def main2():
a = continued_fraction(Integer(17) / Integer(23))
print(a)
b = continued_fraction(Integer(6) / Integer(23))
print(b)
A = CFF(a)
B = CFF(b)
print(A + B)
print(A.value())
print(a.value())
def main5():
c = continued_fraction(
[Integer(1), Integer(2), Integer(3), Integer(4), Integer(5)])
print(c)
print(c.convergents())
print([c.p(n) for n in range(c.length())])
print([c.q(n) for n in range(c.length())])
def main6():
c = continued_fraction([Integer(1)] * (8))
v = [(i, c.p(i) / c.q(i)) for i in range(c.length())]
P = point(v, rgbcolor=(0, 0, 1), pointsize=40)
L = line(v, rgbcolor=(0.5, 0.5, 0.5))
L2 = line([(Integer(0), c.value()), (c.length() - Integer(1), c.value())],
thickness=0.5, rgbcolor=(0.7, 0, 0))
(L + L2 + P).save(filename="continued_fraction.png")
print("Graph is saved as continued_fraction.png")
def main4():
c = continued_fraction(pi)
for n in range(-1, 13):
print(c.p(n) * c.q(n - Integer(1)) -
c.q(n) * c.p(n - Integer(1)), end=' ')
print()
for n in range(Integer(13)):
print(c.p(n) * c.q(n - Integer(2)) -
c.q(n) * c.p(n - Integer(2)), end=' ')
print()
def wiener(e, n):
m = 12345
c = pow(m, e, n)
list1 = continued_fraction(Integer(e) / Integer(n))
conv = list1.convergents()
for i in conv:
d = int(i.denominator())
m1 = pow(c, d, n)
if m1 == m:
return d
def wiener(e, n):
m = 12345
c = pow(m, e, n)
q0 = 1
list1 = continued_fraction(Integer(e) / Integer(n))
conv = list1.convergents()
for i in conv:
k = i.numerator()
q1 = i.denominator()
for r in range(20):
for s in range(20):
d = r * q1 + s * q0
m1 = pow(c, d, n)
if m1 == m:
return d
q0 = q1
def attack(n, e):
"""
Recovers the prime factors of a modulus and the private exponent if the private exponent is too small.
:param n: the modulus
:param e: the public exponent
:return: a tuple containing the prime factors of the modulus and the private exponent, or None if the private exponent was not found
"""
convergents = continued_fraction(Integer(e) / Integer(n)).convergents()
for c in convergents:
k = c.numerator()
d = c.denominator()
if k == 0 or (e * d - 1) % k != 0:
continue
phi = (e * d - 1) // k
factors = known_phi.factorize(n, phi)
if factors:
return *factors, d
def attack(n, e, max_s=20000, max_r=100, max_t=100):
"""
Recovers the prime factors of a modulus and the private exponent if the private exponent is too small.
More information: Dujella A., "Continued fractions and RSA with small secret exponent"
:param n: the modulus
:param e: the public exponent
:param max_s: the amount of s values to try (default: 20000)
:param max_r: the amount of r values to try for each s value (default: 100)
:param max_t: the amount of t values to try for each s value (default: 100)
:return: a tuple containing the prime factors of the modulus and the private exponent, or None if the private exponent was not found
"""
i_n = Integer(n)
i_e = Integer(e)
threshold = i_e / i_n + (RealNumber(2.122) * i_e) / (i_n * i_n.sqrt())
convergents = continued_fraction(i_e / i_n).convergents()
for i in range(1, len(convergents) - 2, 2):
if convergents[i + 2] < threshold < convergents[i]:
m = i
break
for s in range(max_s):
for r in range(max_r):
k = r * convergents[m + 1].numerator() + s * convergents[m + 1].numerator()
d = r * convergents[m + 1].denominator() + s * convergents[m + 1].denominator()
if k == 0 or (e * d - 1) % k != 0:
continue
phi = (e * d - 1) // k
factors = known_phi.factorize(n, phi)
if factors:
return *factors, d
for t in range(max_t):
k = s * convergents[m + 2].numerator() - t * convergents[m + 1].numerator()
d = s * convergents[m + 2].denominator() - t * convergents[m + 1].denominator()
if k == 0 or (e * d - 1) % k != 0:
continue
phi = (e * d - 1) // k
factors = known_phi.factorize(n, phi)
if factors:
return *factors, d
def main1():
print(continued_fraction(Integer(17) / Integer(23)))
print(continued_fraction(e))
print(continued_fraction_list(e, bits=21))
print(continued_fraction_list(e, bits=30))
def main3():
c = continued_fraction(pi)
print(c.convergents()[:6].list())