Q. State whether the following statement is true or false over a field.

The sum of polynomials of degrees m and n has degree max[m,n].

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Q. Is x^3 + 1 reducible over GF(2)

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Q. Is x^3 + x^2 + 1 reducible over GF(2)

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Q. Is x^4 + 1 reducible over GF(2)

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Q. The result of (x2 ⊗ P), and the result of (x ⊗ (x ⊗ P)) are the same, where P is a polynomial.

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Q. Find the 8-bit word related to the polynomial x^6 + x + 1

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

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

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Q. Find the solution of x^2 ≡ 2 mod 11

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

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

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

(p – 1)/2 elements are QR and
(p – 1)/2 elements are QNR.

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

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

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

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Q. Find the solution of x2≡ 16 mod 23

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Q. Find the solution of x^2 ≡ 3 mod 23

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Q. Find the solution of x^2 ≡ 7 mod 19

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

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

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

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

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