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 calc_brute_force(self):
a = ceil(sqrt(self.n))
for n in xrange(1, 2 ** 20):
p, q = self._calc_factors(add(a, n))
if self._check_sol(p, q):
return p, q
#self.log.print_progress(n)
else:
raise ValueError("Cannot factor {}".format(self.n))
def fermat(n):
a = int(gmpy2.ceil(gmpy2.isqrt(n)))
b2 = (a**2) - n
step = 0
while not gmpy2.is_square(b2):
a += 1
b2 = (a**2) - n
step += 1
print(n, step, a, b2)
return a - gmpy2.isqrt(b2), a + gmpy2.isqrt(b2)
def main():
# predefine these, we will stop overwriting them when ready to run the big numbers
p=mpz('13407807929942597099574024998205846127'
'47936582059239337772356144372176403007'
'35469768018742981669034276900318581848'
'6050853753882811946569946433649006084171'
)
g=mpz('117178298803662070095161175963353670885'
'580849999989522055999794590639294997365'
'837466705721764714603129285948296754282'
'79466566527115212748467589894601965568'
)
y=mpz('323947510405045044356526437872806578864'
'909752095244952783479245297198197614329'
'255807385693795855318053287892800149470'
'6097394108577585732452307673444020333'
)
log.info("Starting Discrete Log Calculation")
log.info("p: %i", p)
log.info("g: %i", g)
log.info("y: %i", y)
m = gmpy2.ceil(gmpy2.sqrt(p))
log.info("m: %i", m)
# custom range since builtin has a size limit
long_range = lambda start, stop: iter(itertools.count(start).next, stop)
chunk_size = 100000
if m < 100000:
chunk_size = m
stop_at = chunk_size
start = 0
while True:
chunk = {}
log.info("Starting chunk from %i to %i", start, stop_at)
for i in xrange(start, stop_at):
chunk[gmpy2.powmod(g,i,p)] = i
for t in long_range(0,m):
expone = mpz(gmpy2.mul(-m,t))
g_term = gmpy2.powmod(g, expone, p)
res = gmpy2.f_mod(gmpy2.mul(y, g_term), p)
if res in chunk:
s = chunk[res]
dc = gmpy2.f_mod(mpz(gmpy2.add(s, gmpy2.mul(m,t))), p)
log.info("DC LOG FOUND")
log.info("dc: %i", dc)
return
log.info("Completed chunk run: %i to %i no DC yet :(", start, stop_at)
start = stop_at
stop_at += chunk_size
def fermat_factor(n):
assert n % 2 != 0 # Odd integers only
a = gmpy2.ceil(gmpy2.sqrt(n))
b2 = gmpy2.square(a) - n
while not is_square(b2):
a += 1
b2 = gmpy2.square(a) - n
factor1 = a + gmpy2.sqrt(b2)
factor2 = a - gmpy2.sqrt(b2)
return int(factor1), int(factor2)
def factor3(n):
a = ceil(sqrt(6 * n)) - 0.5
x_square = a**2 - 6 * n
x = sqrt(x_square)
if (a - x) % 3 == 0: # 3p<2q => p<q
return mpz((a - x) / 3)
elif (a - x) % 2 == 0: # 2q<3p
q = (a - x) / 2
if q < n / q:
return mpz(q)
else:
return mpz(n / q)
def fermat_factor(n):
assert n % 2 != 0 # Odd integers only
a = gmpy2.ceil(gmpy2.sqrt(n))
b2 = gmpy2.square(a) - n
while not is_square(b2):
a += 1
b2 = gmpy2.square(a) - n
factor1 = a + gmpy2.sqrt(b2)
factor2 = a - gmpy2.sqrt(b2)
return int(factor1), int(factor2)
def encode(self, seq):
assert len(seq) == len(self.SNPRefList)
L = [-1, -1]
U = [-1, -1]
L[0] = mpz(0)
U[0] = mpz(self.maxSeed)
n = len(seq)
for i in range(1, n + 1):
probs = self.condProb(seq[:i])
sortInd = np.argsort(probs)
assignedRangeSize = [0] * 3
available = U[(i - 1) % 2] - L[(i - 1) % 2] + 1
if probs[sortInd[0]] * available < self.powOfThree[n - i]:
assignedRangeSize[sortInd[0]] = self.powOfThree[n - i]
else:
assignedRangeSize[sortInd[0]] = mpz(
gmpy2.ceil(probs[sortInd[0]] * available))
available -= assignedRangeSize[sortInd[0]]
probs[sortInd[0]] = 0
probs = probs / sum(
probs) #re-normalize the remaining two probabilities
if probs[sortInd[1]] * available < self.powOfThree[n - i]:
assignedRangeSize[sortInd[1]] = self.powOfThree[n - i]
else:
assignedRangeSize[sortInd[1]] = mpz(
gmpy2.ceil(probs[sortInd[1]] * available))
assignedRangeSize[
sortInd[2]] = available - assignedRangeSize[sortInd[1]]
L[i % 2] = L[(i - 1) % 2] + sum(assignedRangeSize[:seq[i - 1]])
U[i % 2] = L[(i - 1) % 2] + sum(
assignedRangeSize[:(seq[i - 1] + 1)]) - 1
#Don't use random.randint for big integer. Sometimes they might be correct on a system, but sometimes they are not.
#I encounter this problem when running this script with condor on icsil1-cluster.epfl.ch. The function generates very strange and unexpected integers.
#seed = random.randint(L[n%2], U[n%2])
tmp = random.randint(0, 2**30)
randomState = gmpy2.random_state(tmp)
seed = gmpy2.mpz_random(randomState,
U[n % 2] - L[n % 2] + 1) + L[n % 2]
return seed
def decode(self, seed):
L = [-1, -1]
U = [-1, -1]
L[0] = mpz(0)
U[0] = mpz(self.maxSeed)
n = len(self.SNPRefList)
seq = [-1] * n
for i in range(1, n + 1):
probs = self.condProb(seq[:i])
sortInd = np.argsort(probs)
assignedRangeSize = [0] * 3
available = U[(i - 1) % 2] - L[(i - 1) % 2] + 1
if probs[sortInd[0]] * available < self.powOfThree[n - i]:
assignedRangeSize[sortInd[0]] = self.powOfThree[n - i]
else:
assignedRangeSize[sortInd[0]] = mpz(
gmpy2.ceil(probs[sortInd[0]] * available))
available -= assignedRangeSize[sortInd[0]]
probs[sortInd[0]] = 0
probs = probs / sum(
probs) #re-normalize the remaining two probabilities
if probs[sortInd[1]] * available < self.powOfThree[n - i]:
assignedRangeSize[sortInd[1]] = self.powOfThree[n - i]
else:
assignedRangeSize[sortInd[1]] = mpz(
gmpy2.ceil(probs[sortInd[1]] * available))
assignedRangeSize[
sortInd[2]] = available - assignedRangeSize[sortInd[1]]
if seed < L[(i - 1) % 2] + sum(assignedRangeSize[:1]):
seq[i - 1] = 0
elif seed < L[(i - 1) % 2] + sum(assignedRangeSize[:2]):
seq[i - 1] = 1
elif seed < L[(i - 1) % 2] + sum(assignedRangeSize[:3]):
seq[i - 1] = 2
else:
raise ValueError("Invalid Seed")
L[i % 2] = L[(i - 1) % 2] + sum(assignedRangeSize[:seq[i - 1]])
U[i % 2] = L[(i - 1) % 2] + sum(
assignedRangeSize[:(seq[i - 1] + 1)]) - 1
return seq
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 is_good_pair(p: int, q: int) -> Optional[Tuple[int, int]]:
"""Returns the encryption modulus and totient for the given primes.
:param p: One of the two primes to be used for encryption.
:type p: int
:param q: The other prime used for encryption.
:type q: int
:returns: Either the encryption modulus and the totient, or None.
:rtype: Optional[Tuple[int, int]]"""
n = p*q
k = gmpy2.ceil(gmpy2.log2(n))
if abs(p - q) > 2**(k/2 - 100):
return n, n - (p + q - 1)
return None
def q1():
N = 179769313486231590772930519078902473361797697894230657273430081157732675805505620686985379449212982959585501387537164015710139858647833778606925583497541085196591615128057575940752635007475935288710823649949940771895617054361149474865046711015101563940680527540071584560878577663743040086340742855278549092581
rt = gmpy2.sqrt(N)
A = gmpy2.ceil(rt)
A2 = pow(A, 2)
assert A2 > N
x = gmpy2.sqrt(gmpy2.sub(A2, N))
p = gmpy2.sub(A, x)
q = gmpy2.add(A, x)
gotcha(N, p, q)
def q1():
N = 179769313486231590772930519078902473361797697894230657273430081157732675805505620686985379449212982959585501387537164015710139858647833778606925583497541085196591615128057575940752635007475935288710823649949940771895617054361149474865046711015101563940680527540071584560878577663743040086340742855278549092581
rt = gmpy2.sqrt(N)
A = gmpy2.ceil(rt)
A2 = pow(A, 2)
assert A2 > N
x = gmpy2.sqrt(gmpy2.sub(A2, N))
p = gmpy2.sub(A, x)
q = gmpy2.add(A, x)
gotcha(N, p, q)
def q3():
N = 720062263747350425279564435525583738338084451473999841826653057981916355690188337790423408664187663938485175264994017897083524079135686877441155132015188279331812309091996246361896836573643119174094961348524639707885238799396839230364676670221627018353299443241192173812729276147530748597302192751375739387929
rt = gmpy2.sqrt(gmpy2.mul(N, 6))
A = gmpy2.ceil(rt)
A2 = pow(A, 2)
assert A2 > N
x = gmpy2.sqrt(gmpy2.sub(A2, N))
x = gmpy2.mul(x, 3 / 2)
p = gmpy2.sub(A, x)
q = gmpy2.add(A, x)
gotcha(N, p, q)
def question2():
#25464796146996183438008816563973942229341454268524157846328581927885777969985222835143851073249573454107384461557193173304497244814071505790566593206419759
N = gmpy2.mpfr(
"648455842808071669662824265346772278726343720706976263060439070378797308618081116462714015276061417569195587321840254520655424906719892428844841839353281972988531310511738648965962582821502504990264452100885281673303711142296421027840289307657458645233683357077834689715838646088239640236866252211790085787877",
1050)
A = gmpy2.ceil((gmpy2.sqrt(N)))
#print("A:",A)
i = 1
while i < 2**20:
#print("K:",k)
x = gmpy2.ceil(gmpy2.sqrt(gmpy2.square(A) - N))
#print("x:",x)
p = A + x
q = A - x
if (N == gmpy2.mul(p, q)):
if (p > q):
p, q = q, p
print("found pq:", p, q)
break
A += 1
i += 1
def FermatFactor(N):
A = gmpy2.mpz( gmpy2.ceil( gmpy2.sqrt(N) ) )
B2 = gmpy2.sub( gmpy2.square(A), N )
while not gmpy2.is_square(B2):
A = gmpy2.add( A, gmpy2.mpz("1") )
B2 = gmpy2.sub( gmpy2.square(A), N )
B = gmpy2.sqrt(B2)
P = gmpy2.mpz( gmpy2.sub( A, B ) )
Q = gmpy2.mpz( gmpy2.add( A, B ) )
if not checkFactors(P,Q,N):
raise Exception("Bad factors generated")
return ( P, Q )
def q3():
N = 720062263747350425279564435525583738338084451473999841826653057981916355690188337790423408664187663938485175264994017897083524079135686877441155132015188279331812309091996246361896836573643119174094961348524639707885238799396839230364676670221627018353299443241192173812729276147530748597302192751375739387929
rt = gmpy2.sqrt(gmpy2.mul(N, 6))
A = gmpy2.ceil(rt)
A2 = pow(A, 2)
assert A2 > N
x = gmpy2.sqrt(gmpy2.sub(A2, N))
x = gmpy2.mul(x, 3/2)
p = gmpy2.sub(A, x)
q = gmpy2.add(A, x)
gotcha(N, p, q)
def ex3():
N = mpz(
720062263747350425279564435525583738338084451473999841826653057981916355690188337790423408664187663938485175264994017897083524079135686877441155132015188279331812309091996246361896836573643119174094961348524639707885238799396839230364676670221627018353299443241192173812729276147530748597302192751375739387929
)
sqrt_6N = sqrt(6 * N)
sqrt_6N_decimal = sqrt_6N % 1
if sqrt_6N_decimal < 0.5:
A = floor(sqrt_6N) + 0.5
else:
A = ceil(sqrt_6N)
x = sqrt(A**2 - 6 * N)
# P = 3 * p
# Q = 2 * q
P = A - x
p = mpz(P / 3)
print(p)
def challenge1():
print("> Challenge 1")
N = 179769313486231590772930519078902473361797697894230657273430081157732675805505620686985379449212982959585501387537164015710139858647833778606925583497541085196591615128057575940752635007475935288710823649949940771895617054361149474865046711015101563940680527540071584560878577663743040086340742855278549092581
N_sqrt = gmpy2.sqrt(N)
# A = round(√N)
A = gmpy2.ceil(N_sqrt)
# x = √(A^2 - N)
x = gmpy2.sqrt(pow(A, 2) - N)
p = A - x
q = A + x
printSmallest(p, q)
return p, q, N
def q2():
N = 648455842808071669662824265346772278726343720706976263060439070378797308618081116462714015276061417569195587321840254520655424906719892428844841839353281972988531310511738648965962582821502504990264452100885281673303711142296421027840289307657458645233683357077834689715838646088239640236866252211790085787877
rt = gmpy2.sqrt(N)
A = gmpy2.ceil(rt)
q = 0
p = 0
while not gotcha(N, p, q):
A2 = pow(A, 2)
assert A2 > N
x = gmpy2.sqrt(gmpy2.sub(A2, N))
p = gmpy2.sub(A, x)
q = gmpy2.add(A, x)
assert N == gmpy2.mul(p, q)
A += 1
def q2():
N = 648455842808071669662824265346772278726343720706976263060439070378797308618081116462714015276061417569195587321840254520655424906719892428844841839353281972988531310511738648965962582821502504990264452100885281673303711142296421027840289307657458645233683357077834689715838646088239640236866252211790085787877
rt = gmpy2.sqrt(N)
A = gmpy2.ceil(rt)
q = 0
p = 0
while not gotcha(N, p, q):
A2 = pow(A, 2)
assert A2 > N
x = gmpy2.sqrt(gmpy2.sub(A2, N))
p = gmpy2.sub(A, x)
q = gmpy2.add(A, x)
assert N == gmpy2.mul(p, q)
A += 1
def challenge2():
"""
Uses Fermant factorisation
"""
print("> Challenge 2")
N = 648455842808071669662824265346772278726343720706976263060439070378797308618081116462714015276061417569195587321840254520655424906719892428844841839353281972988531310511738648965962582821502504990264452100885281673303711142296421027840289307657458645233683357077834689715838646088239640236866252211790085787877
N_sqrt = gmpy2.sqrt(N)
A = gmpy2.ceil(N_sqrt)
b2 = A * A - N
while not gmpy2.is_square(gmpy2.mpz(b2)):
A = A + 1
b2 = A * A - N
# Works out p and q
q = A - gmpy2.sqrt(b2)
p = N / q
printSmallest(p, q)
def findFactorsCaseOne(self):
print("===========================================")
print("Modulus N is: " + str(self.N))
#A = (p+q)/2 == ceil(sqrt(N))
A = gmpy2.ceil(gmpy2.sqrt(self.N))
print("A is: " + str(A))
#x = sqrt(A^2 - N)
A_square = gmpy2.mul(A, A)
x = gmpy2.sqrt(gmpy2.sub(A_square, self.N))
print("x is: " + str(x))
#p = A - x AND q = A + x
self.p = gmpy2.sub(A, x)
self.q = gmpy2.add(A, x)
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 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 factor(N):
# type: (int) -> int
"""
Computes a factorization of N using pure Fermat approach
Expects N to be odd (i.e. 3, 5, etc.) and throws a ValueError otherwise.
See: https://en.wikipedia.org/wiki/Fermat%27s_factorization_method
Args:
N: the number to factorize
Returns:
A factor of N
"""
if N & 1 == 0:
return gmpy2.mpz(2)
if gmpy2.get_context().precision < PRECISION_BITS:
adjust_gmpy2_precision(PRECISION_BITS)
logger = logging.getLogger(LOGGER)
a = gmpy2.mpz(gmpy2.ceil(gmpy2.sqrt(N)))
b = a * a - N
while not gmpy2.is_square(b):
a = a + 1
b = a * a - N
logger.info("Iteration {} and {}".format(a, b))
# NOTE: must convert to integer before subtraction
result = a - gmpy2.mpz(gmpy2.sqrt(b))
return result
def findFactorsCaseTwo(self):
print("===========================================")
print("Modulus N is: " + str(self.N))
#A = (p+q)/2 AND A can be any where from ceil(sqrt(N)) to ceil(sqrt(N)) + 2 ^ 20
A = gmpy2.ceil(gmpy2.sqrt(self.N))
print("A is: " + str(A))
upperRange = 2 ** 20
for index in range (0, upperRange + 1):
#x = sqrt(A^2 - N)
A_square = gmpy2.mul(A, A)
diff = gmpy2.sub(A_square, self.N)
x = mpz(gmpy2.sqrt(diff))
#p = A - x AND q = A + x
self.p = gmpy2.sub(A, x)
self.q = gmpy2.add(A, x)
prod = gmpy2.mul(self.p, self.q)
if prod == self.N:
print("We have got the factors RIGHT")
return
else:
A = A + 1
print("We didn't get the factors")
self.p = 0
self.q = 0
N1 += '0071584560878577663743040086340742855278549092581'
N2 = '6484558428080716696628242653467722787263437207069762630604390703787'
N2 += '9730861808111646271401527606141756919558732184025452065542490671989'
N2 += '2428844841839353281972988531310511738648965962582821502504990264452'
N2 += '1008852816733037111422964210278402893076574586452336833570778346897'
N2 += '15838646088239640236866252211790085787877'
N3 = '72006226374735042527956443552558373833808445147399984182665305798191'
N3 += '63556901883377904234086641876639384851752649940178970835240791356868'
N3 += '77441155132015188279331812309091996246361896836573643119174094961348'
N3 += '52463970788523879939683923036467667022162701835329944324119217381272'
N3 += '9276147530748597302192751375739387929'
diff = 1
gmpy2.set_context(gmpy2.context(precision = 3000))
N = mpz(N3)
lowerBound = mpz(gmpy2.ceil(gmpy2.mul(2, gmpy2.sqrt(gmpy2.mul(6, N)))))
for i in range(0, diff):
A = gmpy2.div(mpfr(gmpy2.add(lowerBound, i)), 2)
SquareA = gmpy2.square(A)
x = gmpy2.sqrt(gmpy2.sub(SquareA, gmpy2.mul(6, N)))
if gmpy2.is_integer(gmpy2.div(gmpy2.sub(A, x), 3)) and gmpy2.is_integer(gmpy2.div(gmpy2.add(A, x), 2)):
print 'p = ' + str(mpz(gmpy2.div(gmpy2.sub(A, x), 3)))
print 'q = ' + str(mpz(gmpy2.div(gmpy2.add(A, x), 2)))
else:
print 'p = ' + str(mpz(gmpy2.div(gmpy2.add(A, x), 3)))
print 'q = ' + str(mpz(gmpy2.div(gmpy2.sub(A, x), 2)))
# brew install gmp # brew install mpfr # brew install libmpc # bin/easy_install http://gmpy.googlecode.com/files/gmpy2-2.0.0b1.zip # wget http://www.math.umbc.edu/~campbell/Computers/Python/numbthy.py from __future__ import division # division with floats import gmpy2 gmpy2.get_context().precision = 1100 N = 179769313486231590772930519078902473361797697894230657273430081157732675805505620686985379449212982959585501387537164015710139858647833778606925583497541085196591615128057575940752635007475935288710823649949940771895617054361149474865046711015101563940680527540071584560878577663743040086340742855278549092581 A = gmpy2.ceil(gmpy2.sqrt(N)) A2 = A**2 print "N :", int(N) print "sq(A): %.f" % gmpy2.sqrt(N) print "round:", int(gmpy2.ceil(gmpy2.sqrt(N))) print "A^2 :", int(A2) assert A2 > N x = gmpy2.sqrt(A2 - N) print "x :", int(x) p = gmpy2.sub(A, x) q = gmpy2.add(A, x) print "p*q :", int(gmpy2.mul(p, q))
def part1():
A = gmpy2.ceil(gmpy2.sqrt(N))
x = gmpy2.mpz(gmpy2.ceil(gmpy2.sqrt(A**2 - N)))
p = gmpy2.sub(A, x)
q = gmpy2.add(A, x)
return p
q = A - x
return q if p * q == N else -1
def int_to_bytes(value, length): ## Function to be used in Q4.
result = []
for i in range(0, length):
result.append(value >> (i * 8) & 0xff)
result.reverse()
return bytes(result)
# Q1 (Given : |p - q| < 2*N^0.25)
N1 = 179769313486231590772930519078902473361797697894230657273430081157732675805505620686985379449212982959585501387537164015710139858647833778606925583497541085196591615128057575940752635007475935288710823649949940771895617054361149474865046711015101563940680527540071584560878577663743040086340742855278549092581
A1 = int(gmpy2.ceil(gmpy2.sqrt(N1)))
print("Ans 1. : %d" % factor(A1, N1))
# Q2 (Given : |p - q| < 2^11 * N^0.25)
N2 = 648455842808071669662824265346772278726343720706976263060439070378797308618081116462714015276061417569195587321840254520655424906719892428844841839353281972988531310511738648965962582821502504990264452100885281673303711142296421027840289307657458645233683357077834689715838646088239640236866252211790085787877
A2 = int(gmpy2.ceil(gmpy2.sqrt(N2)))
for i in range(2**20):
q = factor(A2 + i, N2)
if q != -1:
print("Ans 2. : %d" % q)
break
# Q3 (Given : |3p - 2q| < N^0.25)
N3 = 720062263747350425279564435525583738338084451473999841826653057981916355690188337790423408664187663938485175264994017897083524079135686877441155132015188279331812309091996246361896836573643119174094961348524639707885238799396839230364676670221627018353299443241192173812729276147530748597302192751375739387929
N1 = 179769313486231590772930519078902473361797697894230657273430081157732675805505620686985379449212982959585501387537164015710139858647833778606925583497541085196591615128057575940752635007475935288710823649949940771895617054361149474865046711015101563940680527540071584560878577663743040086340742855278549092581
# Tests
print("test")
print(169.0 % 1)
print(38.509658609 % 1)
# Challenge 1
gmpy2.get_context().precision = 1030
temp = gmpy2.sqrt(N1)
# print("t = ", temp)
A = gmpy2.ceil(temp)
print("A = ", A)
temp2 = A * A
temp3 = temp2 - N1
x = gmpy2.sqrt(temp3)
p = A - x
p4 = p
print("p= ", p)
q = A + x
q4 = q
print("q= ", q)
def factor1(n):
a = mpz(ceil(sqrt(n)))
x = sqrt(a**2 - n)
return mpz(a - x)
import gmpy2
from gmpy2 import mpz
from gmpy2 import mpfr
gmpy2.get_context().precision = 2000
N1 = mpz("179769313486231590772930519078902473361797697894230657273430081157732675805505620686985379449212982959585501387537164015710139858647833778606925583497541085196591615128057575940752635007475935288710823649949940771895617054361149474865046711015101563940680527540071584560878577663743040086340742855278549092581")
A1 = gmpy2.isqrt(N1) + 1 #A = (p+q/2) and is close to sqrt(N), so sqrt(N) is used to estimate A
x1 = gmpy2.isqrt(A1**2 - N1)
p1 = A1 - x1
q1 = A1 + x1
print p1
N2 = mpz("648455842808071669662824265346772278726343720706976263060439070378797308618081116462714015276061417569195587321840254520655424906719892428844841839353281972988531310511738648965962582821502504990264452100885281673303711142296421027840289307657458645233683357077834689715838646088239640236866252211790085787877")
A2 = gmpy2.isqrt(N2) + 1
while not gmpy2.is_square(A2**2 - N2):
A2 = A2 + 1
print A2 - gmpy2.isqrt(A2**2 - N2)
N3 = mpz("720062263747350425279564435525583738338084451473999841826653057981916355690188337790423408664187663938485175264994017897083524079135686877441155132015188279331812309091996246361896836573643119174094961348524639707885238799396839230364676670221627018353299443241192173812729276147530748597302192751375739387929")
A3 = gmpy2.ceil(gmpy2.sqrt(6*N3)*2)/2
print (A3 - gmpy2.sqrt(A3**2 - 6*N3))/3
e_m = mpz("22096451867410381776306561134883418017410069787892831071731839143676135600120538004282329650473509424343946219751512256465839967942889460764542040581564748988013734864120452325229320176487916666402997509188729971690526083222067771600019329260870009579993724077458967773697817571267229951148662959627934791540")
phi = (p1 - 1)*(q1 - 1)
d = gmpy2.invert(65537, phi)
d_m = gmpy2.powmod(e_m, d, N1)
hex = d_m.digits(16)
hex = hex[hex.find("00")+2:]
print hex.decode("hex")
def fac3(N):
root6N = 2 * gmpy2.sqrt(6 * N)
Ax2 = gmpy2.ceil(root6N)
A = Ax2 / 2
x = gmpy2.sqrt(A**2 - 6 * N)
p = (A - x) / 3
q = (A + x) / 2
print('p:\n' + str(int(p)))
print('q:\n' + str(int(q)))
return (int(p), int(q))
rootN = gmpy2.sqrt(N)
#print('rootN:', rootN)
#A = gmpy2.ceil(gmpy2.sqrt(N))
A = gmpy2.ceil(rootN)
rootNsq = rootN**2
Ncalc = N - rootNsq
#print('A:', A, 'A>N:', A>N, 'A>rootN:', A>rootN, 'Ncalc:', Ncalc)
# Since A is the exact mid-point between p and q there is an integer x
# such that p=A−x and q=A+x.
# But then N=pq=(A−x)(A+x)=A2−x2 and therefore x=A2−N**.5
res1 = gmpy2.mpz(A**2 - N)
#print('res1:', res1)
x = gmpy2.sqrt(A**2 - N)
#print('x:', x)
# Now, given x and A you can find the factors p and q of N since p=A−x and q=A+x.
def good_pair(p: int, q: int) -> int:
n = p * q
k = gmpy2.ceil(gmpy2.log2(n))
if abs(p - q) > 2**(k / 2 - 100):
return n
return 0
def is_prime_by_AKS(n):
"""
使用AKS算法确定n是否是一个素数
True:n是素数
False:n是合数
"""
def __is_integer__(n):
"""
判断一个数是否是整数
"""
i = gmpy2.mpz(n)
f = n - i
return not f
def __phi__(n):
"""
欧拉函数，测试小于n并与n互素的个数
"""
res = gmpy2.mpz(n)
a = gmpy2.mpz(n)
for i in range(2, a + 1):
if a % i == 0:
res = res // i * (i - 1)
while a % i == 0:
a //= i
if a > 1:
res = res // a * (a - 1)
return res
def __gcd__(a, b):
"""
计算a b的最大公约数
"""
if b == 0:
return a
return __gcd__(gmpy2.mpz(b), gmpy2.mpz(a) % gmpy2.mpz(b))
print("步骤1, 确定%d是否是纯次幂" % n)
for b in range(2, gmpy2.mpz(gmpy2.floor(gmpy2.log2(n))) + 1):
a = n**(1 / b)
if __is_integer__(a):
return False
print("步骤2，找到一个最小的r，符合o_r(%d) > (log%d)^2" % (n, n))
maxk = gmpy2.mpz(gmpy2.floor(gmpy2.log2(n)**2))
maxr = max(3, gmpy2.mpz(gmpy2.ceil(gmpy2.log2(n)**5)))
nextR = True
r = 0
for r in range(2, maxr):
if nextR == False:
break
nextR = False
for k in range(1, maxk + 1):
if nextR == True:
break
nextR = (gmpy2.mpz(n**k % r) == 0) or (gmpy2.mpz(n**k % r) == 1)
r = r - 1 # 循环多增加了一层
print("r = %d" % r)
print("步骤3，如果 1 < gcd(a, %d) < %d，对于一些 a <= %d, 输出合数" % (n, n, r))
for a in range(r, 1, -1):
g = __gcd__(a, n)
if g > 1 and g < n:
return False
print("步骤4，如果n=%d <= r=%d，输出素数" % (n, r))
if n <= r:
return True
print("步骤5")
print("遍历a从1到\sqrt{\phi(r=%d)}logn=%d" % (r, n))
print("如果(X+a)^%d != X^%d+a mod {X^%d-1, %d}$输出合数" % (n, n, r, n))
# 构造P = (X+a)^n mod (X^r-1)
print("构造多项式(X+a)^%d,并且进行二项式展开" % n)
X = multi_ysymbols('X')
a = multi_ysymbols('a')
X_a_n_expand = binomial_expand(ypolynomial1(X, a), n)
print(X_a_n_expand)
X.pow(r)
reduce_poly = ypolynomial1(X, ysymbol(value=-1.0))
print("构造消减多项式 %s" % reduce_poly)
print("进行运算 (X+a)^%d mod (X^%d-1)" % (n, r))
r_equ = ypolynomial_mod(X_a_n_expand, reduce_poly)
print("得到余式: %s" % r_equ)
print("进行运算'余式' mod %d 得到式(A)" % n)
A = ypolynomial_reduce(r_equ, n)
print("A = %s" % A)
print("B = x^%d+a mod x^%d-1" % (n, r))
B = ypolynomial1(multi_ysymbols('X', power=31), a)
B = ypolynomial_mod(B, reduce_poly)
print("B = %s" % B)
C = ypolynomial_sub(A, B)
print("C = A - B = %s" % C)
maxa = math.floor(math.sqrt(__phi__(r)) * math.log2(n))
print("遍历a = 1 to %d" % maxa)
print("检查每个'%s = 0 (mod %d)'" % (C, n))
for a in range(1, maxa + 1):
print("检查a = %d" % a)
C.set_variables_value(a=a)
v = C.eval()
if v % n != 0:
return False
print("步骤6 输出素数")
return True
import gmpy2
from gmpy2 import mpz
__author__ = 'Qubo'
N1 = '17976931348623159077293051907890247336179769789423065727343008115'
N1 += '77326758055056206869853794492129829595855013875371640157101398586'
N1 += '47833778606925583497541085196591615128057575940752635007475935288'
N1 += '71082364994994077189561705436114947486504671101510156394068052754'
N1 += '0071584560878577663743040086340742855278549092581'
gmpy2.set_context(gmpy2.context(precision=2000))
N = mpz(N1)
A = gmpy2.ceil(gmpy2.sqrt(N))
SquareA = gmpy2.square(A)
if gmpy2.is_square(mpz(gmpy2.sub(SquareA, N))):
x = gmpy2.sqrt(gmpy2.sub(SquareA, N))
print 'p = ' + str(mpz(gmpy2.sub(A, x)))
print 'q = ' + str(mpz(gmpy2.add(A, x)))
else:
print 'x is not square, must be wrong...'
def random_from_interval(random_state, low, high, m=32):
M = int(g.ceil(g.log2(high)))
random = g.mpz_urandomb(random_state, M + m)
c = g.add(g.f_mod(random, g.add(g.sub(high, low), 1)), low)
return c
def calc_near(self):
a = ceil(sqrt(self.n))
p, q = self._calc_factors(a)
assert self._check_sol(p, q)
assert gmpy2.is_prime(p)
return p, q
from gmpy2 import mpz
__author__ = 'Qubo'
N1 = '17976931348623159077293051907890247336179769789423065727343008115'
N1 += '77326758055056206869853794492129829595855013875371640157101398586'
N1 += '47833778606925583497541085196591615128057575940752635007475935288'
N1 += '71082364994994077189561705436114947486504671101510156394068052754'
N1 += '0071584560878577663743040086340742855278549092581'
N2 = '6484558428080716696628242653467722787263437207069762630604390703787'
N2 += '9730861808111646271401527606141756919558732184025452065542490671989'
N2 += '2428844841839353281972988531310511738648965962582821502504990264452'
N2 += '1008852816733037111422964210278402893076574586452336833570778346897'
N2 += '15838646088239640236866252211790085787877'
diff = 2 ** 20
gmpy2.set_context(gmpy2.context(precision = 3000))
N = mpz(N2)
lowerBound = mpz(gmpy2.ceil(gmpy2.sqrt(N)))
for i in range(0, diff):
A = gmpy2.add(lowerBound, i)
SquareA = gmpy2.square(A)
if gmpy2.is_square(gmpy2.sub(SquareA, N)):
x = mpz(gmpy2.sqrt(gmpy2.sub(SquareA, N)))
print 'p = ' + str(gmpy2.sub(A, x))
print 'q = ' + str(gmpy2.add(A, x))
def factor(A, N): # Assuming A = (p + q)/2 ## Function to be used in Q1,Q2.
A_square = int(gmpy2.ceil(gmpy2.square(A)))
x = int(gmpy2.ceil(gmpy2.sqrt(A_square - N)))
p = A + x
q = A - x
return q if p * q == N else -1
tmp_2 = gmpy2.powmod(tmp_1, i, p)
tmp_3 = gmpy2.powmod(gmpy2.mul(b, tmp_2), 1, p)
giant_x.append(tmp_3)
return giant_x
def get_result(baby_x, giant_x, m):
for x_b in baby_x:
for x_g in giant_x:
if x_b == x_g:
return gmpy2.add(gmpy2.mul(giant_x.index(x_g), m),
baby_x.index(x_b))
if __name__ == '__main__':
p = int(input('输入p:'))
a = int(input('输入α：'))
b = int(input('输入β：'))
ord = get_ord(p)
print('ord:' + str(ord))
m = int(gmpy2.ceil(gmpy2.sqrt(ord)))
print('m=' + str(m))
baby_x = get_baby_x(a, p, m)
print('x_b:' + str(baby_x))
giant_x = get_giant_x(a, b, m, p)
print('x_g:' + str(giant_x))
result = get_result(baby_x, giant_x, m)
print(result)
def computeA(N):
return gmpy2.ceil(gmpy2.sqrt(N))
def max_bits_in_digits(d):
"""
Returns maximum number of bits fitting in given number of digits eg. 2 digits (99) fits in 7 bits
:param d: number of digits
"""
return gmpy2.mpz(gmpy2.ceil(d * (gmpy2.log(10) / gmpy2.log(2))))
import numpy import gmpy2 import math from collections import defaultdict from gmpy2 import mpz gmpy2.get_context().precision=10000 N = 179769313486231590772930519078902473361797697894230657273430081157732675805505620686985379449212982959585501387537164015710139858647833778606925583497541085196591615128057575940752635007475935288710823649949940771895617054361149474865046711015101563940680527540071584560878577663743040086340742855278549092581 N=179769313486231590772930519078902473361797697894230657273430081157732675805505620686985379449212982959585501387537164015710139858647833778606925583497541085196591615128057575940752635007475935288710823649949940771895617054361149474865046711015101563940680527540071584560878577663743040086340742855278549092581 N=648455842808071669662824265346772278726343720706976263060439070378797308618081116462714015276061417569195587321840254520655424906719892428844841839353281972988531310511738648965962582821502504990264452100885281673303711142296421027840289307657458645233683357077834689715838646088239640236866252211790085787877 N=720062263747350425279564435525583738338084451473999841826653057981916355690188337790423408664187663938485175264994017897083524079135686877441155132015188279331812309091996246361896836573643119174094961348524639707885238799396839230364676670221627018353299443241192173812729276147530748597302192751375739387929 N= mpz(N)*6 A=gmpy2.ceil(gmpy2.sqrt(gmpy2.mpz(N))) B=numpy.powmod(2,20,N) for i in range(0,B+1): print i #x=mpz(A*A) -N A=mpz(A) z=mpz(A)*mpz(A) #print mpz(z)+1 y=mpz(z)-mpz(N) x=gmpy2.sqrt(gmpy2.mpz(y)) p=mpz(A)-mpz(x) q=mpz(A)+mpz(x) #print p #print mpz(p)
gmpy2.get_context().precision=500*8
n50b = "543971971236630195076898180273"
n60b = "459344432904829185748347139878533089"
n70b = "810814527278919159798277223576982175301717"
n80b = "662875573009490731045345994740327457245824480677"
n25b = 767971435180333
test = 90283
N = gmpy2.mpz(n70b)
n_smooth = 1500
offset = 800000000
start = 0
listN = []
start = gmpy2.mpz(gmpy2.ceil(gmpy2.sqrt(N)))
def init():
for j in range(0, offset):
listN.append(gmpy2.sub(gmpy2.mul(gmpy2.add(start, j), gmpy2.add(start, j)), N))
return start
def read_prime():
list_primes = []
with open("primes1000.txt") as fp:
for line in fp:
list_primes.extend([int(s) for s in re.split(r'\s+', line.strip())])
print("list prime:")
print(list_primes)
return list_primes