This set of Cryptography Questions and Answers for freshers focuses on “Polynomial and Modular Arithmetic”.

1. 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}+x^{3}

b) x^{6}+x^{4}+x^{2}+x

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

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

View Answer

Explanation: Perform Modular subtraction.

2. 5/3 mod 7 =

a) 2

b) 3

c) 4

d) 5

View Answer

Explanation: 5/3 mod 7 = (5×3

^{-1}) mod 7 = (5×5) mod 7 = 4.

3. The polynomial x^{4+1} can be represented as –

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

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

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

d) None of the mentioned

View Answer

Explanation: (x

^{4}+1) = (x+1)(x

^{3}+x

^{2}+x+1).

4. -5 mod -3 =

a) 3

b) 2

c) 1

d) 5

View Answer

Explanation: -5 mod -3 = -2 mod -3 = 1 mod -3.

5. Multiply the polynomials P1 = x^{5} +x^{2}+ x) by P2 = (x^{7} + x^{4} +x^{3}+x^{2} + x) in GF(28) with irreducible polynomial (x^{8} + x^{4} + x^{3} + x + 1). The result is

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

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

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

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

View Answer

Explanation: On performing polynomial multiplication we get with respect to modulus (x

^{8}+ x

^{4}+ x

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

^{5}+ x

^{3}+x

^{2}+x+1.

6. Multiply 00100110 by 10011110 in GF(2^8) with modulus 100011011.The result is

a) 00101111

b) 00101100

c) 01110011

d) 11101111

View Answer

Explanation: On performing polynomial multiplication with respect to modulus 100011011 we get 00101111.

7.Find the inverse of (x^{7}+x+1) modulo (x^{8} + x^{4} + x^{3}+ x + 1).

a) x^{7}+x

b) x^{6}+x^{3}

c) x^{7}

d) x^{5}+1

View Answer

Explanation: Finding the inverse with respect to (x

^{8}+ x

^{4}+ x

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

^{7}as the inverse.

8. 7x = 6 mod 5. Then the value of x is

a) 2

b) 3

c) 4

d) 5

View Answer

Explanation: 7 x 3 mod 5 = 6 mod 5 = 1.

State whether the following few statement are true or false over a field.

9. The product of monic polynomials is monic.

a) True

b) False

c) Can’t Say

d) None of the mentioned

View Answer

Explanation: This is always true over a field.

10. The product of polynomials of degrees m and n has a degree m+n+1.

a) True

b) False

c) Can’t Say

d) None of the mentioned

View Answer

Explanation: The product of polynomials of degrees m and n has a degree m+n.

11. The sum of polynomials of degrees m and n has degree max[m,n].

a) True

b) False

c) Can’t Say

d) None of the mentioned

View Answer

Explanation: True when m is not equal to n; in that case the highest degree coefficient is of degree max[m,n]. But false in general when m = n, because the highest-degree coefficients might cancel (be additive inverses).

12. (7x + 2)-(x^{2} + 5) in Z_10 =

a) 9x^{2} + 7x + 7

b) 9x^{2}+ 6x + 10

c) 8x^{2} + 7x + 6

d) None of the mentioned

View Answer

Explanation:(7x + 2) – (x

^{2}+ 5) in Z_10 = 9x

^{2}+ 7x +7. We can find this via basic polynomial arithmetic in Z_10.

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