a) Considering the bytes 10001001 and 10101010 as elements of the fiel

Question

a) Considering the bytes 10001001 and 10101010 as elements of the field $F_2[X]/\left \langle g(X) \right \rangle,$ where $g(X)$ is the polynomial $X^{8}+X^{4}+X^{3}+X+1,$ find their product and quotient.

b) Find a recurrence that generates the sequence 110110110110110.

10 Mar 2024

Answer :

Word Count : 657

### Part (a): Finding the Product and Quotient in the Field $ F_2[X]/\langle g(X) \rangle $ We are working in the field $ F_2[X]/\langle g(X) \rangle $, where the elements are polynomials with coefficients in $ F_2 $, i.e., coefficients are 0 or 1. We are given the bytes $ 10001001 $ and $ 10101010 $, and we need to find their product and quotient modulo the polynomial $ g(X) = X^8 + X^4 + X^3 + X + 1 $. 1. Convert the bytes to polynomials: - $ 10001001_2 $ corresponds to the polynomial $ X^7 + X^3 + 1 $, which is written as: \[ p_1(X) = ________ ____ _________ ___ __________.
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