Question

compute the gcd of two polynomials a = x¹² + x⁹ + x⁸ + x⁶ + x⁴ + x+ 1 b = x¹¹ + x¹⁰ + x⁶ +x⁵ + x⁴ + x³ + 1 over GF(2) ring and and verify using mathematica

Answer 1

The greatest common divisor of two polynomials over GF(2) ring can be calculated using the Euclidean algorithm for polynomials. The algorithm involves a series of polynomial long division steps until a remainder of 0 is reached. Mathematica's GCD function can be used to verify the result.

Step-by-step explanation:

To compute the greatest common divisor (gcd) of two polynomials a = x¹² + x¹ + x⁸ + x⁶ + x⁴ + x + 1 and b = x¹¹ + x¹⁰ + x⁶ + x⁵ + x⁴ + x³ + 1 over the GF(2) ring, we can use the Euclidean algorithm for polynomials. This is a similar process to the Euclidean algorithm used for integers, but instead, it applies to polynomial division. In the ring GF(2), coefficients are reduced modulo 2, which means that any coefficient with an even power will be reduced to 0, and coefficients with odd powers will be reduced to 1. Here is a step-by-step process you could follow, although the actual computations can get quite lengthy and are usually done using a software like Mathematica:

1. Perform polynomial long division with a as the dividend and b as the divisor to find the remainder r1.
2. Set a to b, and b to r1, and repeat the division process to find the next remainder r2.
3. Continue this process of replacing a with b, and b with the most recent non-zero remainder until you reach a remainder of 0.
4. The last non-zero remainder before obtaining 0 is the gcd of the two polynomials.

To check the calculation in Mathematica, you can use the GCD function, like this:

GCD[x^12 + x^9 + x^8 + x^6 + x^4 + x + 1, x^11 + x^10 + x^6 + x^5 + x^4 + x^3 + 1]