Add much faster integer factorization (#403 related to #199)

Author: delivrance

telethon/crypto/factorization.py

@@ -1,71 +1,45 @@
 from random import randint
-try:
-    import sympy.ntheory
-except ImportError:
-    sympy = None
 
 
 class Factorization:
-    @staticmethod
-    def find_small_multiplier_lopatin(what):
-        """Finds the small multiplier by using Lopatin's method"""
-        g = 0
-        for i in range(3):
-            q = (randint(0, 127) & 15) + 17
-            x = randint(0, 1000000000) + 1
-            y = x
-            lim = 1 << (i + 18)
-            for j in range(1, lim):
-                a, b, c = x, x, q
-                while b != 0:
-                    if (b & 1) != 0:
-                        c += a
-                        if c >= what:
-                            c -= what
-                    a += a
-                    if a >= what:
-                        a -= what
-                    b >>= 1
-
-                x = c
-                z = y - x if x < y else x - y
-                g = Factorization.gcd(z, what)
-                if g != 1:
+    @classmethod
+    def factorize(cls, pq):
+        if pq % 2 == 0:
+            return 2, pq // 2
+
+        y, c, m = randint(1, pq - 1), randint(1, pq - 1), randint(1, pq - 1)
+        g = r = q = 1
+        x = ys = 0
+
+        while g == 1:
+            x = y
+            for i in range(r):
+                y = (pow(y, 2, pq) + c) % pq
+
+            k = 0
+            while k < r and g == 1:
+                ys = y
+                for i in range(min(m, r - k)):
+                    y = (pow(y, 2, pq) + c) % pq
+                    q = q * (abs(x - y)) % pq
+
+                g = cls.gcd(q, pq)
+                k += m
+
+            r *= 2
+
+        if g == pq:
+            while True:
+                ys = (pow(ys, 2, pq) + c) % pq
+                g = cls.gcd(abs(x - ys), pq)
+                if g > 1:
                     break
 
-                if (j & (j - 1)) == 0:
-                    y = x
-
-            if g > 1:
-                break
-
-        p = what // g
-        return min(p, g)
+        return g, pq // g
 
     @staticmethod
     def gcd(a, b):
-        """Calculates the greatest common divisor"""
-        while a != 0 and b != 0:
-            while b & 1 == 0:
-                b >>= 1
+        while b:
+            a, b = b, a % b
 
-            while a & 1 == 0:
-                a >>= 1
-
-            if a > b:
-                a -= b
-            else:
-                b -= a
-
-        return a if b == 0 else b
-
-    @staticmethod
-    def factorize(pq):
-        """Factorizes the given number and returns both
-           the divisor and the number divided by the divisor
-        """
-        if sympy:
-            return tuple(sympy.ntheory.factorint(pq).keys())
-        else:
-            divisor = Factorization.find_small_multiplier_lopatin(pq)
-            return divisor, pq // divisor
+        return a