Python Coding Test Course, Extended Euclidean Algorithm

In this lecture, we will take a closer look at the extended Euclidean algorithm and examine the process of solving related algorithm problems. The extended Euclidean algorithm is an extension of the algorithm for calculating the greatest common divisor (GCD) of two integers in number theory, allowing us to find linear combinations of two integers. This can help solve various problems.

1. Basic Concept of the Extended Euclidean Algorithm

The Euclidean algorithm is a method for finding the greatest common divisor (GCD) of two integers a and b. The basic process is as follows:

1. Compare the two integers a and b, and if b is not 0, calculate the remainder of a divided by b.
2. Update the values of a and b to b and a % b, respectively.
3. When b becomes 0, the current value of a is the GCD.

The extended Euclidean algorithm goes a step further to find x and y such that the linear combination of the two integers a and b satisfies the following form:

ax + by = gcd(a, b)

2. Algorithm of the Extended Euclidean Algorithm

The algorithm of the extended Euclidean method can be implemented recursively. The basic idea is as follows:

  • Recursively perform the Euclidean algorithm to find the GCD for the two integers a and b.
  • Once the GCD is reached through recursive deduction, calculate the values of x and y recursively in order.

3. Python Implementation

Below is the Python code implementing the extended Euclidean algorithm:

def extended_gcd(a, b):
    if b == 0:
        return a, 1, 0  # gcd, x, y
    gcd, x1, y1 = extended_gcd(b, a % b)
    x = y1
    y = x1 - (a // b) * y1
    return gcd, x, y

# Example execution
a = 30
b = 20
gcd, x, y = extended_gcd(a, b)
print(f"GCD: {gcd}, x: {x}, y: {y}")  # GCD: 10, x: -1, y: 2

4. Algorithm Problem

Now, let’s solve a problem using the extended Euclidean algorithm above.

Problem: For the given two integers a and b, find the GCD along with x and y.

For example, if a = 56 and b = 98:

  • GCD(56, 98) = 14
  • You need to find a solution for 56x + 98y = 14.

5. Problem-Solving Process

To solve the problem above, we will see how to use the extended Euclidean algorithm to find x and y. First, we will take the given two integers a and b as input and call the corresponding function to check the results.

# Given two integers
a = 56
b = 98

# Execute the extended Euclidean algorithm
gcd, x, y = extended_gcd(a, b)
print(f"GCD: {gcd}, x: {x}, y: {y}")

When you execute this code, you will receive the values of GCD alongside x and y for the given integers 56 and 98. An example output could be as follows:

GCD: 14, x: 1, y: -1

As a result, the GCD of 56 and 98 is 14, and it can be expressed as a linear combination with 1 times 56 and -1 times 98.

6. Applications of the Extended Euclidean Algorithm

The extended Euclidean algorithm can be applied in various fields beyond just finding the GCD.

  • Password Decryption: It is used in public key cryptosystems such as RSA encryption.
  • Calculation of Least Common Multiple: The least common multiple (LCM) of two integers can be easily calculated using the GCD.
  • Diophantine Equations: It is useful in solving equations with integer solutions.

7. Conclusion

The extended Euclidean algorithm is one of the very useful algorithms for Python coding tests, and it can help solve various problems. I hope this lecture has helped you understand how to use and apply the extended Euclidean algorithm.

I wish you success in preparing for coding tests while solving various algorithm problems in the future. Thank you!