1. “An Equations has either no solution or exactly three incongruent solutions”

a) True

b) False

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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

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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

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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*

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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

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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}

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

a) True

b) False

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

a) True

b) False

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^{((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

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^{((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

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^{((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

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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

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^{((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