
Python code for weil pairing algorithm for Bitcoin:
import ecc
# Generate public and private keys
private_key = ecc.generate_private_key(curve='secp256k1')
public_key = private_key.pubkey
# Generate a random point on the curve
point = ecc.generate_random_point(curve='secp256k1')
# Perform Weil pairing
result = ecc.weil_pairing(public_key, point)
print("Weil pairing result:", result)
This code uses the ecc
library to generate a private and public key, a random point on the curve, and performs the Weil pairing algorithm for Bitcoin.
The Weil pairing algorithm is a mathematical algorithm used in cryptography to calculate a shared secret between two parties. It’s used in Bitcoin to calculate the public and private keys for a Bitcoin address.
The Weil pairing algorithm is based on elliptic curve cryptography (ECC), which is a type of public-key cryptography. ECC is used in Bitcoin to create a secure, decentralized network.
The Weil pairing algorithm is a complex algorithm, and implementing it correctly requires a deep understanding of cryptography and mathematics. It’s not recommended to try to implement it yourself unless you’re an experienced cryptographer or mathematician.
Weil pairing algorithm in Python:
import hashlib
# Define the parameters
p = 21888242871839275222246405745257275088548364400416034343698204186575808495617
a = -3
b = 41018428876551020177132617442589261862607196850031167551342877870051005937681
G1 = (2, 41018428876551020177132617442589261862607196850031167551342877870051005937681)
G2 = (p - 1, 2)
# Define the functions
def add(x, y):
return (x[0] + y[0], x[1] + y[1])
def double(x):
return (x[0] * 2, x[1] * 2)
def subtract(x, y):
return (x[0] - y[0], x[1] - y[1])
def multiply(x, y):
return ((y[0] + x[0]) * (y[1] + x[1]), (y[0] + x[0]) * (y[1] - x[1]) - (y[0] - x[0]) * (y[1] + x[1]))
def negate(x):
return (x[0], -x[1])
def miller_loop(a, b, P):
Q = (0, 1)
R = P
for i in range(a - 1):
Q = double(Q)
if a % 2 == 1:
R = add(R, Q)
return R
# Define the Weil pairing function
def weil_pairing(P, Q):
R = miller_loop(a, b, P)
T = miller_loop(a, b, Q)
return (T[0] - R[0]) / (T[1] - R
Python implementation of the Weil Pairing algorithm:
import hashlib
import binascii
def miller_loop(a, P, r, s):
n = len(P) - 1
d = P[n] - (a ** (r - 1)) % P[n]
for i in range(r - 2, -1, -1):
d = (d ** 2) % P[i] - (a ** (2 ** (r - 1 - i))) % P[i] + P[i]
return d == 0
def weil_pairing(a, b, P):
n = len(P) - 1
r = 2 ** n
s = (r - 1) // 2
if miller_loop(a, P, r, s):
return pow(b, s, P)
else:
return pow(b, -s, P)
def main():
P = [2, 5, 17, 257, 65537, 4294967297]
a = 5
b = 17
print("Weil Pairing (a, b, P):", weil_pairing(a, b, P))
if __name__ == "__main__":
main()
In this code, we first define the miller_loop
function, which is used to check if a given point is on the elliptic curve. The weil_pairing
function is the main function that performs the Weil Pairing algorithm. It takes in three parameters: a
, b
, and P
, where P
is the elliptic curve. Finally, we define the main
function, which sets the values of a
, b
, and P
and prints the result of the Weil Pairing algorithm.
Python function that implements the Weil pairing algorithm:
import math
def weil_pairing(p, q, g, g_sq, a, b):
k = math.floor((p + 1) // 4)
x = g_sq ** ((p - 1) // 2)
y = a ** (p - 1) % p
z = b ** (q - 1) % q
r = g ** x % p
s = g ** y % p
t = g ** z % q
u = s ** (q - 1) % q
v = t ** (p - 1) % p
w = (u - 1) // q
x = (v - 1) // p
y = math.gcd(w, q)
z = math.gcd(x, p)
if y = 1 or z = 1:
return None
return (r ** (k * w) % p, s ** (k * x) % p, t ** (k * z) % q)
This function takes in the prime numbers p
and q
, the generator g
of the order q
, the square of g
raised to the power of q - 1
(i.e. g_sq
), and the two elements a
and b
of the groups Z_p
and Z_q
respectively. It returns the tuple (r, s, t)
where r
is the pairing value of a
and g
, s
is the pairing value of b
and g
, and t
is the pairing value of a
and b
.
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