Add comprehensive doctests to find_mod_inverse function (#13217)

- Added detailed docstring explaining modular multiplicative inverse
- Included 10 valid test cases with verification calculations
- Added 3 error cases testing ValueError for non-coprime inputs
- Added Wikipedia reference for educational value
- All doctests pass locally (python -m doctest -v)
- Passes ruff, mypy, and pre-commit hooks

Contributes to #9943

Co-authored-by: Christian Clauss <cclauss@me.com>

Author: pranlawate

ciphers/cryptomath_module.py

@@ -12,29 +12,64 @@ def find_mod_inverse(a: int, m: int) -> int:
     """
     Find the modular multiplicative inverse of a modulo m.
 
-    The modular multiplicative inverse of a modulo m is an integer x
-    such that (a * x) % m == 1. This only exists when gcd(a, m) == 1.
+    The modular multiplicative inverse of a modulo m is an integer x such that:
+    (a * x) % m = 1
+
+    This function uses the Extended Euclidean Algorithm to find the inverse.
+    An inverse exists if and only if a and m are coprime (gcd(a, m) = 1).
 
     Args:
-        a: The number to find the inverse of
+        a: The integer to find the inverse of
         m: The modulus
 
     Returns:
         The modular multiplicative inverse of a modulo m
 
     Raises:
-        ValueError: If gcd(a, m) != 1 (inverse doesn't exist)
+        ValueError: If gcd(a, m) != 1 (inverse does not exist)
+
+    Reference:
+        https://en.wikipedia.org/wiki/Modular_multiplicative_inverse
 
+    Examples:
+    >>> find_mod_inverse(3, 7)
+    5
+    >>> (3 * 5) % 7  # Verify: 3 * 5 ≡ 1 (mod 7)
+    1
+    >>> find_mod_inverse(3, 10)
+    7
+    >>> (3 * 7) % 10  # Verify: 3 * 7 ≡ 1 (mod 10)
+    1
+    >>> find_mod_inverse(4, 11)
+    3
+    >>> (4 * 3) % 11  # Verify: 4 * 3 ≡ 1 (mod 11)
+    1
     >>> find_mod_inverse(7, 26)
     15
-    >>> find_mod_inverse(3, 11)
-    4
-    >>> find_mod_inverse(5, 17)
-    7
+    >>> (7 * 15) % 26  # Verify: 7 * 15 ≡ 1 (mod 26)
+    1
     >>> find_mod_inverse(1, 5)
     1
     >>> find_mod_inverse(2, 7)
     4
+    >>> find_mod_inverse(3, 11)
+    4
+    >>> find_mod_inverse(5, 11)
+    9
+    >>> find_mod_inverse(5, 17)
+    7
+    >>> find_mod_inverse(2, 4)
+    Traceback (most recent call last):
+        ...
+    ValueError: mod inverse of 2 and 4 does not exist
+    >>> find_mod_inverse(6, 9)
+    Traceback (most recent call last):
+        ...
+    ValueError: mod inverse of 6 and 9 does not exist
+    >>> find_mod_inverse(10, 20)
+    Traceback (most recent call last):
+        ...
+    ValueError: mod inverse of 10 and 20 does not exist
     """
     if gcd_by_iterative(a, m) != 1:
         msg = f"mod inverse of {a!r} and {m!r} does not exist"