OK i would like to say, till now i get all the puzzles from 1 to 65 faster than let´s say kangaroo, and the code i have, is write in python !!
I would like to put a picture here in, so maybe it explain more, but i don´t know how to do it.
My maths. but something is off. i still feel its not right somewhere.
import random
import hashlib
from sage.crypto.util import ascii_to_bin
from sage.arith.misc import random_prime
from sage.crypto.util import ascii_to_bin
from sage.crypto.util import bin_to_ascii
p = 0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f
n = 0xfffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141
E = EllipticCurve(GF(p), [0, 7])
r = 0x
s = 0x
z = 0x
G = E.point( (0x79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798,0x483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8)) # Base point
def egcd(a, b):
if a == 0:
return (b, 0, 1)
else:
g, y, x = egcd(b % a, a)
return (g, x - (b // a) * y, y)
def modinv(a, m):
g, x, y = egcd(a, m)
if g != 1:
raise Exception('modular inverse does not exist')
else:
return x % m
def make_public(r, s, z):
R = E.lift_x(Integer(r))
w = int(modinv(s, n))
u1 = int((z * w) % n)
u2 = int((r * w) % n)
u_n2 = modinv(u2, n) % n
u_n1 = - u1 * modinv(u2, n) % n
pub = u_n1 * G + u_n2 * R
pub2 = u_n1 * G + u_n2 * (-R)
return pub, pub2
def calc_u(r, s, z):
mod_s = modinv(s, n) % n
u1 = mod_s * z % n
u2 = mod_s * r % n
print("u1 =", hex(u1), "n-u1 =", hex(n - u1))
print("u2 =", hex(u2), "n-u2 =", hex(n - u2))
return u1, u2
u1, u2 = calc_u(r, s, z)
def verify(r, s, z, pub, k):
w = int(modinv(s, n))
u1 = int((z * w) % n)
u2 = int((r * w) % n)
D = u1 * G + u2 * pub
x, y = D.xy()
x = int(x)
if (r % n) == (x % n):
print(f"Signature k matches: {hex(k)} ")
return True
else:
print(f"Signature k is invalid {hex(k)} ")
return False
# Calculate the modular inverse of s (w = s^(-1) mod n)
g, x, y = egcd(s, n)
w = x % n
def find_k(r, s, z, pub):
# Step 1: Calculate w = s^(-1) mod n
w = int(modinv(s, n))
# Step 2: Calculate u1 and u2
u1 = int((z * w) % n)
u2 = int((r * w) % n)
# Step 3: Recover the elliptic curve point R
R = u1 * G + u2 * pub
# Extract x-coordinate from R
x_R, _ = R.xy()
x_R = int(x_R)
# Calculate k
k = (x_R - z) * modinv(w, n) % n
print("k:",k)
return k
# Call the function to find k
k = find_k(r, s, z, pub1)
print("Found k:", hex(k))
pub1, pub2 = make_public(r, s, z)
print("public_key1", pub1)
print("pub1_x=", hex(pub1.xy()[0]))
print("public_key2", pub2)
print("pub2_x=", hex(pub2.xy()[0]))
print("Step 1: Calculate g, x, y using extended Euclidean algorithm")
print("g:", hex(g))
print("x:", hex(x))
print("y:", hex(y))
print("Step 2: Calculate w (modular inverse of s)")
print("Calculated w:", hex(w))
print()
def find_k_bruteforce(r, s, z, pub):
i = 1
while True:
k_candidate = (r * i + z) * modinv(s, n) % n
# Verify the signature using the candidate k
if verify(r, s, z, pub, k_candidate):
print("Found k:", hex(k_candidate))
break
else:
print(f"Attempt {i}: Incorrect k value {hex(k_candidate)}")
i += 1 # Increment the attempt counter
# Call the function to find k using brute force
find_k_bruteforce(r, s, z, pub1)
def find_private_key(r, s, z, k, pub):
# Calculate w = s^(-1) mod n
w = int(modinv(s, n))
# Calculate u1 and u2
u1 = int((z * w) % n)
u2 = int((r * w) % n)
# Recover the elliptic curve point R
R = u1 * G + u2 * pub
# Extract x-coordinate from R
x_R, _ = R.xy()
x_R = int(x_R)
# Calculate private key d
d = (k * x_R - z) * modinv(r, n) % n
return d
def find_private_key_bruteforce(r, s, z, k, pub, i):
k_candidate = (r * i + z) * modinv(s, n) % n
# Calculate private key d
d = (k_candidate * r - z) * modinv(k_candidate, n) % n
return d
# Call the function to find k using the formula-based approach
k_formula = find_k(r, s, z, pub1)
d_formula = find_private_key(r, s, z, k_formula, pub1)
print("Private key (formula-based):", hex(d_formula))
# Call the function to find k using brute force
k_bruteforce = find_k_bruteforce(r, s, z, pub1)
i_bruteforce = 1
d_bruteforce = find_private_key_bruteforce(r, s, z, k_bruteforce, pub1, i_bruteforce)
print(f"Private key (brute-force attempt {i_bruteforce}):", hex(d_bruteforce))
guessed_k_values = []
i = 1
while True:
k = (r * i + z) * modinv(s, n) % n
# Calculate R using the guessed k value
guessed_R = k * G
guessed_k_values.append((i, k, guessed_R))
# Check if the guessed R value matches the original r value
if guessed_R.xy()[0] == r:
print(f"The correct k value ({k}) for iteration {i} is found.")
# Calculate the private key using the correct k value
d_bruteforce = find_private_key_bruteforce(r, s, z, k, pub1, i)
print(f"Corresponding private key: {hex(d_bruteforce)}")
break # Exit the loop if the correct k is found
i += 1 # Increment the iteration counter
These brothers are keys with the same value "x"? Good luck counting your keys. 
