diff --git a/math/modular_division.cpp b/math/modular_division.cpp new file mode 100644 index 000000000..c243c8424 --- /dev/null +++ b/math/modular_division.cpp @@ -0,0 +1,109 @@ +/** + * @file + * @brief Division of two numbers under modulo + * + * @details To calculate division of two numbers under modulo p + * Modulo operator is not distributive under division, therefore + * we first have to calculate the inverse of divisor using + * [Fermat's little theorem](https://en.wikipedia.org/wiki/Fermat%27s_little_theorem) + * Now, we can multiply the dividend with the inverse of divisor + * and modulo is distributive over multiplication operation. + * Let, + * We have 3 numbers a, b, p + * To compute (a/b)%p + * (a/b)%p ≡ (a*(inverse(b)))%p ≡ ((a%p)*inverse(b)%p)%p + * NOTE: For the existence of inverse of 'b', 'b' and 'p' must be coprime + * For simplicity we take p as prime + * Time Complexity: O(log(b)) + * + * Example: ( 24 / 3 ) % 5 => 8 % 5 = 3 --- (i) + Now the inverse of 3 is 2 + (24 * 2) % 5 = (24 % 5) * (2 % 5) = (4 * 2) % 5 = 3 --- (ii) + (i) and (ii) are equal hence the answer is correct. + + * + * @see modular_inverse_fermat_little_theorem.cpp, modular_exponentiation.cpp + * + * @author [Shubham Yadav](https://github.com/shubhamamsa) + */ + +#include +#include + +/** + * @namespace math + */ +namespace math { + /** + * @brief This function calculates a raised to exponent b under modulo c using + * modular exponentiation. + * @param a integer base + * @param b unsigned integer exponent + * @param c integer modulo + * @return a raised to power b modulo c + */ + int power(int a, int b, int c) { + int ans = 1; /// Initialize the answer to be returned + a = a % c; /// Update a if it is more than or equal to c + if (a == 0) { + return 0; /// In case a is divisible by c; + } + while (b > 0) { + /// If b is odd, multiply a with answer + if (b & 1) { + ans = ((ans % c) * (a % c)) % c; + } + /// b must be even now + b = b >> 1; /// b = b/2 + a = ((a % c) * (a % c)) % c; + } + return ans; + } + + /** + * @brief This function calculates modular division + * @param a integer dividend + * @param b integer divisor + * @param p integer modulo + * @return a/b modulo c + */ + int mod_division(int a, int b, int p) { + int inverse = power(b, p-2, p)%p; /// Calculate the inverse of b + int result = ((a%p)*(inverse%p))%p; /// Calculate the final result + return result; + } +} + +/** + * Function for testing power function. + * test cases and assert statement. + * @returns `void` + */ +static void test() { + int test_case_1 = math::mod_division(8, 2, 2); + assert(test_case_1 == 0); + std::cout << "Test 1 Passed!" << std::endl; + int test_case_2 = math::mod_division(15, 3, 7); + assert(test_case_2 == 5); + std::cout << "Test 2 Passed!" << std::endl; + int test_case_3 = math::mod_division(10, 5, 2); + assert(test_case_3 == 0); + std::cout << "Test 3 Passed!" << std::endl; + int test_case_4 = math::mod_division(81, 3, 5); + assert(test_case_4 == 2); + std::cout << "Test 4 Passed!" << std::endl; + int test_case_5 = math::mod_division(12848, 73, 29); + assert(test_case_5 == 2); + std::cout << "Test 5 Passed!" << std::endl; +} + +/** + * @brief Main function + * @param argc commandline argument count (ignored) + * @param argv commandline array of arguments (ignored) + * @returns 0 on exit + */ +int main(int argc, char *argv[]) { + test(); // execute the tests + return 0; +}