Scalar Multiplication on Elliptic Curves In the elliptic curve: $y^2 = x^3 + 20x + 13 \bmod{2111}$.
Using the point $P=(3, 10)$

I am wondering how to multiply this point by the scalar $57$? 

I realize I can write $57*P$ as $2^5*P + 2^4*P + 2^3*P + P = 2(2(2(2(2P)))) + 2(2(2(2P))) +2(2(2P)) + P$.

My question is what is the best way to calculate these doubles and additions?

Some pseudo code or real code would be greatly appreciated although even a tutorial on how to do this by hand would help a lot!
The typical way to do this is the double and add method.

You start with the binary expansion of the scalar $57_{10}=111001_2$

Then scanning right from left, for each bit you double and if the bit is set to $1$ you add.

With elliptic curves it would look like this:

$Q\gets\infty$ (or the point at infinity since it is the identity element)
$P_{tmp}\gets P$
$bin\gets [1,0,0,1,1,1]$ (i.e., the binary expansion of the scalar reversed)
for $i$ in $bin$:
$\ \ $if $i==1$:
$\ \ \ \ \ Q\mathrel{+}=P_{tmp}$
$\ \ P_{tmp}=P_{tmp}+P_{tmp}$
return $Q$