def partial_p(p0, kbits, n):
# Ensure n is an integer (as it should be), and create the polynomial ring over Z/nZ
PR.<x> = PolynomialRing(Zmod(n))
nbits = n.nbits() # Get the number of bits in n
f = Integer(2)^kbits * x + Integer(p0) # Ensure all numbers are integers
f = f.monic() # Make the polynomial monic
# Find small roots of the polynomial
try:
roots = f.small_roots(X=Integer(2)**(nbits // 2 - kbits), beta=RealNumber('0.3'))
except Exception as e:
print(f"Error in small_roots: {e}")
return None
if roots:
x0 = Integer(roots[0]) # Get the first root and ensure it's an integer
p = gcd(Integer(2)^kbits * x0 + p0, n) # Compute gcd to find p
return ZZ(p) # Return p as an integer
return None
def find_p(d0, kbits, e, n):
X = var('X') # Define a symbolic variable X
# Iterate through possible values of k
for k in range(1, e + 1):
# Solve the modular equation for possible values of p0
results = solve_mod([e * d0 * X - k * X * (n - X + 1) + k * n == X], 2^kbits)
for x in results:
p0 = ZZ(x[0]) # Convert the solution to an integer
p = partial_p(p0, kbits, n) # Try to find p using partial_p
if p:
return p # Return p if found
return None
if __name__ == '__main__':
n = 0xcc5b706f373a79c680cec9527aac573fd435129cf16c23334085bf97832e5a6c78b633c2f244b12a62f87ec5295dd89fcf3c808c39e45a9afdbda2f8d2d0b50d61b685c0fe9eb41a7018a40f98892f96d738e2a4e740d4e507bcbd07f68c1ecb2ca10bd780ce65265a7e4da00f1031a5db9d038878a29a5ffefcaf2119720005
e = 65537
d0 = 0x20142ae2802b877eb4dfa8a462e7d017c4d348181c367fd1a661ec9b6bbcca9dcb6601ccb6c10416b7f3c20129527346bbc136ee60f9945125cba03a9bba3720f7411
nbits = n.nbits() # Get the number of bits in n
print(f"nbits: {nbits}")
kbits = d0.nbits() # Number of bits in the given part of d
print(f"kbits: {kbits}")
print(f"lower {kbits} bits (of {nbits} bits) are given")
# Attempt to find p
p = find_p(d0, kbits, e, n)
if p:
print(f"Found p: {p}")
q = n // p # Compute q
print(f"Found q: {q}")
private_exponent = inverse_mod(e, (p - 1) * (q - 1)) # Compute the private exponent
print(f"Private exponent: {private_exponent}")
else:
print("Failed to find p.")