Question 1. If ${ord}_p(2024)={ord}_q(2024)$, then the primes $p$ and $q$ cannot be     .

	i)	$7$ and $13$
	ii)	$11$ and $23$
	iii)	$2$ and $23$
	iv)	none of these

Question 2. The inverse of $7$ in ${𝔽}_{11}^{\ast }$ is     .

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

Question 3. For a Diffie-Hellman key-exchange, let $g=2$ and $p=11$. If Alice sends $3$ to Bob and Bob sends $9$ to Alice, then what is their shared key?

	i)	$9$
	ii)	$8$
	iii)	$6$
	iv)	$3$

Question 4. The discrete logarithm problem for the additive group of integers modulo $p$ is solvable in     time.

	i)	polynomial
	ii)	sub-exponential
	iii)	exponential
	iv)	none of these

Question 5. $2^k\equiv 4(mod77)$ for $k=$     .

	i)	$30$
	ii)	$32$
	iii)	$34$
	iv)	$36$

Question 6. Carmichael numbers need not be     .

	i)	composite
	ii)	odd
	iii)	product of distinct primes
	iv)	none of these

Question 7. Let $P=(11,8)$ be a point on the elliptic curve $E:Y^2=X^3+2X+3$ over ${𝔽}_{17}$. Then the inverse of $P$ is     in $E({𝔽}_{17})$.

	i)	$(11,2)$
	ii)	$(6,9)$
	iii)	$(14,2)$
	iv)	$(11,9)$

Question 8. Which of the following is an elliptic curve?

	i)	$y^2=x^3-12x+16$
	ii)	$y^2=x^3-3x+2$
	iii)	$y^2=x^3-3x-2$
	iv)	$y^2=x^3-2x+3$