add another methode using the first prime divider

algo.py

 import math
+import random
 
 def bachet_bezout(a, b):
     """Does the Bachet-Bezout algorithm on a of b
@@ -34,6 +35,7 @@ def bachet_bezout(a, b):
     
     return a, u[i-2], v[i-2]
 
+
 def inverse_modulaire(a, n):
     """Does the inverse modular of a by n, using Bachet-Bezout
 
@@ -47,6 +49,7 @@ def inverse_modulaire(a, n):
     
     return bachet_bezout(a, n)[1]
 
+
 def exponentiation_rapide(a, exp, n):
     """Does the quick explanation of a pow x modulo n
 
@@ -74,6 +77,7 @@ def exponentiation_rapide(a, exp, n):
     
     return int(r)
 
+
 def is_square(a):
     """Check if a number is a perfect square, based on the "Babylonian algorithm" for square root,
     from : https://stackoverflow.com/questions/2489435/check-if-a-number-is-a-perfect-square 
@@ -98,6 +102,7 @@ def is_square(a):
         
     return True
 
+
 def fermat_factorization(n):
     """Does the Fermat's factorization on n,
     n = a² - b² = (a + b) * (a - b) = p * q <=> b² = a² - n
@@ -138,3 +143,89 @@ def decode_msg(M):
     """
     
     return M.to_bytes((M.bit_length() + 7) // 8, "little").decode("utf-8")
+
+
+def miller_rabin_test(d, n):
+    """Make the test of Miller Rabin (called for all k trials)
+    from : https://www.geeksforgeeks.org/primality-test-set-3-miller-rabin/ 
+
+    Args:
+        d (uint): the exponent d is an odd number such that d*2^r = n-1 for some r >= 1
+        n (uint): the modulo
+
+    Returns:
+        bool: False if n is composite, True if n is probably prime.
+    """
+    
+    # Pick a random number in [2..n-2], corner cases make sure that n > 4
+    a = 2 + random.randint(1, n - 4)
+
+    # Compute a^d % n
+    x = exponentiation_rapide(a, d, n)
+
+    if (x == 1 or x == n - 1):
+        return True
+
+    # Keep squaring x while one of the following doesn't happen
+    #   1. d does not reach n-1
+    #   2. (x^2) % n is not 1
+    #   3. (x^2) % n is not n-1
+    while (d != n - 1):
+        x = (x * x) % n
+        d *= 2
+
+        if (x == 1):
+            return False
+        if (x == n - 1):
+            return True
+
+    return False # Return composite
+
+
+def is_prime(n, k):
+    """Check if a number is prime using the test ofMiller Rabin.
+    from : https://www.geeksforgeeks.org/primality-test-set-3-miller-rabin/ 
+
+    Args:
+        n (uint): the value checked (n must be greater than 4)
+        k (uint): an input parameter that determines accuracy level, higher value of k indicates more accuracy.
+
+    Returns:
+        bool: True if the number is probably prime, False if n is composite
+    """
+
+    # Corner cases
+    if (n <= 1 or n == 4):
+        return False
+    if (n <= 3):
+        return True
+
+    # Find r such that n = 2^d * r + 1 for some r >= 1
+    d = n - 1
+    while (d % 2 == 0):
+        d //= 2
+
+    # Iterate given number of 'k' times
+    for i in range(k):
+        if (miller_rabin_test(d, n) == False):
+            return False
+
+    return True
+    
+
+def get_prime_divider(n):
+    """Get the first prime divider of n, starting with the root of n going to 0
+
+    Args:
+        n (uint): number used
+
+    Returns:
+        uint: the prime divider of n
+    """
+    
+    for i in reversed(range(math.ceil(math.sqrt(n)))):
+        
+        if (n % i == 0 and is_prime(i, 4)):
+            return i
+        
+    return 1
\ No newline at end of file

main.py

@@ -33,6 +33,10 @@ def main():
     p = a + b
     q = a - b
     
+    # - second methode (sqrt(n) -> 0) -
+    # p = get_prime_divider(n)
+    # q = n // p
+    
     fi = (p - 1) * (q - 1)
     d = inverse_modulaire(e, fi) # get private key