# Number Theory in Cryptography and Network Security MCQs

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

A. TRUE.

B. FALSE.

C. Nothing can be said.

D. None of the mentioned.

2. Find the solution of x2? 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.

Answer= x ? 5 mod 11 and x ? 6 mod 11

3. Find the solution of x2? 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.

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*.

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

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.

C. Nothing can be said.

D. None of the mentioned.

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}.

Answer= QR { 1, 3, 4, 9, 10, 12}

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

A. TRUE.

B. FALSE.

C. Nothing can be said.

D. None of the mentioned.

8. Find the solution of x2? 15 mod 23 has a solution..

A. TRUE.

B. FALSE.

C. Nothing can be said.

D. None of the mentioned.

9. Find the solution of x2? 16 mod 23.

A. x = 6 and 17.

B. x = 4 and 19.

C. x = 11 and 12.

D. x = 7 and 16.

Answer= 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.

Answer= x?±16 mod 23

11. Find the solution of x2? 2 mod 11 has a solution..

A. TRUE.

B. FALSE.

C. Nothing can be said.

D. None of the mentioned.

12. Find the solution of x2?7 mod 19.

A. x?±16 mod 23.

B.  x?±11 mod 23.

C.  x?±14 mod 23.

D.  x?±7 mod 23.

Answer=  x?±11 mod 23

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.

C. Nothing can be said.

D. None of the mentioned.

14. ?(231)=

A. 230.

B. 60.

C. 80.

D. 120.

15. n is prime if and only if n divides (2n – 2)..

A. TRUE.

B. FALSE.

C. Nothing can be said.

D. None of the mentioned.

16. Find x for the CRT when x= 2 mod 3; x= 3 mod 5; x = 2 mod 7.

A. 33.

B. 22.

C. 23.

D. 31.

17. Consider a function: f(n) = number of elements in the set {a: 0 <= a < n and gcd(a,n) = 1}. What is this function?.

A. Primitive.

B. Totient.

C. Primality.

D. All of the mentioned.

18. The inverse of 49 mod 37 is

A. 31.

B. 23.

C. 22.

D. 34.

19. Six teachers begin courses on Monday Tuesday Wednesday Thursday Friday and Saturday, respectively, and announce their intentions of lecturing at intervals of 2,3,4,1,6 and 5 days respectively. Sunday lectures are forbidden. When first will all the teachers feel compelled to omit a lecture? Use CRT..

A. 354.

B. 371.

C. 432.

D. 213.

20. How many primitive roots are there for 25?.

A. 4.

B. 5.

C. 7.

D. 8.

21. 17 x2 = 10 ( mod 29 ).

A. x = 3, 22 (mod 29).

B. x = 7, 28 (mod 29).

C. x = 2, 27 (mod 29).

D. x = 4, 28 (mod 29).

Answer= x = 2, 27 (mod 29)

22. x – 4x – 16 = 0 (mod 29).

A. x = 6, 24 (mod 29).

B. x = 9, 24 (mod 29).

C. x = 9, 22 (mod 29).

D. x = 6, 22 (mod 29).

Answer= x = 9, 24 (mod 29)

23. x7 = 17 (mod 29).

A. x = 8, 9, 12, 13, 15, 24, 28 (mod 29).

B. x = 8, 10, 12, 15, 18, 26, 27 (mod 29).

C. x = 8, 10, 12, 15, 17, 24, 27 (mod 29).

D. x = 8, 9, 13, 15, 17, 24, 28 (mod 29).

Answer= x = 8, 10, 12, 15, 18, 26, 27 (mod 29)

24. The inverse of 37 mod 49 is

A. 23.

B. 12.

C. 4.

D. 6.

25. How many primitive roots are there for 19?.

A. 4.

B. 5.

C. 3.

D. 6.

26. Find the order of the group G = <Z12*, ×>?.

A. 4.

B. 5.

C. 6.

D. 2.

27. Find the order of the group G = <Z21*, ×>?.

A. 12.

B. 8.

C. 13.

D. 11.

28. Find the order of group G= <Z20*, x>.

A. 6.

B. 9.

C. 10.

D. 8.

29. Find the order of group G= <Z7*, x>.

A. 6.

B. 4.

C. 3.

D. 5.

30. In the group G = <Zn*, ×>, when the order of an element is the same as order of the group (i.e. f(n)), that element is called the Non – primitive root of the group..

A. TRUE.

B. FALSE.

C. Nothing can be said.

D. None of the mentioned.

31. In the order of group G= <Z20*, x>, what is the order of element 17?.

A. 16.

B. 4.

C. 11.

D. 6.

32. The order of group G= <Z9, x> , primitive roots of the group are

A. 8 , Primitive roots- 2,3.

B. 6 , Primitive roots- 5.

C. 6 , Primitive roots- 2,5.

D. 6 , Primitive roots- 5,7.

Answer= 6 , Primitive roots- 2,5

33. Which among the following values:  17, 20, 38, and 50, does not have primitive roots in the group G = <Zn*, ×>?.

A. 17.

B. 20.

C. 38.

D. 50.

34. Find the number of primitive roots of G=<Z11*, x>?.

A. 5.

B. 6.

C. 4.

D. 10.

35. Find the primitive roots of G=<Z11*, x>?..

A. {2, 6, 8}.

B. {2, 5, 8}.

C. {3, 4, 7, 8}.

D. {2, 6, 7, 8}.

Answer= {2, 6, 7, 8}

36. Find the primitive roots of G = <Z10*, ×>..

A. {2, 6, 8}.

B. {3,6 ,9}.

C. {3, 7, 8}.

D. {3, 7}.

Answer= {3, 7, 8}

37. gcd( 18,300) =

A. 4.

B. 12.

C. 8.

D. 6.

38. ?(37)=

A. 24.

B. 22.

C. 13.

D. 36.

39. ?(35)=

A. 24.

B. 25.

C. 22.

D. 18.

40. ?(21)=

A. 10.

B. 12.

C. 8.

D. 14.

41. 73 mod 19 =

A. 18.

B. 1.

C. 14.

D. 12.

42. 7(3+j) mod 19 =

A. 7j mod 19.

B. 1 mod 19.

C. 73 + 7j mod 19.

D. All of the mentioned are true.

Answer= 7j mod 19

43. What is the period of 7m  mod 19?.

A. 2.

B. 3.

C. 4.

D. 5.

44. ?(19)=

A. 14.

B. 13.

C. 18.

D. 17.

45. What is the period of 11 (mod 19).

A. 2.

B. 3.

C. 4.

D. 5.

46. What is the period of 17 (mod 19).

A. 5.

B. 7.

C. 9.

D. 11.

47. What is the period of 9 (mod 19).

A. 12.

B. 10.

C. 11.

D. 9.

48. How many primitive roots does Z<19> have?.

A. 5.

B. 8.

C. 7.

D. 6.

49. What is the Discrete logarithm to the base 10 (mod 19) for a =7?.

A. 12.

B. 14.

C. 8.

D. 11.

50. 3201 mod 11 =

A. 3.

B. 5.

C. 6.

D. 10.

51. Find a number x between 0 and 28 with x^85 congruent to 6 mod 29..

A. 22.

B. 12.

C. 6.

D. 18.

52. What is the Discrete logarithm to the base 13 (mod 19) for a =13?.

A. 14.

B. 1.

C. 8.

D. 17.

53. What is the Discrete logarithm to the base 15 (mod 19) for a =9?.

A. 3.

B. 7.

C. 12.

D. 4.

54. Find a number x between 0 and 28 with x85 congruent to 6 mod 35..

A. 6.

B. 32.

C. 8.

D. 28.

55. Find a number 'a' between 0 and 72 with 'a' congruent to 9794 mod 73..

A. 53.

B. 29.

C. 12.

D. 37.

56. What is the Discrete logarithm to the base 2 (mod 19) for a =7?.

A. 3.

B. 4.

C. 6.

D. 9.

57. ?(41)=

A. 40.

B. 20.

C. 18.

D. 22.

58. ?(27)=

A. 6.

B. 12.

C. 26.

D. 18.

59. Find a number 'a' between 0 and 9  such that 'a' is congruent to 7^1000 mod 10..

A. 2.

B. 1.

C. 3.

D. 4.

60. ?(440)=

A. 200.

B. 180.

C. 160.

D. 220.