This set of Cryptography Multiple Choice Questions & Answers (MCQs) focuses on “Number Theory”.

1. Equations have either no solution or exactly three incongruent solutions

a) True

b) False

View Answer

Explanation: Equations has either no solution or exactly two incongruent solutions.

2. Find the solution of x^{2}≡ 3 mod 11

a) x ≡ -9 mod 11 and x≡ 9 mod 11

b) x ≡ 9 mod 11

c) No Solution

d) x ≡ 5 mod 11 and x ≡ 6 mod 11

View Answer

Explanation: On finding the quadratic congruencies we get x ≡ 5 mod 11 and x ≡ -5 mod 11.

3. Find the solution of x^{2}≡ 2 mod 11

a) No Solution

b) x ≡ 9 mod 11

c) x ≡ 4 mod 11

d) x ≡ 4 mod 11 and x ≡ 7 mod 11

View Answer

Explanation: There is no solution possible on solving the congruency.

4. Find the set of quadratic residues in the set –

Z11* = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

a) QR set = {1, 2, 4, 5, 9} of Z11*

b) QR set = {1, 3, 6, 5, 9} of Z11*

c) QR set = {1, 3, 4, 9,10} of Z11*

d) QR set = {1, 3, 4, 5, 9} of Z11*

View Answer

Explanation: QR set = {1, 3, 4, 5, 9} of Z11* is the set of quadratic residues. The values which have solutions fall under the QR set.

5. In Zp* with (p-1) elements exactly:

(p – 1)/2 elements are QR and

(p – 1)/2 elements are QNR.

a) True

b) False

View Answer

Explanation: The statement is true concerning elements of Zp* with (p-1) elements.

6. Find the set of quadratic residues in the set –

Z13* = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,11,12}

a) QR { 1, 2, 4,5, 10, 12}

b) QR { 2, 4, 5, 9, 11, 12}

c) QR { 1, 2, 4,5,10, 11}

d) QR { 1, 3, 4, 9, 10, 12}

View Answer

Explanation: QR { 1, 3, 4, 9, 10, 12}of Z13* is the set of quadratic residues. The values which have solutions fall under the QR set.

7. Euler’s Criterion can find the solution to x2 ≡ a (mod n).

a) True

b) False

View Answer

Explanation: Euler’s Criterion cannot find the solution to x2 ≡ a (mod n).

8. Find the solution of x^{2}≡ 15 mod 23 has a solution.

a) True

b) False

View Answer

Explanation: a=15 (15)

^{((23-1)/2)}≡(15)

^{11}≡-1 (QNR and no solution).

9. Find the solution of x^{2}≡ 16 mod 23

a) x = 6 and 17

b) x = 4 and 19

c) x = 11 and 12

d) x = 7 and 16

View Answer

Explanation: a=16 (16)

^{((23+1)/4)}≡ (16)

^{6}≡1 (QR and there is solution).

x ≡ ±16(23 + 1)/4 (mod 23) ≡±4 i.e. x = 4 and 19.

10. Find the solution of x^2≡3 mod 23

a) x≡±16 mod 23

b) x≡±13 mod 23

c) x≡±22 mod 23

d) x≡±7 mod 23

View Answer

Explanation: a=3 3

^{((23+1)/4)}≡3

^{6}≡1 (QR and there is solution).

x ≡ ±3(23 + 1)/4 (mod 23) ≡±16 i.e. x = 7 and 16.

11. Find the solution of x^{2}≡ 2 mod 11 has a solution.

a) True

b) False

View Answer

Explanation: 2 is a QNR.

12. Find the solution of x^{2}≡7 mod 19

a) x≡±16 mod 23

b) x≡±11 mod 23

c) x≡±14 mod 23

d) x≡±7 mod 23

View Answer

Explanation: a=7 7

^{((19+1)/4)}≡7

^{5}≡1 (QR and there is solution)

x ≡ ±7(19 + 1)/4 (mod 19) ≡±11 i.e. x = 11 and 12.

13. If we use exponentiation to encrypt/decrypt, the adversary can use logarithm to attack and this method is very efficient.

a) True

b) False

View Answer

Explanation: The first part of the statement is true. But this method is very inefficient as it uses the exhaustive search method.

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

To practice all areas of Cryptography and Network Security, __here is complete set of 1000+ Multiple Choice Questions and Answers__.