C++ Coding Test Course, Extended Euclidean Algorithm

Introduction

In modern development environments, various algorithms are necessary to solve numerous problems. Among them, algorithms that utilize mathematical techniques, particularly the “Extended Euclidean Algorithm,” are very useful tools. In this course, we will deeply explore the theory of the Extended Euclidean Algorithm and problems that frequently appear in coding tests. Through this content, you will systematically understand the Extended Euclidean Algorithm and develop the ability to solve practical problems using C++.

What is the Extended Euclidean Algorithm?

The Euclidean Algorithm is an algorithm for finding the greatest common divisor (GCD) of two integers a and b. The Extended Euclidean Algorithm goes a step further by finding integers x and y that express the GCD of a and b as a linear combination of a and b.

Mathematically, this can be expressed as follows:

gcd(a, b) = ax + by

Here, x and y are integers, and gcd(a, b) represents the greatest common divisor of a and b.

Necessity of the Extended Euclidean Algorithm

The Extended Euclidean Algorithm is mainly needed to solve problems such as:

• Calculating modular multiplicative inverses
• Solve linear Diophantine equations
• Applications in cryptographic algorithms and hash algorithms

Problem Definition

Now, let’s define an algorithmic problem utilizing the Extended Euclidean Algorithm. The problem is as follows:

Given two integers a and b, find their greatest common divisor along with integers x and y that represent a and b as a linear combination.

For example, for a = 30 and b = 21, the greatest common divisor is 3, and it can be expressed as a linear combination of 30 and 21 as follows:

3 = 1 * 30 + (-1) * 21

Problem Solving Process

1. Algorithm Design

The basic idea of the Extended Euclidean Algorithm is to extend the Euclidean Algorithm to calculate the GCD along with the required x and y. The algorithm can be designed in the following manner:

1. Recursively call the function to calculate x and y values along with the GCD.
2. Set up the following relationship using the two numbers a and b:
   

   gcd(a, b) = gcd(b, a % b)

3. As a base case, when b is 0, gcd(a, 0) = a and set x=1, y=0.
4. After the recursive call, update x and y to obtain the linear combination.

2. C++ Implementation

Here is the code implementing the above algorithm in C++:

#include <iostream>

using namespace std;

// Extended Euclidean Algorithm
int extended_gcd(int a, int b, int &x, int &y) {
    // Base case
    if (b == 0) {
        x = 1;
        y = 0;
        return a;
    }
    
    int x1, y1; // Values after recursive call
    int gcd = extended_gcd(b, a % b, x1, y1);
    
    // Update x and y
    x = y1;
    y = x1 - (a / b) * y1;
    
    return gcd;
}

int main() {
    int a, b;
    int x, y;
    
    cout << "Enter two numbers (a b): ";
    cin >> a >> b;
    
    int gcd = extended_gcd(a, b, x, y);
    
    cout << "Greatest Common Divisor: " << gcd << endl;
    cout << "x: " << x <<, " y: " << y << endl;
    
    return 0;
}
        

3. Code Explanation

The above code implements the Extended Euclidean Algorithm to return the greatest common divisor along with x and y for the given two numbers a and b.

• extended_gcd function: Takes two numbers and x, y as parameters and calculates the GCD along with x and y values.
• main function: Receives input from the user for two numbers, calls the extended_gcd function, and outputs the results.

Test Cases

To test the implemented algorithm, let’s use several test cases. For example:

Conclusion

In this course, we explored the theory of the Extended Euclidean Algorithm along with the problem-solving process using C++. The Extended Euclidean Algorithm is not only used for finding the greatest common divisor but is also widely applicable to solving various mathematical problems. Based on this fundamental theory, I hope you develop the ability to tackle more complex algorithmic problems.