Add Extended Euclidean Algorithm implementation #[HACKTOBERFEST 2025]

[piyushkumar0707]

🚀 Overview

This PR implements the Extended Euclidean Algorithm - an advanced number theory algorithm that finds GCD and Bézout coefficients, with applications in cryptography and modular arithmetic.

✨ Features

• ✅ Recursive & iterative versions - Two implementation approaches
• ✅ Bézout coefficient calculation - Find x, y such that ax + by = gcd(a,b)
• ✅ Modular multiplicative inverse - Essential for cryptography
• ✅ Diophantine equation solver - Solve ax + by = c
• ✅ Range-based solutions - Find all solutions in given bounds
• ✅ Practical applications - Real-world problem examples

🎯 Why This Matters

Extended Euclidean Algorithm is crucial for:

• RSA Cryptography - Computing modular inverses for key generation
• Modular arithmetic - Solving congruences and inverse calculations
• Number theory - Fundamental tool for mathematical proofs
• Competitive programming - Advanced mathematical contests
• Computer algebra - Symbolic computation systems
• Error correction codes - Coding theory applications

📚 Implementation Details

• Time Complexity: O(log(min(a, b))) - same as regular GCD
• Space Complexity: O(log(min(a, b))) recursive, O(1) iterative
• Algorithm: Extended version of Euclidean GCD algorithm
• Output: GCD plus coefficients satisfying Bézout's identity

🔧 Advanced Functions

• extended_gcd_recursive() - Classic recursive approach
• extended_gcd_iterative() - Memory-efficient iterative version
• modular_inverse() - Find multiplicative inverse mod m
• solve_diophantine() - Solve linear Diophantine equations
• find_diophantine_solutions_in_range() - Bounded solution finder

🧮 Mathematical Applications

Bézout's Identity: For any integers a, b, there exist integers x, y such that:

[Copilot]

Pull Request Overview

This PR implements the Extended Euclidean Algorithm in R, which computes the GCD of two integers along with Bézout coefficients that satisfy the identity ax + by = gcd(a,b). The implementation provides both recursive and iterative approaches with practical applications in cryptography and number theory.

• Implements recursive and iterative Extended Euclidean Algorithm functions
• Adds modular multiplicative inverse calculation for cryptographic applications
• Includes Diophantine equation solver with range-based solution finding
• Provides comprehensive examples and test cases demonstrating real-world applications

Reviewed Changes

Copilot reviewed 2 out of 2 changed files in this pull request and generated 2 comments.

File	Description
mathematics/extended_euclidean_algorithm.r	Complete implementation of Extended Euclidean Algorithm with multiple variants and practical applications
DIRECTORY.md	Added entry for the new Extended Euclidean Algorithm file in the mathematics section

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[siriak]

Looks good, thanks!