt(n)
A = init_guess
# print('intial guess:')
# print(guess)
#print('N2/guess:\n'+ str(int(N2/guess)) + '\nN2%guess:\n'+ str(int(N2%guess)))
mindiv = n
for i in range(1048576):
x = gmpy2.sqrt(A**2 - n)
p = A - x
q = A + x
mod = gmpy2.fmod(n, p)
div = gmpy2.div(n, p * q)
if div < mindiv:
mindiv = div
if not mod:
print('p:\n' + str(int(p)))
print('q:\n' + str(int(q)))
break
A += 1
def william_p1(n, process_id):
# this algorithm technically works but literally none of the modulos were p+1 so idk
# choose a B (work limit) to start with
if gmpy2.bit_length(gmpy2.mpz(n)) <= 2044:
work_limit = pow(n, (1 / 6))
else:
work_limit = gmpy2.div(n, 27)
threshold = 3
previous_sub2 = 2
# A needs to be greater than 2 to start with so we therefore start with 3
A = process_id + 3
previous = A
counter = 0
# if the counter ever reaches m, terminate
while counter != threshold:
counter += 1
# current = (a^(current-1) - current-2) % n)
current = (((A**previous) - previous_sub2) % n)
# move the previous variables forward
previous_sub2 = previous
previous = current
d = gmpy2.gcd(current - 2, n)
if d != 1 and d != n:
# calculate the factorial of m
mult = gmpy2.fac(threshold)
if DEBUG:
print(d, mult)
# check to see if we've found the terminating conditions for p + 1
if gmpy2.f_mod(mult, d):
return d
else:
# increment threshold by 1 if we haven't found anything
threshold += 1
if threshold > work_limit:
return 1
return 1
def pollard_p1(n, process_id):
# choose a B (work limit) to start with
if gmpy2.bit_length(gmpy2.mpz(n)) <= 2044:
work_limit = pow(n, (1 / 6))
else:
work_limit = gmpy2.div(n, 27)
a = 2
# break up how many things we have to check so we can split over cores
process_range = (math.floor(work_limit) // PROCESS_COUNT)
start = process_range * process_id
end = process_range * (process_id + 1) - 1
# assign default values here so we are an excellent, coding standard following coder
q1, q2, q3 = 0, 0, 0
if DEBUG:
# lock processes so we can print cleanly (thanks 421)
processLock.acquire()
print("CORE", process_id, "checking p1 from", start, "to", end)
processLock.release()
# calculate the quarter ranges
q1 = ((process_range // 4) * 1) + start
q2 = ((process_range // 4) * 2) + start
q3 = ((process_range // 4) * 3) + start
for i in range(start, end):
# print informative debug statements
if DEBUG:
if i == q1:
print("CORE", process_id, "p1 25%")
elif i == q2:
print("CORE", process_id, "p1 50%")
elif i == q3:
print("CORE", process_id, "p1 75%")
# check to see if we've found the terminating conditions for p - 1
a = gmpy2.powmod(a, i, n)
d = gmpy2.gcd(a - 1, n)
if d != 1 and d != n:
return d
return 1
def test_binary(a, b):
t = a + b
t = a - b
t = a * b
try:
t = a / b
except:
pass
try:
t = a // b
except:
pass
try:
t = divmod(a, b)
except:
pass
t = gmpy2.add(a, b)
t = gmpy2.sub(a, b)
t = gmpy2.mul(a, b)
t = gmpy2.div(a, b)
t = gmpy2.agm(a, b)
t = gmpy2.atan2(a, b)
def rsa_(n,e,c,b):
T = pow(2,b)
n=gmpy2.mpz(n)
c=gmpy2.mpz(c)
cs=[gmpy2.mpz()]*(T)
# counter = 0
# T_ = T/100
# percent= 1
for r in range(1,T):
cs[r] = gmpy2.c_mod(gmpy2.div(c,gmpy2.powmod(r,e,n)),n)
# counter += 1
# if (counter >= T_):
# print percent
# percent +=1
# counter = 0
#cs.sort()
print cs
for s in range (1, T):
try:
index=cs.index(gmpy2.powmod(s,e,n))
#print (gmpy2.c_mod(index*s,n),cs[index],gmpy2.powmod(s,e,n) )
return index*s
except ValueError:
def pi_calc():
global calcs, y, u, q, r, t, j, calc_uy, startTime
digitstring = ''
dpm = 0
strPi = ''
loop = 1
elapsed = 0
elapsedStart = time.time()
while loop:
digitCalcTime = time.time()
j3 = mul(3, j)
u, y = mpz(3 * (j3 + 1) * (j3 + 2)), mpz(div((add(mul(q, (mul(27, j)) - 12), mul(5, r))), mul(5, t)))
strPi = str(y)
digitstring += strPi
q, r, t, j = mpz(mul((20 * j ** 2 - mul(10, j)), q)), \
mpz(mul(mul(10, u), (q * (mul(5, j) - 2) + r - mul(y, t)))), \
mpz(mul(t, u)), add(j, 1)
# dpm = digits per minute
now = time.time()
elapsed = now - elapsedStart
# if the elapsed calculation time exceeds .5 seconds break the loop and return the digits calculated
if elapsed >= 1:
break
elif (j - 2) % 1000 == 0: # break also every 1000nth digit to report stats
break
dps = (1.0 / elapsed) * len(digitstring)
info = {
"digits": digitstring,
"digitcount": int(j) - 2,
"dpm": round(dps * 60),
"dps": round(dps, 2)
}
if (j - 2) % 1000 == 0:
info['mark'] = {"digitmark": (int(str(j)) - 2),
"runtime": time.time() - startTime}
return info
def findFactorsCaseThree(self):
print("===========================================")
print("Modulus N is: " + str(self.N))
#A = ceil(2 * sqrt(6N))
A = gmpy2.ceil(gmpy2.mul(mpz(2), gmpy2.sqrt(gmpy2.mul(mpz(6), self.N))))
print("A is: " + str(A))
#X = sqrt(A^2 - 24N)
A_square = gmpy2.mul(A, A)
X = gmpy2.sqrt(gmpy2.sub(A_square, gmpy2.mul(mpz(24), self.N)))
print("X is: " + str(X))
#q = (A + X)/4 AND p = N/q
self.q = gmpy2.f_div(mpz(gmpy2.add(A, X)), mpz(4))
self.p = gmpy2.div(self.N, self.q)
prod = gmpy2.mul(self.p, self.q)
print("Product of pq is: " + str(prod))
if prod == self.N:
print("We have got the factors RIGHT")
else:
print("We didn't get the factors")
print(self.p)
print(self.q)
self.p = 0
self.q = 0
print("===========================================")
def odd_factorial(n, primes):
if n < 2: return 1
tmp = odd_factorial(div(n, 2), primes)
return mul(mul(tmp, tmp), swing(n, primes))
def quadform(a, b, c):
det = gmpy2.isqrt(b ** 2 - 4 * a * c)
r1 = gmpy2.div((-b + det), 2 * a)
r2 = gmpy2.div((-b - det), 2 * a)
return (r1, r2)