Some confusions about Repeated Doubling Algorithm?
This algorithm is a small optimization to Algorithm 3.21 of the same book on page 91. Algorithm 3.21 is the original algorithm for computing elliptic curve doubling using Jacobian coordinates (with $a=-3$). If you have a close look at both of the algorithms, you will notice many similarities. For example, Algorithm 3.21 computes $A \leftarrow 3(X_1-Z_1^2)(X_1+Z_1^2)$, whereas Algorithm 3.23 computes $A \leftarrow 3(X^2-W)$ (note that $W=Z^4$!). Therefore I would suggestion breaking problem (1) down into two parts:


For $m=1$ prove that Algorithm 3.23 computes the same thing as Algorithm 3.21. Since we are using Jacobian coordinates, this means showing that $X_3/Z_3^2=X/Z^2$ and $Y_3/Z_3^3=(Y/2)/Z^3$ (using the notation of the respective algorithms, and of course with equal input).
Use induction to prove the result for $m&gt;1$.


I am not sure whether I understand part (2) of your question correctly. The statement $2Y$ for $4P$ is not really well-defined. Since we are working with projective coordinates (Jacobian), what you would probably want to check is whether we are computing a $X,Y,Z$ such that $4P=(X/Z^2 : (Y/2)/Z^3: 1)$. Since the first part of your question proves that the algorithm correctly computes $2^mP$ for any $m&gt;1$, the second statement follows by applying the definition of Jacobian coordinates to Line 4 of Algorithm 3.21.