Cryptography Problem:

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Problem 1 - Binary polynomial arithmetic, 20 marks In this problem, we consider different types of modular arithmetic on polynomial with coef- ficient in GF(2), the set {0, 1) with arithmetic modulo 2. (a) Recall that a polynomial is irreducible if it does not have a factorization into polynomials of smaller positive degree, and reducible otherwise. (2 marks) List all the polynomials of degree 3 with coefficients in GF(2) lexicographical order. i. 0,1 in ii. (3 marks) List all the reducible polynomials of degree 3 with coefficients in GF (2) For each of these polynomials, provide a proof of reducibility (3 marks) List all the irreducible polynomials of degree 3 with coefficients in GF(2) For each of these polynomials, provide a proof of irreducibility. Problem 1 - Binary polynomial arithmetic, 20 marks In this problem, we consider different types of modular arithmetic on polynomial with coef- ficient in GF(2), the set {0, 1) with arithmetic modulo 2. (a) Recall that a polynomial is irreducible if it does not have a factorization into polynomials of smaller positive degree, and reducible otherwise. (2 marks) List all the polynomials of degree 3 with coefficients in GF(2) lexicographical order. i. 0,1 in ii. (3 marks) List all the reducible polynomials of degree 3 with coefficients in GF (2) For each of these polynomials, provide a proof of reducibility (3 marks) List all the irreducible polynomials of degree 3 with coefficients in GF(2) For each of these polynomials, provide a proof of irreducibility