This set of tricky Cryptography questions and answers focuses on “Polynomial and Modular Arithmetic – IV”.

1. (6x^{2} + x + 3)x(5x^{2} + 2) in Z_10 =

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

b) 5x^{3} + 7x^{2} + 2x + 6

c) x^{3} + 7x^{2} + 2x + 6

d) None of the mentioned

View Answer

Explanation:(6x

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

^{2}+ 2) in Z_10 = 5x

^{3}+ 7x

^{2}+ 2x + 6. We can find this via basic polynomial arithmetic in Z_10.

2. Is x^{3} + 1 reducible over GF(2)

a) Yes

b) No

c) Can’t Say

d) Insufficient Data

View Answer

Explanation: Reducible: (x + 1)(x

^{2}+ x + 1).

3. Is x^{3} + x^{2} + 1 reducible over GF(2)

a) Yes

b) No

c) Can’t Say

d) Insufficient Data

View Answer

Explanation: Irreducible. On factoring this polynomial, one factor is x and the other is (x + 1), which gives us the roots x = 0 or x = 1 respectively. By substitution of 0 and 1 into this polynomial, it clearly has no roots.

4. Is x^{4} + 1 reducible over GF(2)

a) Yes

b) No

c) Can’t Say

d) Insufficient Data

View Answer

Explanation: Reducible: (x + 1)

^{4}.

5. The result of (x2 ⊗ P), and the result of (x ⊗ (x ⊗ P)) are the same, where P is a polynomial.

a) True

b) False

View Answer

Explanation: The statement is true and this is the logic used behind the multiplication of polynomials on a computer. This reduces computation time.

6. The GCD of x^{3}+ x + 1 and x^{2} + x + 1 over GF(2) is

a) 1

b) x + 1

c) x^{2}

d) x^{2} + 1

View Answer

Explanation: The GCD of x

^{3}+ x + 1 and x

^{2}+ x + 1 over GF(2) is 1.

7. The GCD of x^{5}+x^{4}+x^{3} – x^{2} – x + 1 and x^{3} + x^{2} + x + 1 over GF(3) is

a) 1

b) x

c) x + 1

d) x^{2} + 1

View Answer

Explanation: The GCD of x

^{5}+x

^{4}+x

^{3}– x

^{2}– x + 1 and x

^{3}+ x

^{2}+ x + 1 over GF(3) is x + 1.

8. The GCD of x^{3} – x + 1 and x^{2} + 1 over GF(3) is

a) 1

b) x

c) x + 1

d) x^{2} + 1

View Answer

Explanation: The GCD of x

^{3}– x + 1 and x

^{2}+ 1 over GF(3) is 1.

9. Find the 8-bit word related to the polynomial x6 + x + 1

a) 01000011

b) 01000110

c) 10100110

d) 11001010

View Answer

Explanation: The respective 8-bit word is 01000011.

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

a) x^{7}+x^{5}+x^{4}

b) x^{7}+x^{5}+x^{4}+x^{3}+x

c) x^{4}+x^{2}+x+1

d) x^{6}+x^{4}+x^{2}+x+1

View Answer

Explanation: Perform Modular addition.

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

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