Question 1. The number of primitive roots in ${𝔽}_{23}$ is     .

	i)	$10$
	ii)	$11$
	iii)	$22$
	iv)	none of these

Question 2. The inverse of $7$ modulo $11$ is     .

	i)	$3$
	ii)	$4$
	iii)	$5$
	iv)	$8$

Question 3. If there is a constant $c>0$ such that for inputs of size $O(k)$ bits, there is an algorithm to solve a problem in $O(e^{ck})$ steps, then the problem is said to be solvable in     time.

	i)	linear
	ii)	quadratic
	iii)	sub-exponential
	iv)	exponential

Question 4. Which of the following is not explicitly based on the difficulty of the discrete logarithm problem?

	i)	Diffie-Hellman key exchange
	ii)	Elgamal public key cryptosystem
	iii)	RSA public key cryptosystem
	iv)	none of these

Question 5. Carmichael numbers are     .

	i)	even
	ii)	primes
	iii)	composite
	iv)	finitely many

Question 6. Let $\pi (X)=(\#\text{ of primes }p\text{ satisfying }2\le p\le X).$ Then $\pi (X)\ne \varphi (X)$ if $X=$    .

	i)	$4$
	ii)	$8$
	iii)	$10$
	iv)	none of these

Question 7. Which of the following elliptic curves has no non-trivial points over ${𝔽}_3$?

	i)	$Y^2=X^3+X$
	ii)	$Y^2=X^3-X$
	iii)	$Y^2=X^3-X+1$
	iv)	$Y^2=X^3+2X+2$

Question 8. The fastest known algorithm to solve ECDLP in $E({𝔽}_p)$ takes approximately     steps.

	i)	$\log <!--FUNCTION\; APPLICATION-->p$
	ii)	$\sqrt{p}$
	iii)	$\log <!--FUNCTION\; APPLICATION-->\sqrt{p}$
	iv)	$\sqrt{\log <!--FUNCTION\; APPLICATION-->p}$