Question 1. Which of the following is a primitive root modulo $11$?

	i)	$3$
	ii)	$5$
	iii)	$7$
	iv)	$9$

Question 2. Let $a,m\in \mathbb{N}$. Consider the following statements.

(i)

$\exists <!--FUNCTION\; APPLICATION-->b\in \mathbb{N}$ such that $ab\equiv 1(modm)$.

(ii)

$\gcd <!--FUNCTION\; APPLICATION-->(a,m)=1$.

Which of the following is true?

	i)	(i) $\Rightarrow$ (ii) but (ii) $\nRightarrow$ (i)
	ii)	(i) $\Rightarrow$ (ii) and (ii) $\Rightarrow$ (i)
	iii)	(ii) $\Rightarrow$ (i) but (i) $\nRightarrow$ (ii)
	iv)	(i) $\nRightarrow$ (ii) and (ii) $\nRightarrow$ (i)

Question 3. Let $p$ be a prime satisfying $p\equiv 3(mod4)$ and $a$ be an integer. Then     a solution of the congruence $x^2\equiv a(modp)$.

	i)	$a^{(p+1)∕4}(modp)$ is
	ii)	$a^{(p+1)∕2}(modp)$ is
	iii)	$a^{(p+3)∕4}(modp)$ is
	iv)	there need not be

Question 4. An attack in which an eavesdropper has access to an oracle that decrypts arbitrary ciphertexts is known as a     attack.

	i)	chosen ciphertext
	ii)	man-in-the-middle
	iii)	known plaintext
	iv)	chosen plaintext

Question 5. A number whose prime factors are all less than or equal to $k$ is called a     number.

	i)	composite
	ii)	Miller-Rabin
	iii)	$k$-smooth
	iv)	Carmichael

Question 6. $\varphi (100)=$     .

	i)	$4$
	ii)	$10$
	iii)	$40$
	iv)	$99$

Question 7. The discriminant of an elliptic curve $E:Y^2=X^3+AX+B$ is given by     .

	i)	$27A^2+4B^3$
	ii)	$27A^3+4B^2$
	iii)	$4A^2+27B^3$
	iv)	$4A^3+27B^2$

Question 8. Let $E:Y^2=X^3+X+2$ be an elliptic curve over ${𝔽}_5$. Then $\#E({𝔽}_5)=$     .

	i)	$3$
	ii)	$4$
	iii)	$25$
	iv)	none of these