docs, test: fit modular inverse fermat little theorem to contributing guidelines (#2779)

* Update modular_inverse_fermat_little_theorem.cpp

* Update modular_inverse_fermat_little_theorem.cpp

* Update modular_inverse_fermat_little_theorem.cpp

* Update modular_inverse_fermat_little_theorem.cpp

* Update math/modular_inverse_fermat_little_theorem.cpp

Co-authored-by: realstealthninja <68815218+realstealthninja@users.noreply.github.com>

* Update math/modular_inverse_fermat_little_theorem.cpp

Co-authored-by: realstealthninja <68815218+realstealthninja@users.noreply.github.com>

* Update modular_inverse_fermat_little_theorem.cpp

Add time complexity in comment

* Update modular_inverse_fermat_little_theorem.cpp

* Update modular_inverse_fermat_little_theorem.cpp

---------

Co-authored-by: realstealthninja <68815218+realstealthninja@users.noreply.github.com>

math/modular_inverse_fermat_little_theorem.cpp

@@ -30,8 +30,8 @@
  * a^{m-2} &≡&  a^{-1} \;\text{mod}\; m
  * \f}
  *
- * We will find the exponent using binary exponentiation. Such that the
- * algorithm works in \f$O(\log m)\f$ time.
+ * We will find the exponent using binary exponentiation such that the
+ * algorithm works in \f$O(\log n)\f$ time.
  *
  * Examples: -
  * * a = 3 and m = 7
@@ -43,56 +43,98 @@
  * (as \f$a\times a^{-1} = 1\f$)
  */
 
-#include <iostream>
-#include <vector>
+#include <cassert>  /// for assert
+#include <cstdint>  /// for std::int64_t
+#include <iostream> /// for IO implementations
 
-/** Recursive function to calculate exponent in \f$O(\log n)\f$ using binary
- * exponent.
+/**
+ * @namespace math
+ * @brief Maths algorithms.
  */
-int64_t binExpo(int64_t a, int64_t b, int64_t m) {
-    a %= m;
-    int64_t res = 1;
-    while (b > 0) {
-        if (b % 2) {
-            res = res * a % m;
-        }
-        a = a * a % m;
-        // Dividing b by 2 is similar to right shift.
-        b >>= 1;
+namespace math {
+/**
+ * @namespace modular_inverse_fermat
+ * @brief Calculate modular inverse using Fermat's Little Theorem.
+ */
+namespace modular_inverse_fermat {
+/**
+ * @brief Calculate exponent with modulo using binary exponentiation in \f$O(\log b)\f$ time.
+ * @param a The base
+ * @param b The exponent
+ * @param m The modulo
+ * @return The result of \f$a^{b} % m\f$
+ */
+std::int64_t binExpo(std::int64_t a, std::int64_t b, std::int64_t m) {
+  a %= m;
+  std::int64_t res = 1;
+  while (b > 0) {
+    if (b % 2 != 0) {
+      res = res * a % m;
     }
-    return res;
+    a = a * a % m;
+    // Dividing b by 2 is similar to right shift by 1 bit
+    b >>= 1;
+  }
+  return res;
 }
-
-/** Prime check in \f$O(\sqrt{m})\f$ time.
+/**
+ * @brief Check if an integer is a prime number in \f$O(\sqrt{m})\f$ time.
+ * @param m An intger to check for primality
+ * @return true if the number is prime
+ * @return false if the number is not prime
  */
-bool isPrime(int64_t m) {
-    if (m <= 1) {
-        return false;
-    } else {
-        for (int64_t i = 2; i * i <= m; i++) {
-            if (m % i == 0) {
-                return false;
-            }
-        }
+bool isPrime(std::int64_t m) {
+  if (m <= 1) {
+    return false;
+  } 
+  for (std::int64_t i = 2; i * i <= m; i++) {
+    if (m % i == 0) {
+      return false;
     }
-    return true;
+  }
+  return true;
+}
+/**
+ * @brief calculates the modular inverse.
+ * @param a Integer value for the base
+ * @param m Integer value for modulo
+ * @return The result that is the modular inverse of a modulo m
+ */
+std::int64_t modular_inverse(std::int64_t a, std::int64_t m) {
+  while (a < 0) {
+    a += m;
+  }
+
+  // Check for invalid cases
+  if (!isPrime(m) || a == 0) {
+    return -1; // Invalid input
+  }
+
+  return binExpo(a, m - 2, m);  // Fermat's Little Theorem
+}
+} // namespace modular_inverse_fermat
+} // namespace math
+
+/**
+ * @brief Self-test implementation
+ * @return void
+ */
+static void test() {
+  assert(math::modular_inverse_fermat::modular_inverse(0, 97) == -1);
+  assert(math::modular_inverse_fermat::modular_inverse(15, -2) == -1);
+  assert(math::modular_inverse_fermat::modular_inverse(3, 10) == -1);
+  assert(math::modular_inverse_fermat::modular_inverse(3, 7) == 5);
+  assert(math::modular_inverse_fermat::modular_inverse(1, 101) == 1);
+  assert(math::modular_inverse_fermat::modular_inverse(-1337, 285179) == 165519);
+  assert(math::modular_inverse_fermat::modular_inverse(123456789, 998244353) == 25170271);
+  assert(math::modular_inverse_fermat::modular_inverse(-9876543210, 1000000007) == 784794281);
 }
 
 /**
- * Main function
+ * @brief Main function
+ * @return 0 on exit
  */
 int main() {
-    int64_t a, m;
-    // Take input of  a and m.
-    std::cout << "Computing ((a^(-1))%(m)) using Fermat's Little Theorem";
-    std::cout << std::endl << std::endl;
-    std::cout << "Give input 'a' and 'm' space separated : ";
-    std::cin >> a >> m;
-    if (isPrime(m)) {
-        std::cout << "The modular inverse of a with mod m is (a^(m-2)) : ";
-        std::cout << binExpo(a, m - 2, m) << std::endl;
-    } else {
-        std::cout << "m must be a prime number.";
-        std::cout << std::endl;
-    }
+  test();  // run self-test implementation
+  return 0;
 }