$10\mathrm{.}10.$ We now consider the Elgamal signature scheme. You are given Bob$$s private key Kpr $=($d$)=(67)$ and the corresponding public key Kpub $=($p$,\backslash$alpha $,\backslash$beta $)=$

$(97,23,15).$

$1.$ Calculate the Elgamal signature $($r$,$s$)$ and the corresponding verification for a

message from Bob to Alice with the following messages x and ephemeral keys

kE:

a$.$ x $=17$ and kE $=31$

b$.$ x $=17$ and kE $=49$

c$.$ x $=85$ and kE $=77$

$2.$ You receive two alleged messages x$1,$x$2$ with their corresponding signatures

$($ri$,$si$)$ from Bob. Verify whether the messages $($x$1,$r$1,$s$1)=(22,37,33)$ and

$($x$2,$r$2,$s$2)=(82,13,65)$ both originate from Bob.

$3.$ Compare the RSA signature scheme with the Elgamal signature scheme. Where

are their relative advantages and drawbacks?