python implementation of miller’s algorithm for bitcoin. here is an example:
def miller_rabin(n, k):
"""
miller-rabin primality test.
n: integer to test for primality.
k: number of iterations.
"""
if n < 2:
return false
d = n - 1
while d % 2 == 0:
d //= 2
r = 0
x = pow(2, d, n)
while k > 0:
x = pow(x, 2, n)
if x == 1:
return true
if x == n - 1:
r += 1
k -= 1
return r == 0
this implementation takes two arguments: n
and k
. n
is the number to test for primality, and k
is the number of iterations. it returns true
if n
is probably prime, and false
otherwise.
to use this function to test whether a bitcoin private key is valid, you can generate a random 256-bit number and test it with miller’s algorithm. if the number passes the test, it is probably prime and can be used as a private key. here is an example:
import random
def miller_rabin(n, k):
# implementation of miller-rabin primality test
while true:
private_key = random.getrandbits(256)
if miller_rabin(private_key, 10):
break
print(f"valid private key: {private_key}")
this code generates a random 256-bit number, tests it with miller’s algorithm with 10 iterations, and prints it if it passes the test.
python code for miller’s algorithm to solve the discrete logarithm problem. however, i must remind you that miller’s algorithm is not very secure, and it’s not used in practice anymore.
here’s an example code:
def miller_rabin(n, k):
# base case
if n == 1:
return true
if n == 2:
return false
# find the factors of n
factors = []
for i in range(1, n):
if n % i == 0:
factors.append(i)
# loop through the factors
for factor in factors:
if factor == n - 1:
continue
x = pow(factor, (n - 1) // factor, n)
if x == 1 or x == n - 1:
continue
for i in range(k - 1):
x = pow(x, 2, n)
if x == n - 1:
return false
return false
return true
in this code, n
is the number whose discrete logarithm we want to find, and k
is the number of iterations. the function returns true
if n
is a probable prime, and false
otherwise.
note that this code is not optimized for speed or efficiency, and it may not work for very large values of n
and k
. also, as i mentioned earlier, miller’s algorithm is not very secure, and it’s not used in practice anymore.
Miller’s algorithm for solving the discrete logarithm problem in Python.
def miller_rabin(n, k):
# Check if n is a prime number
if n <= 1:
return False
if n <= 3:
return True
if n % 2 == 0 or n % 3 == 0:
return False
i = 5
while i * i <= n:
if n % i == 0 or n % (i + 2) == 0:
return False
i += 6
return True
# Function to find the discrete logarithm of a number
def discrete_logarithm(n, k):
# Check if n is a prime number
if not miller_rabin(n, k):
return -1
# Calculate the value of b
b = 2
while b <= n:
if n % b == 1:
break
b += 1
# Calculate the value of x
x = 2
while x <= n:
if n % x == 1:
break
x += 1
# Calculate the value of a
a = (b - 1) // x
# Calculate the value of y
y = 1
while y <= n:
if y == 1:
break
y = y * y % n
# Calculate the value of r
r = 1
while r <= n:
if r == n - 1:
break
r = r * r % n
# Calculate the value of s
s = 1
while s <= n:
if s == n - 1:
break
s = s * s % n
# Calculate the value of t
t = (s - 1) // r
# Calculate the value of u
u = t * r % (n - 1)
# Calculate the value of v
v = (u - 1) // x
Python implementation of Miller’s algorithm:
def miller_algorithm(base, exponent, n):
if exponent == 0:
return 1
elif exponent % 2 == 0:
x = miller_algorithm(base, exponent // 2, n)
return (x * x) % n
else:
y = miller_algorithm(base, (n - 1) // exponent, n)
return (base * y ** (n - 1)) % n
This function takes as input the base, exponent, and modulus n, and returns the solution to the discrete logarithm problem (i.e., the value of x such that base ** x = exponent mod n). The algorithm works by recursively computing the value of x using the formula:
x = (base ** (exponent // 2)) % n
If exponent is even, and (base ** (exponent // 2)) % n is not equal to 1, then we can continue recursing. Otherwise, we can compute the value of x as:
x = (base ** ((n - 1) // exponent)) % n
And then return x.
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