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functions.py

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+"""This module implements the Miller-Rabin primality test and the Extended Euclidean Algorithm."""
+import random
+
+
+# Python implementation of the Miller-Rabin test according to the pseudocode at
+# https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test#Pseudocode:~:text=prime%E2%80%9D%20otherwise-,let%20s%20%3E%200%20and%20d%20odd%20%3E%200%20such%20that%20n%20%E2%88%92%201,1%20then%0A%20%20%20%20%20%20%20%20return%20%E2%80%9Ccomposite%E2%80%9D%0Areturn%20%E2%80%9Cprobably%20prime%E2%80%9D,-Complexity%5Bedit
+# n is the number to be tested; k is the number of testing rounds
+def primality_test(n: int, k: int = 100) -> bool:
+  """Miller-Rabin primality test"""
+  if n <= 1:
+    return False
+  if n in (2, 3):
+    return True
+  if n % 2 == 0:
+    return False
+  s = 0
+  d = n - 1
+  while d % 2 == 0:
+    d //= 2
+    s += 1
+  for _ in range(k):
+    a = random.randint(2, n - 2)
+    x = pow(a, d, n)
+    if x in (1, n - 1):
+      continue
+    for _ in range(s - 1):
+      x = pow(x, 2, n)
+      if x == n - 1:
+        break
+    else:
+      return False
+  return True
+
+
+# Python implementation of the Extended Euclidean Algorithm according to the pseudoode at
+# https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm#Pseudocode:~:text=function%20extended_gcd(a,gcd%3A%22%2C%20(t%2C%20s)
+# Returns a tuple (gcd, x, y) such that a * x + b * y = gcd, with gcd being the greatest common divisor of a and b.
+def extended_gcd(a: int, b: int) -> tuple[int, int, int]:
+  """Extended Euclidean Algorithm"""
+  old_r, r = a, b
+  old_s, s = 1, 0
+  old_t, t = 0, 1
+
+  while r != 0:
+    quotient = old_r // r
+    old_r, r = r, old_r - quotient * r
+    old_s, s = s, old_s - quotient * s
+    old_t, t = t, old_t - quotient * t
+
+  return (abs(old_r), old_s, old_t)