This set of tough Cryptography Questions focuses on “Number Theory – II”.

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

a) 4

b) 5

c) 6

d) 2

View Answer

Explanation: It can be obtained using Euler Phi function, i.e. f(n).

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

a) 12

b) 8

c) 13

d) 11

View Answer

Explanation: |G| = f(21) = f(3) × f(7) = 2 × 6 =12

There are 12 elements in this group:

G = <Z21*, ×> = {1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20}. All are relatively prime with 21.

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

a) 6

b) 9

c) 10

d) 8

View Answer

Explanation: |G| = f(20) = f(4) × f(5) = f(22) × f(5) = (22-21)(51-50) = 8.

G = <Z20 *, x> = { 1, 3, 7, 9, 11, 13, 17, 19 }.

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

a) 6

b) 4

c) 3

d) 5

View Answer

Explanation: |G| = f(7) = (71-70) = 6

G = <Z20, x> = { 1, 2, 3, 4, 5, 6 }.

5. 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

View Answer

Explanation: Such a group is called the primitive root of the group.

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

a) 16

b) 4

c) 11

d) 6

View Answer

Explanation:

17 17 9 13 1 ord(17) = 4

n? 1 2 3 4 5 6 7 order

7. 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

View Answer

Explanation: |G| = f(9) = (32-31) = 6

G = <Z20, x> = { 1, 2, 4, 5, 7, 8 }.

8. 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

View Answer

Explanation: The group G = <Zn*, ×> has primitive roots only if n is 2, 4, pt, or 2pt

‘p’ is an odd prime and‘t’ is an integer.

G = <Z17*, ×> has primitive roots, 17 is a prime.

G = <Z20*, ×> has no primitive roots.

G = <Z38*, ×> has primitive roots, 38 = 2 × 19 prime.

G = <Z50*, ×> has primitive roots, 50 = 2 × 52 and 5 is a prime.

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

a) 5

b) 6

c) 4

d) 10

View Answer

Explanation: Number of primitive roots = f(f(11))=f((111-110)) = f(10) = f(2). f(5)

= (21-20)(51-50) = 1 x 4 = 4

The primitive roots of this set {2, 6, 7, 8}.

10. 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}

View Answer

Explanation: Number of primitive roots = f(f(11))=f((111-110)) = f(10) = f(2). f(5)

= (21-20)(51-50) = 1 x 4 = 4

The primitive roots of this set {2, 6, 7, and 8}.

11. If a group has primitive roots, it is a cyclic group

a) True

b) False

View Answer

Explanation: Yes, a group which has primitive roots is a cyclic group.

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

a) {2, 6, 8}

b) {3,6 ,9}

c) {3, 7, 8}

d) {3, 7}

View Answer

Explanation: Number of primitive roots = f(f(11))=f((111-110)) = f(10) = f(2). f(5)

= (21-20)(51-50) = 1 x 4 = 4

The primitive roots of this set are {3, 7}.

13. The group G = <Zp*, ×> is always cyclic.

a) True

b) False

View Answer

Explanation: G = <Zp*, ×> is always cyclic.

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

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