Consider the elliptic curve $E$:$Y^2=x^3+x+18$ over $F_p,$ with $p=31.($While the numbers are small,

you still need to use methods that also work well for larger numbers, unless otherwise instructed. When

computing multiples, use the double$-$and$-$add algorithm or its variant based on ternary expansions.$)$

$($a$)$ Verify that there is a point $P$ in $E(F_p)$ with $x-$coordinate $10.$ Suppose the $Y-$coordinate is encoded

by just one extra bit $0,$ as we explained in class. Determine the exact $Y-$coordinate.

$($b$)$ Verify that the point $P$ in $($a$)$ has order $15$ in $E(F_p).($Do not compute all multiples of $P.)$ Based on

this verification, determine #$E(F_p)$ without performing any further group operations in $E(F_p).$ Is

it true that each element of $E(F_p)$ is of the form $mP$ for some minZ $?$