# Symmetric Ciphers Questions and Answers – Polynomial and Modular Arithmetic- III

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

1. If f(x)=x7+x5+x4+x3+x+1 and g(x)=x3+x+1, find f(x) – g(x).
a) x7+x5+x4+x3
b) x6+x4+x2+x
c) x4+x2+x+1
d) x7+x5+x4

Explanation: Perform Modular subtraction.

2. 5/3 mod 7 =
a) 2
b) 3
c) 4
d) 5

Explanation: 5/3 mod 7 = (5×3-1) mod 7 = (5×5) mod 7 = 4.

3. The polynomial x4+1 can be represented as –
a) (x+1)(x3+x2+1)
b) (x+1)(x3+x2+x)
c) (x)(x2+x+1)
d) None of the mentioned

Explanation: (x4+1) = (x+1)(x3+x2+x+1).
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4. -5 mod -3 =
a) 3
b) 2
c) 1
d) 5

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

5. Multiply the polynomials P1 = x5 +x2+ x) by P2 = (x7 + x4 +x3+x2 + x) in GF(28) with irreducible polynomial (x8 + x4 + x3 + x + 1). The result is
a) x4+ x3+ x+1
b) x5+ x3+x2+x+1
c) x5+ x4+ x3+x+1
d) x5+ x3+x2+x

Explanation: On performing polynomial multiplication we get with respect to modulus (x8 + x4 + x3 + x + 1) we get x5+ x3+x2+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

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

7.Find the inverse of (x7+x+1) modulo (x8 + x4 + x3+ x + 1).
a) x7+x
b) x6+x3
c) x7
d) x5+1

Explanation: Finding the inverse with respect to (x8 + x4 + x3+ x + 1) we get x7 as the inverse.

8. 7x = 6 mod 5. Then the value of x is
a) 2
b) 3
c) 4
d) 5

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

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

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

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)-(x2 + 5) in Z_10 =
a) 9x2 + 7x + 7
b) 9x2+ 6x + 10
c) 8x2 + 7x + 6
d) None of the mentioned

Explanation:(7x + 2) – (x2 + 5) in Z_10 = 9x2 + 7x +7. We can find this via basic polynomial arithmetic in Z_10.

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