$5\mathrm{.}10.$ Use the double$-$and$-$add algorithm $($Table $5\mathrm{.}3)$ to compute nP in E$($Fp$)$ for each of the following curves and points, as we did in Figure $5\mathrm{.}4.$

$($a$)$ E: Y$^2=$ X$^3+23$X $+13,$ p $=83,$ P $=(24,14),$ n $=19$;

Y$2=$ X$3($b$)$ E:Y $+143$X $+367,$ p $=613,$ P $=(195,9),$ n $=23$;

$($c$)$ E: Y$^2=$ X$^3$

$+1828$X$+1675,$ p $=1999,$

P

$(1756,348),$

n $=11$;

$($d$)$ E: Y$^2=$ X$32+1541$X$+1335,$ p$=3221,$ P$=(2898,439),$ n $=3211.$

Input. Point P in E$($FP$)$ and integer n $>=1.$

$1.$ Set QP and R $=$ O$.$

$2.$ Loop while n $>0.$

$3.$ If n $=1($mod $2),$ set R $=$ R $+$ Q$.$

$4.$ Set Q $2$Q and n $=[$n$/2].$

$5.$ If n $>0,$ continue with loop at Step $2.$

$6.$ Return the point R$,$ which equals nP$.$

Table $5\mathrm{.}3$: The double$-$and$-$add algorithm for elliptic curves