Add lab13

lab13/zad1.py

@@ -0,0 +1,55 @@
+# Aleksander Kluczka
+
+import random
+
+# wynik:
+# 561 - z
+# 1729 - z
+# 6601 - z
+# 881 - p
+# 9677 - p
+# 17371 - p
+# 37579 - p
+
+
+def witness(a, n):
+    t, u = 0, n - 1
+    while u % 2 == 0:
+        t += 1
+        u //= 2
+    assert t >= 1
+    assert u % 2 == 1
+    assert n - 1 == pow(2, t) * u
+    x_0 = pow(a, u, n)
+    x_i1 = x_0
+    for _ in range(t):
+        x_i = pow(x_i1, 2, n)
+        if (x_i == -1) and (x_i1 != 1) and (x_i1 != (n - 1)):
+            return True
+        x_i1 = x_i
+
+    if x_i != 1:
+        return True
+    return False
+
+
+def miller_rabin(n, s):
+    for j in range(s):
+        a = random.randint(1, n - 1)
+        if witness(a, n):
+            return False
+    return True
+
+
+def zad1():
+    input = [561, 1729, 6601, 881, 9677, 17321, 37579]
+    for number in input:
+        print(f"{number} - {'p' if miller_rabin(number, 50) else 'z'}")
+
+
+def main():
+    zad1()
+
+
+if __name__ == "__main__":
+    main()

lab13/zad2.py

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+# Aleksander Kluczka
+
+
+def nwd_expanded(j, k):
+    if j == 0:
+        return k, 0, 1
+    r = k % j
+    d, xp, yp = nwd_expanded(r, j)
+    x = yp - (k // j) * xp
+    y = xp
+    return d, x, y
+
+
+def pollard_p_minus_1(n: int, b: int) -> int | None:
+    a: int = 2
+    for j in range(2, b + 1):
+        a = pow(a, j, n)
+    d: int = nwd_expanded(a - 1, n)[0]
+    if 1 < d < n:
+        return d
+    else:
+        return None
+
+
+def pollard_p_minus_1_with_search(n: int) -> tuple[int, int]:
+    b = 2
+    result = None
+    while True:
+        result = pollard_p_minus_1(n, b)
+        if result is not None:
+            break
+        b += 1
+    return result, b
+
+
+def zad2():
+    number_1 = 262063
+    number_2 = 9420457
+
+    result_1, b_1 = pollard_p_minus_1_with_search(number_1)
+    print(f"pollard_p1({number_1}) = {result_1}, minimum b = {b_1}")
+
+    result_2, b_2 = pollard_p_minus_1_with_search(number_2)
+    print(f"pollard_p1({number_2}) = {result_2}, minimum b = {b_2}")
+
+
+def main():
+    zad2()
+
+
+if __name__ == "__main__":
+    main()

lab13/zad3.py

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+# Aleksander Kluczka
+
+from typing import Callable
+
+
+def nwd_expanded(j: int, k: int) -> tuple[int, int, int]:
+    if j == 0:
+        return k, 0, 1
+    r: int = k % j
+    d, xp, yp = nwd_expanded(r, j)
+    x: int = yp - (k // j) * xp
+    y: int = xp
+    return d, x, y
+
+
+def pollard_rho(n: int, x1: int, f_x: Callable) -> tuple[int | None, int]:
+    x: int = x1
+    xp: int = f_x(x) % n
+    p: int = nwd_expanded(abs(x - xp), n)[0]
+    iter: int = 0
+    while p == 1:
+        iter += 1
+        x = f_x(x) % n
+        xp = f_x(xp) % n
+        xp = f_x(xp) % n
+        p = nwd_expanded(abs(x - xp), n)[0]
+    print(f"pollard_rho({n}, {x1}) = {p} * {n // p}, iter = {iter}")
+    print(f"    x_{iter} = {x}, x_{iter *2} = {xp}")
+    if p == n:
+        return None, iter
+    else:
+        return p, iter
+
+
+def zad3():
+    number_1 = 262063
+    number_2 = 9420457
+    number_3 = 181937053
+    func = lambda x: x**2 + 1
+
+    pollard_rho(number_1, 1, func)
+    pollard_rho(number_2, 1, func)
+    pollard_rho(number_3, 1, func)
+
+
+def main():
+    zad3()
+
+
+if __name__ == "__main__":
+    main()