Question: This is a Python implementation related to elliptic curve cryptography $($ECC$).$ I need help implementing an "optimized" Double$-$and$-$Add algorithm for scalar multiplication on an elliptic curve.

Background:

The goal is to compute scalar multiplications of a generator point G on the secp$256$k$1$ elliptic curve.

The hint provided is that $31$P $=2(2(2(2(2$P$))))$P $.$

Note that it's effortless to find the inverse point $($P$)$ on a curve, and the algorithm should take advantage of this.

Code:

Below is the current Python code. The optimized$\_$double$\_$and$\_$add function is incomplete. Please help me fill in the missing implementation to make it work efficiently. Ensure that it counts the number of double and add operations performed.

# Python implementation for ECC scalar multiplication using secp$256$k$1$ curve.

from ecdsa import ellipticcurve

# Global curve parameters for secp$256$k$1$

P $=0$xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC$2$F

A $=0$x$0000000000000000000000000000000000000000000000000000000000000000$

B $=0$x$0000000000000000000000000000000000000000000000000000000000000007$

GX $=0$x$79$BE$667$EF$9$DCBBAC$55$A$06295$CE$870$B$07029$BFCDB$2$DCE$28$D$959$F$2815$B$16$F$81798$

GY $=0$x$483$ADA$7726$A$3$C$4655$DA$4$FBFC$0$E$1108$A$8$FD$17$B$448$A$68554199$C$47$D$08$FFB$10$D$4$B$8$

GZ $=0$x$0000000000000000000000000000000000000000000000000000000000000001$

N $=0$xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE$6$AF$48$A$03$BBFD$25$E$8$CD$0364141$

H $=0$x$01$

# Initialize the curve and generator point

curve $=$ ellipticcurve.CurveFp$($P$,$ A$,$ B$,$ H$)$

G$\_$POINT $=$ ellipticcurve.PointJacobi$($curve$,$ GX$,$ GY$,$ GZ$,$ N$,$ generator$=$True$)$

INFINITY $=$ ellipticcurve.INFINITY

# Convert a point to hexadecimal format

def point$\_$to$\_$hex$($point$)$:

$"""$Convert a PointJacobi object to hexadecimal representation."""

if point $==$ INFINITY:

return "INFINITY"

return f$"\{$point$.$x$().$to$\_$bytes$(32,$ 'big'$).$hex$()\}\{$point$.$y$().$to$\_$bytes$(32,$ 'big'$).$hex$()\}"$

def optimized$\_$double$\_$and$\_$add$($multiplier$,$ base$\_$point$)$:

$"""$Optimized Double$-$and$-$Add algorithm that simplifies sequences of consecutive $1\text{'}$s$."""$

# TODO: Implement the optimized algorithm here

result $=$ INFINITY

num$\_$doubles $=0$

num$\_$additions $=0$

return result, num$\_$doubles, num$\_$additions

if $\_\_$name$\_\_=="\_\_$main$\_\_"$:

input$\_$value $=31$ # Scalar multiplier

# Perform the multiplication

result$\_$point, doubles, additions $=$ optimized$\_$double$\_$and$\_$add$($input$\_$value, G$\_$POINT$)$

# Print the results

print$($f$"$Resulting Point: $\{$point$\_$to$\_$hex$($result$\_$point$)\}")$

print$($f$"$Number of Doubles: $\{$doubles$\}")$

print$($f$"$Number of Additions: $\{$additions$\}")$