def number_odd_period(N):
total = 0
for n in tqdm(range(1, N + 1)):
continued_fraction = continued_fraction_periodic(0, 1, n)
if len(continued_fraction) > 1 and len(continued_fraction[1]) % 2 == 1:
total += 1
return total
def get_lowest_cf(p, d, x):
#find convergent cfs
cfp = continued_fraction_periodic(p, d)
fracs = list(continued_fraction_convergents(cfp))
#get list of denominators
denoms = [int(f.q) for f in fracs]
try:
val = max([num for num in denoms if num < x])
except ValueError:
raise ('List is empty!')
return val
def wiener_attack_rsa(public_key):
N = public_key[0]
e = public_key[1]
#
cf = continued_fraction_periodic(e, N)
convergents = list(continued_fraction_convergents(cf))
polynomial_power = 0
for i in convergents:
polynomial_power += 1
n, d = fraction(i)
if n != 0:
exp = ((e * d) - 1) / n
n2, d2 = fraction(exp)
#if d2 == 1:
factors = (np.roots([1, -((N - n2) + 1), N]))
return factors
def viner_attack(e, n):
# Find period of fraction
drob = continued_fraction_periodic(e, n)
# Find all convergent fraction k_1/d_1, k_i/d_i, k_n/d_n
# Calculation based on Euler recurrent formulas
convergents = []
convergents.append((drob[0], 1))
convergents.append((drob[0] * drob[1] + 1, drob[1]))
i = 2
while i < len(drob):
p = convergents[i - 1][0] * drob[i] + convergents[i - 2][0]
q = convergents[i - 1][1] * drob[i] + convergents[i - 2][1]
convergents.append((p, q))
i = i + 1
i = 0
for (k, d) in convergents:
# check if d is actually the key
# We are try to calculate: f_i = (e*d_i - 1) / k_i
# 1. So k_i != 0
# 2. We don't want irrational f_i, so check (e * d - 1) % k == 0
if k != 0 and (e * d - 1) % k == 0:
f = (e * d - 1) // k
# Original equation: x^2 - ((N - f) + 1)*x + N = 0
# So we are replace s = ((N - f) + 1)
s = n - f + 1
# check if the equation x^2 - s*x + n = 0
# has integer roots
discr = s * s - 4 * n
# Discriminant exist
if (discr >= 0):
# Find t = sqrt(discr)
t = _is_perfect_square(discr)
if t != -1 and (s + t) % 2 == 0:
return int(d)
def main(): # generate a list of square numbers to skip for convenience n = 1 squares = set() while n**2 <= MAXN: squares.add(n**2) n+=1 N = 1 count = 0 while N <= MAXN: if N not in squares: if len(continued_fraction_periodic(0,1,N)[1]) % 2 == 1: count += 1 # print(N, "counted") # else: #print(N, "not counted") N += 1 print(count)
def test_p064():
assert continued_fraction_periodic(0, 1, 23) == [4, [1, 3, 1, 8]]
assert number_odd_period(13) == 4
from sympy.ntheory.continued_fraction import continued_fraction_periodic print continued_fraction_periodic(3, 2, 7)