This set of Advanced Cryptography Questions and Answers focuses on “Polynomial and Modular Arithmetic – II”.

1. If f(x)=x^{7}+x^{5}+x^{4}+x^{3}+x+1 and g(x)=x^{3}+x+1, find f(x) x g(x).

a) x^{12}+x^{5}+x^{3}+x^{2}+x+1

b) x^{10}+x^{4}+1

c) x^{10}+x^{4}+x+1

d) x^{7}+x^{5}+x+1

View Answer

Explanation: Perform Modular Multiplication.

2. If f(x)=x^{7}+x^{5}+x^{4}+x^{3}+x+1 and g(x)=x^{3}+x+1, find the quotient of f(x) / g(x).

a) x^{4}+x^{3}+1

b) x^{4}+1

c) x^{5}+x^{3}+x+1

d) x^{3}+x^{2}

View Answer

Explanation: Perform Modular Division.

3. Primitive Polynomial is also called a ____

i) Perfect Polynomial

ii) Prime Polynomial

iii) Irreducible Polynomial

iv) Imperfect Polynomial

a) ii) and iii)

b) only iii)

c) iv) and ii)

d) None

View Answer

Explanation: Irreducible polynomial is also called a prime polynomial or primitive polynomial.

4. Which of the following are irreducible polynomials?

i) X^{4}+X^{3}

ii) 1

iii) X^{2}+1

iv) X^{4}+X+1

a) i) and ii)

b) only iv)

c) ii) iii) and iv)

d) All of the options

View Answer

Explanation: All of the mentioned are irreducible polynomials.

5. The polynomial f(x)=x^{3}+x+1 is a reducible.

a) True

b) False

View Answer

Explanation: f(x)=x

^{3}+x+1 is irreducible.

6. Find the HCF/GCD of x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1 and x^{4}+x^{2}+x+1.

a) x^{4}+x^{3}+x^{2}+1

b) x^{3}+x^{2}+1

c) x^{2}+1

d) x^{3}+x^{2}+1

View Answer

Explanation: Use Euclidean Algorithm and find the GCD. GCD = x

^{3}+x

^{2}+1.

7. On multiplying (x^{5} + x^{2} + x) by (x^{7} + x^{4} + x^{3} + x^{2} + x) in GF(28) with irreducible polynomial (x^{8} + x^{4} + x^{3} + x + 1) we get

a) x^{12}+x^{7}+x^{2}

b) x^{5}+x^{3}+x^{3}

c) x^{5}+x^{3}+x^{2}+x

d) x^{5}+x^{3}+x^{2}+x+1

View Answer

Explanation: Multiplication gives us (x

^{12}+ x

^{7}+ x

^{2}) mod (x

^{8}+ x

^{4}+ x

^{3}+ x + 1).

Reducing this via modular division gives us, (x

^{5}+x

^{3}+x

^{2}+x+1)

8. On multiplying (x^{6}+x^{4}+x^{2}+x+1) by (x^{7}+x+1) in GF(2^{8}) with irreducible polynomial (x^{8} + x^{4} + x^{3} + x + 1) we get

a) x^{7}+x^{6}+ x^{3}+x^{2}+1

b) x^{6}+x^{5}+ x^{2}+x+1

c) x^{7}+x^{6}+1

d) x^{7}+x^{6}+x+1

View Answer

Explanation: Multiply and Obtain the modulus we get the polynomial product as x

^{7}+x

^{6}+1.

9. Find the inverse of (x^{2} + 1) modulo (x^{4} + x + 1).

a) x^{4}+ x^{3}+x+1

b) x^{3}+x+1

c) x^{3}+ x^{2}+x

d) x^{2}+x

View Answer

10. Find the inverse of (x^{5}) modulo (x^{8}+x^{4} +x^{3}+ x + 1).

a) x^{5}+ x^{4}+ x^{3}+x+1

b) x^{5}+ x^{4}+ x^{3}

c) x^{5}+ x^{4}+ x^{3}+1

d) x^{4}+ x^{3}+x+1

View Answer

Explanation: Finding the inverse with respect to (x

^{8}+x

^{4}+x

^{3}+ x + 1) we get x

^{5}+ x

^{4}+ x

^{3}+1 as the inverse.

**Sanfoundry Global Education & Learning Series – Cryptography and Network Security.**

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