Using Any Programming Language Of Your Choice Implement The

Using Any Programming Language Of Your Choice Implement The Extende

1) Using any programming language of your choice implement the Extended Euclidean algorithm

2) Specifications: The program should take two inputs: 1) An integer a, which is the modulus; 2) A non-negative integer b that is less than a. The program should output three values: 1) gcd(a,b); 2) Integer x; and 3) Integer y, such that ax + by = gcd(a,b)

Test 1: Run your program with a = 1759 and b = (value not specified). What are your outputs? What is the modular multiplicative inverse of 550 mod 1759?

Test 2: Run your program with a = 43 and b = (value not specified). What are your outputs? What is the modular multiplicative inverse of 17 mod 43?

Test 3: Run your program with a = 400 and b = (value not specified). What are your outputs? What is the modular multiplicative inverse of 10 mod 400? Be mindful of the gcd value to answer this question.

Paper For Above instruction

The Extended Euclidean Algorithm is a fundamental method in number theory used to find the greatest common divisor (gcd) of two integers while also providing coefficients that satisfy Bézout's identity. This algorithm not only computes the gcd but also finds integers x and y such that ax + by = gcd(a, b), which has practical applications in modular arithmetic, cryptography, and solving Diophantine equations.

In implementing the Extended Euclidean Algorithm, the primary goal is to develop a program that accepts two inputs: an integer a representing the modulus and a non-negative integer b less than a. The output should include three values: the gcd of a and b, and the coefficients x and y that satisfy Bézout’s identity. These coefficients are critical in calculating modular inverses when the gcd is 1, which is a key operation in cryptosystems like RSA.

To elucidate the utility of the algorithm, consider testing with specific input values such as a = 1759, b = (unspecified). The program calculates gcd(1759, b) and the corresponding coefficients. Similarly, for a = 43 and b = (unspecified), and a = 400 with b = (unspecified), the algorithm outputs reveal the relationships between these pairs of integers. Importantly, when gcd equals 1, the inverse of b modulo a exists, which is essential in modular arithmetic applications.

For example, to find the modular multiplicative inverse of 550 mod 1759, the Extended Euclidean Algorithm determines the coefficients such that 1759x + 550y = 1. The value of y, taken modulo 1759, provides the inverse if it exists. When the gcd exceeds 1, no inverse exists because the number shares factors with the modulus, violating the conditions necessary for invertibility under modular arithmetic.

Implementing this algorithm involves iterative or recursive procedures. A common approach is to use a recursive function that returns the gcd along with the coefficients based on the division and rest steps of Euclidean’s algorithm. The recursive relations help in back-substituting to find the coefficients x and y.

Practically, the implementation can be carried out in various programming languages such as Python, Java, or C++. Python, for example, offers simplicity and readability, making it suitable for demonstrating the algorithm's logic. Using Python’s function definitions, the extended_gcd() function can be designed to return the gcd and coefficients efficiently.

In conclusion, implementing the Extended Euclidean Algorithm is an essential exercise in understanding modular arithmetic. It facilitates the computation of modular inverses, which are fundamental in cryptography, coding theory, and algorithm design. By testing with specific examples like a=1759, 43, and 400, the algorithm's robustness and practical relevance are effectively demonstrated.

References

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