# Symmetric Ciphers Questions and Answers – Polynomial and Modular Arithmetic – II

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

1. If f(x)=x7+x5+x4+x3+x+1 and g(x)=x3+x+1, find f(x) x g(x).
a) x12+x5+x3+x2+x+1
b) x10+x4+1
c) x10+x4+x+1
d) x7+x5+x+1

Explanation: Perform Modular Multiplication.

2. If f(x)=x7+x5+x4+x3+x+1 and g(x)=x3+x+1, find the quotient of f(x) / g(x).
a) x4+x3+1
b) x4+1
c) x5+x3+x+1
d) x3+x2

Explanation: Perform Modular Division.

3. Primitive Polynomial is also called a ____
i) Perfect Polynomial
ii) Prime Polynomial
iii) Irreducible Polynomial
iv) Imperfect Polynomial

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a) ii) and iii)
b) only iii)
c) iv) and ii)
d) None

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

4. Which of the following are irreducible polynomials?
i) X4+X3
ii) 1
iii) X2+1
iv) X4+X+1

a) i) and ii)
b) only iv)
c) ii) iii) and iv)
d) All of the options

Explanation: All of the mentioned are irreducible polynomials.

5. The polynomial f(x)=x3+x+1 is a reducible.
a) True
b) False

Explanation: f(x)=x3+x+1 is irreducible.

6. Find the HCF/GCD of x6+x5+x4+x3+x2+x+1 and x4+x2+x+1.
a) x4+x3+x2+1
b) x3+x2+1
c) x2+1
d) x3+x2+1

Explanation: Use Euclidean Algorithm and find the GCD. GCD = x3+x2+1.

7. On multiplying (x5 + x2 + x) by (x7 + x4 + x3 + x2 + x) in GF(28) with irreducible polynomial (x8 + x4 + x3 + x + 1) we get
a) x12+x7+x2
b) x5+x3+x3
c) x5+x3+x2+x
d) x5+x3+x2+x+1

Explanation: Multiplication gives us (x12 + x7 + x2) mod (x8 + x4 + x3 + x + 1).
Reducing this via modular division gives us, (x5+x3+x2+x+1)

8. On multiplying (x6+x4+x2+x+1) by (x7+x+1) in GF(28) with irreducible polynomial (x8 + x4 + x3 + x + 1) we get
a) x7+x6+ x3+x2+1
b) x6+x5+ x2+x+1
c) x7+x6+1
d) x7+x6+x+1

Explanation: Multiply and Obtain the modulus we get the polynomial product as x7+x6+1.

9. Find the inverse of (x2 + 1) modulo (x4 + x + 1).
a) x4+ x3+x+1
b) x3+x+1
c) x3+ x2+x
d) x2+x

Explanation:

10. Find the inverse of (x5) modulo (x8+x4 +x3+ x + 1).
a) x5+ x4+ x3+x+1
b) x5+ x4+ x3
c) x5+ x4+ x3+1
d) x4+ x3+x+1

Explanation: Finding the inverse with respect to (x8+x4 +x3+ x + 1) we get x5+ x4+ x3+1 as the inverse.

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