#### Karatsuba Algorithm for Fast Multiplication

```/**
* @file
* @brief Implementation of the [Karatsuba algorithm for fast
* multiplication](https://en.wikipedia.org/wiki/Karatsuba_algorithm)
* @details
* Given two strings in binary notation we want to multiply them and return the
* value. Simple approach is to multiply bits one by one which will give the time
* complexity of around O(n^2). To make it more efficient we will be using
* Karatsuba algorithm to find the product which will solve the problem
* O(nlogn) of time.
* @author [Swastika Gupta](https://github.com/Swastyy)
* @author [Ameer Carlo Lubang](https://github.com/poypoyan)
*/

#include <cassert>   /// for assert
#include <cstring>   /// for string
#include <iostream>  /// for IO operations
#include <vector>    /// for std::vector

/**
* @namespace divide_and_conquer
* @brief Divide and Conquer algorithms
*/
namespace divide_and_conquer {
/**
* @namespace karatsuba_algorithm
* @brief Functions for the [Karatsuba algorithm for fast
* multiplication](https://en.wikipedia.org/wiki/Karatsuba_algorithm) implementation
*/
namespace karatsuba_algorithm {
/**
* @param first, the input string 1
* @param second, the input string 2
* @returns the sum binary string
*/
std::string add_strings(std::string first, std::string second) {
std::string result;  // to store the resulting sum bits

// make the string lengths equal
int64_t len1 = first.size();
int64_t len2 = second.size();
std::string zero = "0";
if (len1 < len2) {
for (int64_t i = 0; i < len2 - len1; i++) {
zero += first;
first = zero;
zero = "0"; // Prevents CI from failing
}
} else if (len1 > len2) {
for (int64_t i = 0; i < len1 - len2; i++) {
zero += second;
second = zero;
zero = "0"; // Prevents CI from failing
}
}

int64_t length = std::max(len1, len2);
int64_t carry = 0;
for (int64_t i = length - 1; i >= 0; i--) {
int64_t firstBit = first.at(i) - '0';
int64_t secondBit = second.at(i) - '0';

int64_t sum = (char(firstBit ^ secondBit ^ carry)) + '0';  // sum of 3 bits
result.insert(result.begin(), sum);

carry = char((firstBit & secondBit) | (secondBit & carry) |
(firstBit & carry));  // sum of 3 bits
}

if (carry) {
result.insert(result.begin(), '1');  // adding 1 incase of overflow
}
return result;
}

/**
* @brief Wrapper function for substr that considers leading zeros.
* @param str, the binary input string.
* @param x1, the substr parameter integer 1
* @param x2, the substr parameter integer 2
* @param n, is the length of the "whole" string: leading zeros + str
* @returns the "safe" substring for the algorithm *without* leading zeros
* @returns "0" if substring spans to leading zeros only
*/
std::string safe_substr(const std::string &str, int64_t x1, int64_t x2, int64_t n) {
int64_t len = str.size();

if (len >= n) {
return str.substr(x1, x2);
}

int64_t y1 = x1 - (n - len);  // index in str of first char of substring of "whole" string
int64_t y2 = (x1 + x2 - 1) - (n - len);  // index in str of last char of substring of "whole" string

if (y2 < 0) {
return "0";
} else if (y1 < 0) {
return str.substr(0, y2 + 1);
} else {
return str.substr(y1, x2);
}
}

/**
* @brief The main function implements Karatsuba's algorithm for fast
* multiplication
* @param str1 the input string 1
* @param str2 the input string 2
* @returns the product number value
*/
int64_t karatsuba_algorithm(std::string str1, std::string str2) {
int64_t len1 = str1.size();
int64_t len2 = str2.size();
int64_t n = std::max(len1, len2);

if (n == 0) {
return 0;
}
if (n == 1) {
return (str1[0] - '0') * (str2[0] - '0');
}

int64_t fh = n / 2;     // first half of string
int64_t sh = n - fh;  // second half of string

std::string Xl = divide_and_conquer::karatsuba_algorithm::safe_substr(str1, 0, fh, n);   // first half of first string
std::string Xr = divide_and_conquer::karatsuba_algorithm::safe_substr(str1, fh, sh, n);  // second half of first string

std::string Yl = divide_and_conquer::karatsuba_algorithm::safe_substr(str2, 0, fh, n);   // first half of second string
std::string Yr = divide_and_conquer::karatsuba_algorithm::safe_substr(str2, fh, sh, n);  // second half of second string

// calculating the three products of inputs of size n/2 recursively
int64_t product1 = karatsuba_algorithm(Xl, Yl);
int64_t product2 = karatsuba_algorithm(Xr, Yr);
int64_t product3 = karatsuba_algorithm(

return product1 * (1 << (2 * sh)) +
(product3 - product1 - product2) * (1 << sh) +
product2;  // combining the three products to get the final result.
}
}  // namespace karatsuba_algorithm
}  // namespace divide_and_conquer

/**
* @brief Self-test implementations
* @returns void
*/
static void test() {
// 1st test
std::string s11 = "1";     // 1
std::string s12 = "1010";  // 10
std::cout << "1st test... ";
assert(divide_and_conquer::karatsuba_algorithm::karatsuba_algorithm(
s11, s12) == 10);
std::cout << "passed" << std::endl;

// 2nd test
std::string s21 = "11";    // 3
std::string s22 = "1010";  // 10
std::cout << "2nd test... ";
assert(divide_and_conquer::karatsuba_algorithm::karatsuba_algorithm(
s21, s22) == 30);
std::cout << "passed" << std::endl;

// 3rd test
std::string s31 = "110";   // 6
std::string s32 = "1010";  // 10
std::cout << "3rd test... ";
assert(divide_and_conquer::karatsuba_algorithm::karatsuba_algorithm(
s31, s32) == 60);
std::cout << "passed" << std::endl;
}

/**
* @brief Main function
* @returns 0 on exit
*/
int main() {
test();  // run self-test implementations
return 0;
}
```