Asymmetric Ciphers Questions and Answers – Number Theory – V

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

1. ᶲ(231)=
a) 230
b) 60
c) 80
d) 120
View Answer

Answer: d
Explanation: ᶲ(231) = ᶲ(3) x ᶲ(7) x ᶲ(11) = 2 x 6 x 10 = 120.
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2. n is prime if and only if n divides (2n – 2).
a) True
b) False
View Answer

Answer: b
Explanation: This isn’t true for all cases. Take for example 341 which is non prime.

3. 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
View Answer

Answer: c
Explanation: We have M = 3 x 5 x 7 = 105; M/3 = 35; M/5 = 21; M/7 = 15.
The set of linear congruences
35 x b1 = 1 (mod 3); 21 x b2 = 1 (mod 5); 15 x b3 = 1 (mod 7)
has the solutions b1 = 2; b2 = 1; b3 = 1. Then,
x = 2 x 2 c 35 + 3 x 1 x 21 + 2 x 1 x 15 = 233 (mod 105) = 23.

4. 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
View Answer

Answer: b
Explanation: Such a set is known as Totient.

5. The inverse of 49 mod 37 is –
a) 31
b) 23
c) 22
d) 34
View Answer

Answer: d
Explanation: 49-1 mod 37 = 34.
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6. 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
View Answer

Answer: b
Explanation: Use CRT to get the answer as 371.

7. How many primitive roots are there for 25?
a) 4
b) 5
c) 7
d) 8
View Answer

Answer: d
Explanation: 2, 3, 8, 12, 13, 17, 22, 23 are the primitive roots of 25.

Given 2 as a primitive root of 29, construct a table of discrete algorithms and solve for x in the following –

8. 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)
View Answer

Answer: c
Explanation: On solving we get x = 2, 27 (mod 29).
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9. 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)
View Answer

Answer: b
Explanation: On solving we get x = 9, 24 (mod 29).

10. 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)
View Answer

Answer: b
Explanation: On solving we get x = 8, 10, 12, 15, 18, 26, 27 (mod 29).

11. The inverse of 37 mod 49 is –
a) 23
b) 12
c) 4
d) 6
View Answer

Answer: c
Explanation: 37-1 mod 49 = 4.

12. How many primitive roots are there for 19?
a) 4
b) 5
c) 3
d) 6
View Answer

Answer: d
Explanation: 2, 3, 10, 13, 14, 15 are the primitive roots of 19.
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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He is Linux Kernel Developer & SAN Architect and is passionate about competency developments in these areas. He lives in Bangalore and delivers focused training sessions to IT professionals in Linux Kernel, Linux Debugging, Linux Device Drivers, Linux Networking, Linux Storage, Advanced C Programming, SAN Storage Technologies, SCSI Internals & Storage Protocols such as iSCSI & Fiber Channel. Stay connected with him @ LinkedIn