complete algo.py

algo.py

+import math
+
+def bachet_bezout(a, b):
+    """Does the Bachet-Bezout algorithm on a of b
+
+    Args:
+        a (uint): the number
+        b (uint): the modulo
+
+    Returns:
+        uint: An array with the pgdc and the coefficients of bachet_bezout, [pgdc, u, v]
+    """
+    
+    if (a == 0):
+        return 0
+    if (b == 0):
+        return -1
+    
+    u = [1, 0]
+    v = [0, 1]
+    q = 1
+    r = 1
+    i = 2
+    while (r > 0):
+        q = a // b
+        r = a % b
+        
+        u.append(u[i-2] - q * u[i-1])
+        v.append(v[i-2] - q * v[i-1])
+        
+        a = b
+        b = r
+        i += 1 
+    
+    return a, u[i-2], v[i-2]
+
+def exponentiation_rapide(a, exp, n):
+    """Does the quick explanation of a pow x modulo n
+
+    Args:
+        a (uint): the number
+        exp (uint): the exponent of the number
+        n (uint): the modulo
+
+    Returns:
+        uint: An array with the pgdc and the coefficients of bachet_bezout, [pgdc, u, v]
+    """
+    
+    if (a == 0):
+        return 0
+    if (exp == 0):
+        return 1
+    
+    r = 1
+    b = a % n
+    while (exp > 0):
+        y = exp % 2
+        r = (r * b**y) % n
+        b = (b**2) % n
+        exp = exp // 2
+    
+    return r
+
+def is_square(a):
+    """Check if a number is a perfect square, using the Newton methode
+    from https://stackoverflow.com/questions/2489435/check-if-a-number-is-a-perfect-square 
+        
+    Args:
+        a (uint): number checked
+
+    Returns:
+        bool: true if the number is a perfect square, otherwise flase
+    """
+    
+    x = a // 2
+    seen = set([x])
+    
+    while x * x != a:
+        x = (x + (a // x)) // 2
+        
+        if x in seen: 
+            return False
+        
+        seen.add(x)
+        
+    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
+
+    Args:
+        n (uint): number used
+
+    Returns:
+        tuple of uint: the two coeficient a and b
+    """
+    
+    a = math.ceil(math.sqrt(n))
+    b = 0
+    while(True):
+        b2 = a**2 - n
+        
+        if (is_square(b2)):
+            b = math.sqrt(b2)
+            break
+        
+        a += 1
+        
+    return (a, b)
+